Set Functions

Set Functions HAHN & ROSENTHAL

SET FUNCTIONS By

HANS HAHN Late Professor of Mathematics University of Vienna and

ARTHUR ROSENTHAL Professor of Mathematics Purdue University

THE UNTVERsITy OF NEW MEXICO PRESS ALBUQUERQUE, NEW MEXICO

COPYRIGHT 1948

THE UNIVERSITY OF NEW MEXICO PRESS

AU Rights Reserved

PREFACE This book was written on the basis of manuscripts left by the late HANG KAHN, formerly Professor of Mathematics at the University of Vienna, who died in 1934. Hans Hahn published the first volume of his well-known work Theorie der reellen Funktionen at Berlin (publisher: Julius Springer) in 1921. Because of its originality and its high and general point of view, this Volume I formed a great advance in the Theory of Real Functions and had a great influence on the further development of this theory. In 1932, instead of a second edition of this first volume, Hans Hahn published quite a new book in a thoroughly new form, namely, Volume I of the Reelle Funktionen, at Leipzig (publisher: Akademische Verlagsgesellschaft). In all this time Hans Hahn worked also at the second volume, never published by him. At his death he left large manuscripts for Volume II. In 1937, his widow and his publisher asked me to work out and edit Volume II of the Reelle Funklionen on the basis of those manuscripts. But soon the political situation in Germany grew worse and it became obvious that publishing this Volume II there was impossible for me. Thus the Akademische Verlagsgesellschaft and I agreed to cancel our contract and, then residing at Amsterdam, I made a new contract for Volume II with the Noordhollandsche Uitgevers i%-fij., Amsterdam, at the end of 1939. I finished the German manuscript for Volume II of the Reelle Funktionen in the first part of 1942. But at that time it was already evident that there would be no possibility of publishing this German manuscript. Therefore, I soon started writing the present book on Set Functions, considering only the perhaps most interesting parts (about two-thirds) of the German manuscript and presenting the subject independently of Volume I of Hahn's Reelle Funktionen. This book on Set Functions, the manuscript of which I completed in the spring of 1945, when I was a member of the staff of the University of New Mexico, is now being published by the University of New Mexico Press, with the consent of the N oordhollandsche Uitgevers Mij. The remainder of the German manuscript for Volume II of the Reelle Funktionen belongs essentially to the theory of point functions, mostly of one real variable or partly of several real variables, and hence has not been considered in this book on Set Functions (although there are some relations to, and applications of, the theory of set functions). As to the notations in the field of point sets or point functions, we mostly apply the notations used in H. Hahn's Reelle Funktionen, Vol. I, sometimes even in such caries where other mathematicians have preferred different notations. In our discussion the domain of the set functions is mostly quite general (i.e., any arbitrary set), sometimes restricted to topological or metric spaces, and only in a few cases specialized to Euclidean spaces. On the other hand, real num-

bers are always taken as the range of the set functions--thus showing a trace

of the originally intended volume on real functions; but this certainly has the essential advantage of making the presentation rather unified. This book on Set Functions is not a textbook in the customary sense, but it can surely be used for graduate courses. The systematic development and the very detailed proofs will facilitate the study of the book. The subjects discussed in this work and its structure and arrangement can easily be seen from the copious table of contents. The book is subdivided into chapters, sections (§), and articles (Nos.). Usually at the close of each No. the reader will find a bibliography with references related to this No. At the end of this work a list of symbols and signs and a list of the books quoted is found. Numbers in [...J refer to this list of books. The theorems are designated by numbers like 4.3.21, where the first and second

parts (4 and 3) indicate the section (§) and the article (No.), while the third part (21) shows the place of the theorem in that No. and has to be read like a decimal fraction. Thus 4.3.21 is subsequent to 4.3.1, 4.3.11, 4.3.2 and preceding 4.3.22, 4.3.3. In a similar way the formulas are numbered: (4.11) designates a formula of No. 4, subsequent to the formulas (4) and (4.1) and preceding the formulas (4.12), (4.2). §4,3 means No. 3 of §4, and §4(2) or §4(2.1) refers to formulas in No. 2 of §4. But we simply write No. 3 or (2.1) if this quotation is made somewhere in the same §4.

I wish to thank Dr. Charles B. Barker, who read almost the whole manuscript of this book, and Professor Lincoln LaPaz, who read about half of the manuscript; both of them helped me in improving the English style. Professor J. F. Randolph read all the galleys of the book and Professor C. V. Newsom read about one-third of those galleys; to both of them I am grateful for their pains and indebted for some useful suggestions. Finally, to the University of New Mexico Press, and in particular to its Director, Mr. Fred E. Harvey, and to Professor Dudley Wynn, formerly Director of Publications at the University of New Mexico, I wish to express my thanks for the publication of this work. ARTHUR RosENTHAL

Albuquerque, New Mexico

August 10, 1947

CONTENTS Page PRZFACE .........................................................................

V

INTRODUCTION

Section, No. 11. Sets and systems of sets

1. Fundamental relations for sets ............................................... 2. Systems of sets ..............................................................

1

2

§2. Topological and metric spaces 1. Topological spaces ..........................................................

4

2. SoparaWe sets in topological spaces ...........................................

7

3. Metric spaces ...............................................................

8

CHAPTER I ADDITIVE AND TOTALLY ADDITIVE SET FUVL"rIoas

§3. Basic properties of additive and totally additive set functions

1. Additive act functions ....................................................... 2. Totally additive set functions......... ..... ..............................

11

14

3. T xtremes of totally additive act functions ..................................... tli 4. Positive-, negative-, absolute-function ..............

5. Theorems about limits ............... ....................................... 6. Sequences of set functions ...................................................

18

22 23

§4. Zero-sets and complete fields 1. Zero-sets ..........

.....................................

..

2. Neglecting zero-sets ............................................

..

......

27

.........

28

3. Zero-sets in the wider sense. ................................................ 4. Complete fields.

.

-

-

-

-

....................................... ...........

§5. Regular and singular parts of totally additive set functions

1. Singular sets............

33 34

.......... 36

2. Regular and singular sot functions.... ...... 3. Continuous and purely discontinuous set functions ............................

40 43

5. Atomlese and atomic set functions ...................

45 48

............................... ... ........ .... ................ ........... .................... ..... ......... 6. Intermediate Values. .. 7. k-continuous set functions .. ............................................... 4. p-Atoms ................. .....

.

.

8. Sequences of 1G-continuous set functions ......................................

51

54

56

CHAPTER 11 MEASURE

§6. Measure functions

1. Measurable sets .............................................................

2. Outer and inner gyp-measure .......... ........................................ 3. Measure-covers and measure-kernels ......................................... 4. Regular measure functions ................................................... 5. A method for construction of regular measure functions ........................

6. Extension of a totally additive set function ............................ vii

.

...

61

64

68 72 74

so

CONTENTS

Page

Section, No.

;7. Content functions 1. Ordinary measure functions ...................... ....................... 2. Content functions ........................................................... 3. Non-w-measurable sets ............................. ........................ 4. Content-like set functions ........................

.........................

5. A first general method for the construction of content functions ................

81

84

87 88

90

§8. Content functions in the space R. ..

90

2. The n-dimensional measure of a point set of R......... .......................

93

1. Interval functions........................ ...........................

3. Basis of the real numbers .................................................... 100 4. Onedimensionally non-measurable sets ........................................ 103

5. A second general method for the construction of content functions .............. 104

6. The k-dimensional measure of a point set of R ................................. 106 CHAPTER III MEASURABLE FUNCTIONS

19. Measurable Functions 1. Basic properties ofo-measurable functions ................ ................... . 110 2. Relations ofo-measurable functions .......................................... 113

3. Non-µmeasurable functions ................................................ 116 4. c-equivalent functions ....................................................... 118

§10. Sequences of measurable functions 1. Sequences of measurable functions ........................................... 119 2. n-convergent sequences ...................................................... 121 3. Asymptotic convergence .......... .......................................... 126 4. Compact sets offo-measurable functions ...................................... 135

§1l. Measurable and continuous functions

1. Measurability of continuous functions ........................................ 139

2.p-continuousfunctions ......... ....... .................................... 141 3. Continuity properties of measurable functions ................................ 144 CHAPTER IV INTEGRATION

112. Integrable functions 1. They-integral of a function .................................................. 149 2. Other properties of v-integrable functions ..................................... 158 3. Sununablefunctions ......................................................... 165

4. Characterization of the set functions that are a-integrals .......................

168

5. Upper and lower integral .................................................... 171

6. Content-like determining function.. ......................................... 175 7. Improper integrals .......................................................... 176

§13. Lebesgue and Riemann sums 1. Lebesgue sums .............................................................. 179 2. Riemann sums .............................................................. 188

3. Distinguished sequences of decomposition ...................................

186

4. Darboux sums ................................................. ............ 190

CONTENTS

Section, No.

ix Page

§14. Mean value theorems and inequalities

1. The first mean value theorem ....... ........................................ 194

2. The second mean value theorem ... .......................................... 195 3. The Holder inequality .... .................................................. 198 4. The H Olderinequalityforintegrals ........................................... 202

§15. Theorems on convergence 1. Mean convergence.... .... .. ..... ....................................... 208 2. Integrable and completely integrable sequences ............................... 213

§16. Multiplication of set functions 1. Product of monotone set functions. .......................................... 223 2. Product of non-monotone set functions ....................................... 234

3. Geometric interpretation of the integral ...................................... 237 4. Fuhini's theorem ............................................................ 238 5. Iterated integrals ........................................................... 241

CHAPTER V DIFFERENTIATION

§17. Derivation of set functions 1. Derivates ofa set function ................................................... 246 ....... ................................................... 247 3. Paving derivates ............................................................ 254 4. The regular derivative ....................................................... 257 5. Covering systems ........................................................... 261 2. Vitali derivates .

6. Tile derivates ........................................................

...... 268

§18. Applications 1. 2. 3. 4.

Density ofa set ............................................................. 274 Density ofa set in R .......................................................... 281 Approximately continuous functions ......................................... 286 Integration and differentiation ............................................... 289

5. The limit functions of a derivate ............................................. 294 6. Transformation of integrals .................................................. 296 §19. Interval functions 1. Interval functions and associated set functions ................................ 297 2. Derivatives of interval functions ............................................. 302

BIBLIOGRAPHY (LIST of BOOKS QuOTED) ............................................. 812 LIST OF SYMBOLS AND SIGNS ......................................................... 313 INDEX .............................................................................. 317

INTRODUCTION

§1. Sets and systems of sets 1. Fundamental relations for sets. If a is an element of the set M, we write a e M; if a is not an element of M, we write a t M. The set consisting of the , ak single element a is written jai; the set consisting of the elements at, a2 , , ak . We denote the empty set (containing no element) is written { a,, as, by A. A set consisting of a finite number of elements is called a finite set. A set M is denoted as denumerable if a one-to-one correspondence can be established }

between the elements of M and the positive integers. A set which is either denumerable or finite is called countable.' If a set A is a subset of the set B, that is, if every element of A is an element of B, then we write A C B or B a A. If A Q B, but not B C A, then A is called a proper subset of B, written symbolically: A C B or B D A.

A + B designates the sum of the sets A and B, that is, the set consisting of all the elements that belong to at least one of the sets A and B.

A B or AB designates the intersection of the sets A and B, that is, the set consisting of all the elements that belong to both A and B. Immediately we have the distributive laws:

(A + B) C = AC + BC, (A + B) (A + C) = A + BC. A - B designates the difference of the sets A and B, that is, the set of the

(1)

elements of A not belonging to B. We have: (1.1)

A - A = A,

(1.11)

A- B =A - AB,

(1.12)

AB = A - (A - B),

(1.13)

A-(B-C)=(AB)+AC. In

More generally, we write A, +- A2 +

+

S Ak and A, + A2 + k-,

+ Ak +

= SAk for the sum consisting of all the elements that belong m

k

to at least one of the sets Ak; and we write A , - A 2 -

. . .

A,,, = D Ak and k-1

A,-As-

Ak . . . = DA k. for the intersection consisting of all the elements k

that belong to all of the sets Ak. Let K be a set of elements k and to every k e Ii' let a set Ak be attached; then we define their sum S Ak and their int.ei ection k6X

D AA, analogously.

k.x

1 This distinction of the expressions "denumerable" and "countable" has consistently been used by R. L. Wilder in his classes. I

INTRODUCTION

2

As an immediate consequence of the definitions, we have:

A - (B, + B2) = (A - &) (A - B:), A - B,B2 = (A - B1) + (A - B2),

(1.2)

and more generally: (1.21)

A -DBt= Skix(A-Bk). k4K

A-SBk=D(A-Bk), ked

k,K

The sets A and B are called disjoint if AB = A. Then we say also: A is disjoint from B. A system of sets is called disjoint if every two different members of this system are disjoint. of elements and ((A,)) for , ak, We write ((ak)) for a sequence al, as, of sets. Almost all terms of a sequence means a sequence A,, A 2 , , A k, all terms except at most a finite number of them. A sequence ((Ak)) of sets is called monotone increasing if Ak C Ak+a, and monotone decreasing if Ak Ak+,

_

((Ak)) being a given sequence of sets, Lim Ak designates the set of all elements k

k

belonging to infinitely many AA;, and Lim Ak designates the set of all elements belonging to almost all Ak . It follows that

k

Lim Ak C Lim Ak .

(1.3)

k

If we have Lim A k = Lim A k , then we write simply Lim Ak for this set. k

If Ak Q Bk for almost all k, then (1.31)

Lim Ak a Lim Bk and Lim-Ak C Lim Bit. k

k

k

k

This implies:

1.1.1. If At

Bk Q Ck for almost all k and Lim Ak = Lim Ck, then we have k

also Lim Ak = Lim BA; = Lim Ck . k

k

k

Let us set s, = S A, - Ak+Ak+1+ Ikk

andDk = D Aa = l

k

then we have:

(1.32)

Lim A k = SDk and Lim A k - DSk . A

k

k

k

This implies: 1.1.2. If ((A,)) is monotone increasing, then Lim At = SAk ; if ((Ak)) is monk

k

tone decreasing, then Lim Ak = DAk. k

k

2. Systems of sets. By a system of sets is meant a set of sets. A system T1 of sets is called closed with respect to addition (or intersection or subtraction)

SETS AND SYSTEMS OF SETS

§1]

3

if the sum (or intersection or difference) of any two sets of !7l is again a set of S1?.

A system 9? of sets which is closed with respect to addition and intersection is called a ring. A system of sets which is closed with respect to addition and subtraction is called a field. Because of (1.12) we have: 1.2.1. Every field is also a ring. Because of (1.1) we have: 1.2.11. If a is a non-empty field,, then A e ZJ.

1.2.12. If

is a field, then every sum SAk (with Ak a ) can be represented in k

the form SBk (with Bk a a) where the Bk are disjoint. k

k-1

For we can set: B1 = A, and Bk = Ak - fiA.fork> 1. i-i

A system V of sets, in which for every sequence of sets.-1, , A,, , A k, belonging to fi? we have Ak a 112 (or DAk e 9.T2), is called a d-system (8-system).

k If we extend a given system

0

k

of sets by adding the sums SAk (or the interk

sections DAk) of all the sequences ((Ak)) belonging to ID?, then we designate k

the new system by fi?, (or by fi?a) 1.2.2. 91 is the smallest o-system ()Y s is the smallest 5-system) over fi?' V. is contained in every o-system over R. But also 94 itself is a o-system ;

for let ((Ak)) be any sequence of sets belonging to fit, and write A = SAk ; then k

((Ak)) can be represented as a double sequence of sets belonging to fit, which caii be ordered into a simple sequence; thus A e V.. Instead of (Dl,)a we simply write Tl,a, and instead of (9)2a), we write M. . A ring which is also a o-system (or a 5-system) is called a d-ring (5-ring). A field which is also a o-system (or a 5-system) is called a d-field (5-field). 1.2.3. If 9? is a ring, then 9?, is a o-ring and 92a is a 6-ring. Because of 1.2.21??, is a a-system; thus we have to prove only that with A e SR, B e'J?, also AB e 9?,. Let A = SAi (Ai e 9?), B = SBk (Bk e J?). Then we i

k

have AB = S A ;Bk , where A tBk a fi; therefore, ordering this double sequence i.k

into a simple sequence, we get AB a 4R, . 1.2.4. Every o f eld a is also a d-field.

Let Ak a a; we write Al = Bi , Al - A2 ... At - Bk ;then Bk a Bk+1 . Thus DAk - DBk = Bi - S (Bk -- Bk+1); therefore, since Bk a a as a result of 1.2.1, k k k it follows that DAk a. k Again let fit be any system of sets. The smallest2 system over fi? which is both a o-system and a 5-system is called the Borel system over lZ and is desigEvery set belonging to V a is called a Borel set over V.

nated by 1X12 B.

"That is. containing U. f That such a smallest system always exists, has been shown by F. Hausdorff (2),

p. 84. Of. also H. Hahn [21, p. 260.

I A TRone CTI O`

4

§2. Topological and metric spaces 1. Topological spaces. A set E is called a topological space if there is a certain

system (i of sets 0 Q E which are designated as open sets and which satisfy the following conditions (the "topological axioms")':

t, The empty set A is an open set. 2, For every a e E there exists an open set containing a. 3, The intersection of two open sets is an open set. 4, The sum of every system of open sets is an open set. 5, To any two different points a e E, b e E there exist two disjoint open sets, one of which contains a, the other b.

If E contains more than one point, then 1, and 2, are immediate consequences of 5, and 3,.

2, and 4, imply: 2.1.1. The space E is an open set.

2.1.11. IfaeE,thenE - (a) is open. This follows from 5, and 4,. 2.1.12. If G is open and a e E, then G - {a) also is open. Since G - {a) = G- (E - (a)), this follows from 2.1.11 and 3,.

2.1.13. If G is open and a e E (k = 1, 2, ,m),thenG- tai ,a,, ... a.,) is also open. This follows from 2.1.12 and 3,.

If A Q E, then the sets AG (with G e 0) satisfy the topological axioms (with A instead of E); thus if we call these sets AG open in A, then A also becomes a topological space. Every open set containing the point a is called a neighborhood of a; we desig-

nate such an open set by U.. A point a is called an inner point of the set A if there exists a neighborhood U. a A. The set of all the inner points of A is called the open kernel A (,) of A.

Obviously A (;) is the sum of all the open sets

G C A. The set A (.) = A - A(;) is called the border of A; the points of A(,) are called the border points of A ; the sets A for which A = Aw and thus A called border sets.

A, are

If A C E, then E - A is called the complement of A ; instead of E - A we

also write - A. The sum of the borders of A and E - A constitute the frontier of A ; its points are called the frontier points of A.

A set F (cE) is called closed if its complement E - F is open. Because of It, 2.1.1, 2.1.13 we have: E and A are closed; every finite set (cE) is closed. Because of §1 (1.2), (1.21) and 3,, 4, we have: The sum of two closed sets is a closed set. The intersection of every system of closed sets is a closed set. Thus the intersection of all closed sets Q A is the smallest closed set A and is called the closure of A; we write A ° for it. A is closed if and only if A = A °. ' According to H. Hahn [21, p. 46.

TOPOLOGICAL AND METRIC SPACES

§21

5

Since the closed sets Q A are the complements of the open sets a (-A), we get from §1 (1.21): A° 2.1.2. a e A°, if and only if every neighborhood U. contains at least One Point of A

(1)

According to (1), a e A° means a ti e (-A)(,) ; that is, no U. c (-A). If A Q; E, then the sets A F (where the F are closed) are called closed in A. 2.1.3. In order that the set B a A be closed in A, it is necessary and sufficient that A - B be open in A.

NECESSITY: If B = A F, where F is closed, then A - B = A - AF =

A (E - F) with (E - F) 0.

Suvpxcrm4cy: If A - B = A G with G e (s), then

B = A - (A - B) = A - AG = A (E - G), where E - G is closed. An immediate consequence of the definitions is: 2.1.31. A closed (open) set in a closed (open) set is a closed (open) set.

The intersection of a sequence of open sets is called a Gj-set (or, more conveniently, a Gs); the sum of a sequence of closed sets is called an F,-set (or an F,). Analogously Ga-set in A (F,-set in A) is defined by means of sequences of sets open (closed) in A. Because of §1 (1.21) we get: 2.1.4. The complement of a Gs is an F, ; the complement of an F. is a G, . 2.1.41. Every Gs is the intersection of a monotone decreasing sequence of open sets; every F. is the sum of a monotone increasing sequence of closed sets.

Let A be a Gs-set; thus A = D Gk, where the Gk are open sets. We write k

Qt = Gi -G:

Gk; then Ok is open, Ok+x a Ok , and A = DOk. k

A set A is called dense (in E) if (E - A) o = A, that is, if E - A is a border set. A set A is called nowhere dense (in E) if (E - A)(Q is dense (in E). From (1) we get immediately: 2.1.5. A is nowhere dense if and only if A° is a border set. 2.1.51. A is dense if and only if A° = E. Because of (1), (-A) c,, = A is the same as A ° = E. 2.1.52. In order that A be dense, it is necessary and sufficient that every open set G 0 A contain at least one point of A. NECESSITY: This follows from 2.1.51, 2.1.2. SumcxzNcr: The open set (-A)(0 contains no point of A; thereforewe have (-A)(0 = AandthusA isdense. If the set A Q B is a stun of countably many sets nowhere dense in B, then A is called a set of the first category in B; if the set A C B is not a sum of countably many sets nowhere dense in B, then A is called a set of the second category in B. If A is a set of the first (second) category in A, then A is called a set of the first (second) category in itself. A point a is called a point of accumulation of the set A if every neighborhood U. contains infinitely many points of A. A point a is a point of accumulation of A if and only if every neighborhood U. contains at least one point of A different from a. (For if a is not a point of

6

INTRODUCTION

accumulation of A, there is a neighborhood U. containing only a finite number

of points a,, a2,

U. - (a, ,

a2 ,

, a, (0a) of A [if any at all]; then, because of 2.1.13,

, a,

}

is a neighborhood of a, containing no point of A

other than a.) A point a is called a limit point of the sequence ((ak)), and we write him ak = a

or ak -+ a if every neighborhood U. contains almost all

ax.

A sequence

((ak)) is called convergent if there is a point a, such that him ak = a.

The set of all points of accumulation of A is called the I derivative A' of A.

2.1.6. A°=A+A'.

Because of 2.1.2 every point a e (A + A') is also a point of A°. Conversely: e A, then, again by 2.1.2, a is a point of accumulation of A, If a e A ° and a

that is, a e A'. Since A = A° is characteristic for the closed sets A, we obtain from 2.1.6: 2.1.61. A set A is closed if and only if A contains all its points of accumulation.

Replacing the space E by a set B C E we obtain from 2.1.61: 2.1.611. Let A C B. Then A is closed in B if and only if A contains all its points of accumulation which belong to B. A set A is called compact if for every infinite set C C A the derivative C' Pd A.

Every subset of a compact set is also compact. A set A 9. B is called compact in B if for every infinite set C C A we have C' B # A. A set A which is compact in A is called self-compact; such a set is also compact. 2.1.62. In order that A be compact, it is necessary and sufficient that for every denumerable set B Q A the derivative B' 76 A. NECESSITY: Trivial, SUFFICIENCY: Every infinite set C C; A contains a de-

numerable subset B and by assumption B' P6 A; thus since B' C C', also C' ' A; that is, A is compact. 2.1.63. If A is self-compact, then every subset B C A, closed in A, is also selfcompact.

Let C be any infinite subset of B. By 2.1.611 C'A Q B'A G B, and hence C'A C C'B. But since B is compact in A, we have C'A 0 A. Thus we have also C'B 0 A. The points of A which are not points of accumulation are called isolated points of A. If every point of A is a point of accumulation, then A is called dense-in-itself. (Thus also A is dense-in-itself). A set without any subset D A which is dense-in-itself is called scattered. A set which is closed and dense-initself is called a perfect set. The sum of all the subsets of A which are dense-in-themselves is called the

nucleus Air of A; it is the largest subset of A which is dense-in-itself. We set

A - Ax = As, where As (being scattered) is called the scattered part of A. Thus for every set A We have: (1.1)

A = Ax + A8 (with A,rAe - A).

TOPOLOGICAL AND METRIC SPACES

§2]

7

2.1.7. The nucleus Ax of A is closed in A. Because of (1.1) we have (1.11)

A (A ,r)° - (Ar + As) (A1) ° = Al + As. (A N)°.

Here we have (1.12)

A;

for if there were a point a e A8 (A,,)', we would have a e As, and thus a ti e A 1; therefore, by 2.1.6, a e (A1)'; that is: a would be a point of accumulation of A,1 and therefore Al + {a} would be dense-in-itself, while AK is the largest subset of A which is dense-in-itself. Then (1.11) and (1.12) give .4 a = A (A1)°, which proves 2.1.7, since (Ax)- is closed. The theorems 2.1.7 and 2.1.31 imply: 2.1.71. If A is closed, then the nucleus As is perfect.

2. Separable sets in topological spaces. Again let E be a topological space. B a system t of non-empty sets H, open in A, is called a disFor any set .-1 tinguished system of sets, open in A,' if for every a e A and for every set G, open in A and containing a, there is a set H e with a e H and H G. Then every set G, open in A, is the sum of a partial system of t. The set A (Q E) is called separable if there is a countable distinguished system of sets H, (r - 1, 2, ), open in A. 2.2.1. Every subset B of a separable set A is separable.

If ((AG,)) is a distinguished system of sets, open in A, then the non-empty sets BG, form a distinguished system of sets, open in B. 2.2.11. Every separable set A contains a countable subset which is dense in A." Choose a, a H. (with H, a., v = 1, 2, ) ; every set G P6 A, open in A, contains at least one of the sets H, and thins at least one of the points a,. Therefore, because of 2.1.52, the set of the points a, is dense in A. Let 41 be a system of sets; if every point a e A is contained in at least one set of the system 6'i, then we say: (9 covers A. 2.2.2. If A is separable, then every system ( of sets, open in A, which covers A contains a countable subsystem which covers A also. Let us keep only those sets H., a which are contained in at least one set of 4. To every such H,, only one set G., e (( with G.. H,, may be attached. Every point a e A is contained in a certain G e (9, therefore also in a certain H,, e .V and thus in G,., too. Hence the G,, form a countable subsystem of 9 which covers A also. 4 According to H. Hahn [2], p. 77.

' If the space E is metric (No. 3), then also the converse of 2.2.11 is valid; that is: If A contains a countable subset which is dense in A, then A is separable. For instance, cf. H. Hahn 121, p. 78 (theorem 13.1.31).

INTRODUCTION

8

3. Metric spaces. A set E is called a metric space if to every two elements a, b of E a (finite) number ab (the distance of a, b) is attached, satisfying the following conditions (the "metric axioms"): 1,.

ab = 0 if and only if a = b.

ab + cb ("triangle inequality"). ac are: If a b, then ab > 0. (For if we set Immediate consequences of 1 we get because of 1,,: 0 < 2ab.) Furthermore: c = a in 2,,,

ac = ca.

(3)

(For if we set b = a in 2.,, we get because of 1,,: ac S ca; and interchanging a and c we also obtain ca 5 ac.) If a s E and p > 0, then the set of all points x e E for which za < p (or xa 5 p), is called the sphere S (or the closed sphere Sa,) with centre a and radius p. We define a set G Q E as open if to every a e G there is a sphere Sa, Q G. Then, because of 2,a, every sphere is an open set and the open sets satisfy the topological axioms (No. 1); therefore every metric space is also a topological space. An example of a metric space is the n-dimensional Euclidean space R. , that is, , x,.) with the following the set of all n-tuples of real numbers (x, , xt , , a,.), b = (b, , b, , - , b,,); then definition of the distance: let a = (a, , as , we set

ab = -%/(a,-b1)'+(at-b,)'+... In particular RI is formed by the set of all real numbers with the distance

(a-b)'= Ia - bI.

ab =

In a metric space E the distance aB of a point a from a set B (0 A) is defined

by : aB - inf ab.' b.5

From 2., and (3) we get:

aC 5 ab + bC.

(3.1)

Analogously, in a metric space E the distance AB of two sets A (0 A) and B (P6 A) is defined by: AB = inf ab. aeA,b.B

From 2., and (3) we obtain:

AC 5 bA + bC. If A (;-, A) C; E and p > 0, then the set of all points x for which zA < p,

(3.11)

Here and throughout the remainder of the book we shall use the notation inf (i.e., infnum) rather than g.i.b. (-. greatest lower bound) and sup (i.e., supremum) rather than l.u.b. (- least upper bound). Thus if A (Pf A) is a set of real numbers, then sup x is the :.A

smallest number to which no x e A is superior and inf a is the greatest number to which no s.A

x e A is inferior.

9

TOPOLOGICAL AND METRIC SPACES

§21

Because of

is called the neighborhood p of the set A; we designate it by UA, . (3.1) every neighborhood UA, is an open set. For the set A (0 A) we define the diameter d(A) of A by:

d(A) = sup ab.

(3.2)

If AB

o.A.b.A

A, then it follows from 2, that

d(A + B) 5 d(A) + d(B).

(3.21)

A set A is called bounded if its diameter d(A) is finite. Since every neighborhood U. contains spheres Sa, , the relation lim ak = a is k

equivalent to lim aka = 0. k

2.3.1. If a is a point of accumulation of the set A of a metric space, then A containe a sequence ((at)) of different points with lim ak - a. k

Let ((S.,.)) with pk --+ 0 be a sequence of spheres contracting to a. Every Sa,k

contains at least one point at a A, distinct from a, and we can assume these points ak to be different; then lim ak = a. k

2.3.11. In order that the set A of a metric space be compact (§2, 1), it is necessary and sufficient that in every sequence ((a.)) of points of A a convergent subsequence be contained.

NECEssITY: If infinitely many a., (i = 1, 2, ) represent the same point a, then lim a,4 = a. Otherwise ((a,)) forms an infinite and compact subset of A, and hence ((a,)) determines at least one point of accumulation, say a. Then by 2.3.1 ((a,)) contains a subsequence ((a,,)) of different points with lim a,,, = a. k

SUFFICIENCY: If A is not compact, then by 2.1.62 there is a denumerable set B C A

with B' = A. Let a, (v = 1, 2,

) be the points of B; then ((a,)) determines no point of accumulation, contrary to the assumption. Let B C; A Q E and p> 0. The set B is called a e-net in A if to every a e A there is a b e B with ab < p and if for any pair of different points bL , b2 of B we have bib, ?= p. 2.3.12. In a compact set A (P6 A) of a metric space there is a finite p-net for every

p>0.

Let b, e A ; if ab, < p for every a e A, then (b, j is a p-net in A. Otherwise there is a b,. e A with bib: z p; if for every a e A either abc < p or ab2 < p, then { bl , b2 {

is a p-net in A. Otherwise there is a bs e A with bibs Z p and bibs ? p; if for every a e A either ab, < p or ab2 < p or abs < p, then { b 1 , b2 , bs} is a p-net in A ; and so on. This procedure must stop after a finite number of steps. For otherwise we would obtain a sequence ((b,)) with b,;b,, z p for v; 0 vk , contrary to 2.3.11. A sequence ((ak)) of points of E is called a Cauchy sequence if to every b > 0 there is an index ks . rich that 6 for k' ? ka , k" > ka .

INTRODUCTION

10

2.3.2. If ((ak)) is a Cauchy sequence with a as a point of accumulation, then we have lim ak = a. k

To every 6 there is a kt , such that

6 for k' Z ka, k" Z ka ; and there

is a certain k;; >__ k, with aak,' < 6; thus, because of 2., , we have:

aak; <26fork' ? k6. A set A is called complete if every Cauchy sequence in A has a limit point belonging to A A.

23.3. In a complete metric space E every set A of the first category (§2, 1) is a border set.

Let A = B, + B2 +

+ Bk +

, where the Bk are nowhere dense. Then (according to 2.1.52) in S there is a Let S be any sphere with radius p. closed sphere fit , with radius p, < e, which does not contain any point of B1.

And in , , there is a closed sphere &, with radius p2

any point of B2 . Continuing in this way, we get a sequence (($k)) of closed

spheres with radius pk 5 2k , such that S

, ,

and

Sk BA; = A; thus also Sk . (B1 + ... + Bk) = A. Let ak a Sk (k = 1, 2, ...) , then ((sk)) is a Cauchy sequence and hence has a limit point a e E. Since the Sk are closed, by 2.1.61 we have: a e DSk and hence a tie A. Thus every sphere S contains a point a not belonging to A; therefore A(,) = A and A is a border set. From 2.3.3 we obtain immediately: 2.3.31. Every open set G (;,& A) in a complete metric space is of the second category

in itself.

CHAPTER I

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS §3. Basic properties of additive and totally additive set functions 1. Additive set functions. Let TZ be a system of sets; a relation by which to every set .4 e 972 a (finite or infinite') real number p(A) is attached defines a set function p in 97l2

co for all We assume that neither ,p(A) = + oo for all A e 9 ? nor p(A) A e 971; then the set function p is called additive if for every two disjoint sets A e 971 and B e 932, for which also A + B e 972, we have:

cp(A + B) = p(A) + e(B).

(1)

Here p(A) and p(B) cannot be infinite values of different signs, since otherwise the right hand side of (1) would be meaningless.

3.1.1. If fit is closed with respect to addition, p is additive in 9R, and A ; , m) are disjoint sets of 972, them. (i = 1, 2, p(Al + A2 + ... + A.,) = p(A1) + p(A2) + ... + p(A,,.). This follows from (1) by induction.

3.1.11. Ifpisadtlitivein9)1,if AeP,Be9?,A - Be93?,BCA,andV(B) is finite, then p(A - B) = p(A) -- p(B). For since B and A - B are disjoint, according to (1) we have: (1.1)

p(A) = v(B) + p(A -- B).

A, and p(B) is 3.1.2. If pie additive in 972, if A e 932, B e 972, A - B e 972, B infinite, then p(A) = p(B). This follows immediately from (1.1). From 3.1.2 we obtain : 3.1.21. If p is additive in fit, if A e fit, B e 992, A - B e fit, B Q; A, and p (A) is finite, then p(B) is also finite. 3.1.22. If A e 972 and p is additive in 9D2, then p(A) = 0. If yo(A) + oo (or = - oo ), then p(A) = p(A + A) = p(A) -i p(A) = + ac (or = - oc) for all A e 992, contradictory to the definition of the additive set functions; thus p(A) is finite. Since s is additive, we have p(A) + p(A) _ p(A). Thereforep(A) = 0.

3.1.23. If p is additive in the field W2, then fit cannot contain two sets A and B

with p(A) = + o , p(B) = - oo. Because of (1), this is true for disjoint sets A and B. If A and B are any sets This is to say that +co and -.0 are included. = This is analogous to the usual notion of a point function f on a given set A : such a point function is defined by a relation which to every point a e A attaches a (finite or infinite') real numberf(a). 11

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

12

[CHAP. I

of nit, then A B, A - B, B - A are disjoint sets of 5171; thus among the three values cp(AB), po(A

-- B), 5p(B - A) there cannot be both the values + oo and - oo;

hence, as according to (1)

,p(A) = p(AB) + p(A - B),

p(B) = cv(AB) + cp(B - A),

we cannot have p(A) = + oo, q(B)

oo.

Generalizing 3.1.1 we get: , m) are sets of S2, and 3.1.3. If cp is additive in the field 5711, if A, (i = 1, 2, Ck is the set of all the elements belonging to exactly k sets A{ , then m

wi

i-t

k_I

k s (Ck).

.9

Let S = Si_tAi and A; = S -- A; ; if it < is <

< it are 1 different of the

, m and j, < f2 < ... < jm_t the other m - t numbers, then we 1j.-,; because of 1.2.1 we have A it A;, A;, set : Ail,, ... it = Ail A i, A,,,, ... WE 911; the 2m sets Ai,i, ... i, are disjoint. Ck is the sum of those Ai,i, ... i, which have exactly k indices; thus Ck a 91 and, according to 3.1.1, cp(Ck) is the sum of the corresponding cp(Ai,i, ... ,k). Furthermore, A; is the sum of those Aii, ... it among the indices of which i occurs, and, by 3.1.1, cp(Ai) is the sum of the corresponding cp(Ai,i, ... i,). If we substitute this sum

numbers 1, 2,

m

for every term cp(Ai) in E cp(Ai), then every cp(Ai,i, ... ik) occurs in exactly k i-I , p(Aik)); from this follows the conterms cp(Ai) (namely in p(Ai,), cp(Ai,), tention. , m) are sets of zR, 3.1.31. If rp is additive in the field SJ1'l, if Ai (i = 1, 2, and Bit is the set of all the elements belonging at least to k sets A,, then

cp(Ai) _ E cp(Bk). k.-t

+ Cm and C, e 91, we have Bk E [2. As the Ci are dis-

Since Bib = Ck +

joint, we have cp(Bk) = V(Ck) +

+ cp(C,.); thus the proposition follows

immediately from 3.1.3.

For m = 2 we get the particular result that for any two sets A and B of the field TI: (1.2)

v(A) + q(B) = cp(A + B) + cp(AB).

The set function cp in SD2 is called monotone increasing (or decreasing) if for

every two sets A E sOl, B E 9)l it follows from B C A that p(B) S cp(A) (or ca(B) ! cp(-'1)) 3.1.4. If cp is additive in 52)1 and A E 911, then in order that' be monotone increasing

(or decreasing), it is necessary and, if 212 is closed with respect to subtraction, also sufrie vt that ..(A) ? 0 (or 5 0) for every A e 9W' .

BASIC PROPERTIES

§3]

13

NEcxssrrY: Because of 3.1.22, from A A and A e T1 it follows that a(A) z 0. SUFFICIENCY: If B C A, A e T2, B s 9J?, then we have A = B + (A - B), thus

p(A) = v(B) + p(A - B) ; therefore, since (p(A - B) z 0, we get. V(A) z V(B).

3.1.41. If w is additive and monotone increasing (or decreasing) in the field 0, then for every two sets A e SD2, B s fit we have: (p (A + B) S cp(A) + p(B) (or z se(A) + cp(B)). Because of 3.1.4, 1.2.11, this follows immediately from (1.2). 3.1.42. If fp is additive and monotone increasing (or decreasing) in the field TZ, then for every two sets A e 0, B e fit, for which not both of the values cp(A), ,p(B) are infinite, we have: (1.3)

p(A - B) z So(A) - p(B)

(or S cp(A) - p(B)).

If cp(B) is infinite, then (1.3) is trivial; thus let co(B) be finite. Since A AB + (A -- B), as a result of 3.1.41 we have p(A) S ,p(AB) + cp(A -B) ;because V(AB) < p(B); thus:,p(A) -- p(B) 5 p(A) - cp(AB) 5 of 3.1.4, we have 0 jp(A -- B). 3.1. S. If y, is additive in 9)2 and a is a finite number 0 0, then also Gp is additive in OR.

3.1.51. If rp, and qn are additive in fit, then in order that (p, + V, (or in - 02) be also additive in 9)l, it is necessary and sufficient that there be no set A e M for which qp,(A) and co,(A) are infinite of different (or equal) sign, and that for not all A a 9 gyp, + c2 (or 01 - toy) be infinite with the same sign. NECEssITY: co + v, must not be meaningless nor always infinite with the same sign. SUFFICIENCY: For two disjoint sets A e fit, B e TZ with A + B e fit we

have: ,p,(A + B) = vi(A) + gpi(B) ; p2(A + B) _ cp,(A) + c(B). Then ,pi + V3 is additive, if one may always add both these equations without getting anything meaningless. This is the case indeed: for instance, if p,(A) = + 00,

qft(B) = - co, then pp,(A + B) = + co, 4(A + B) = - co, contradictory to the stated condition of the theorem.

3.1.52. If A e A2 atld gyp, , rpz are additive in fit, then in order that also rp, + co, (or p, - gyp,) be additive in 912, it is necessary and sufficient that there be no set A e fit for which p,(A) and o (A) are infinite of different (or equal) sign. This follows from 3.1.51 because of 3.1.22. 3.1.6. If 9, (v = 1, 2, . ) are additive in TZ, if p(A) = lim cp,(A) exists for

every A s V, and p(A) is not infi-'hite with the same sign for all A e 9Pt, then in order that p be also additive in fit, it is necessary and sufficient that there be no two disjoint sets A e fit, B e fit with A + B e 912 for which cp(A) and io(B) are infinite of different sign.

For a field 0 we obtain from 3.1.6, because of 3.1.22 and 3.1.23: 3.1.61. If co. (v = 1, 2, ) are additive in the field 92 and p(A) = lim c,(A) exists for every A e 9l, then in order that co be also additive in fit, it is necessary and sufficient that fo does not have infinite values of different sign.

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

14

[C,IAP. I

2. Totally additive set functions. A set function cp, additive in D2, is called totally additive` in Di if for every denumerable system of disjoint sets A. s 9% (m = 1, 2, ) for which also SAM e 7Ji, we haveM

p(SA.) = F, v(A.n)

(2)

m

,a

Example of a totally additive set function': Let D2 be the system of all subsets of an arbitrary set E and, for every A e D2, let v(A) be the number of elements of A (in particular v(4) = + co if A is an infinite set). 3.2.1. If v is totally additive in the or-field fit and ((Am)) is a sequence of disjoint sets of D2, for which E p(AM) is finite, then E E sp(A,,,) I is also finite. M

M

We denote by A', those A m for which w(A m) x 0, by A;' those for which

w(AM) < 0,

and we set SAM = A, SA; = A', SA; = A"; then we have A e 0, A' a D2, ,p(A;'). Accord4" a D2, and,p(A) _ Ejp(A.,). v(A') = S;-p(.4;), p(A") = ing to the assumption, p(A) i finite, and thus because of 3.1.21 y,(A') and O(All) are also finite.

Since

the proposition follows. 3.2.2. If 9 is closed with respect to subtraction and v is totally additive in TI, of sets of D2, for which then for every monotone increasing sequence Lim AM a D2, we have: (p(Lim A) = lim p(A..).

If any cp(A.,), for instance ,(A,,), is infinite, then since A.. C A. (m k me), A.,, C; Lim A M , the contention follows from 3.1.2. Thus suppose all p(AM) ELI

to be finite. Since ((A.,)) is monotone increasing, because of 1.1.2 we have:

Lim A M - SAM = A, + S (A M - A.-,); hence as Al , A2 -- A 1, w

M

M>1

.

. ,

are disjoint sets of 0, we have as a result of 3.1.11:

AM - A.,._: ,

rp(Lim AM) = v(A1) + E ((v(A.,.) - ,p(AM_L)) = lim v(AM). M

M>1

M

3.2.21. If D2 is closed with respect to subtraction and v is totally additive in 0, ' Other expressions used for it are: absolutely additive, completely additive, countably additive. ' Example of an additive, but not totally additive, set function: Let R be the system of

all subsets of an infinite set E and let p(A) a 0 if A is finite, '(A) -+.0 if A is infinite. if now B is a denumerable subset of B, consisting of the elements b (m 1, 2, ...), and if we set Ib,,,l = B.., then we haveB - SBm,,v(BM) - 0,w(B) - +-e.

15

BASIC PROPERTIES

31

then for every monotone decreasing sequence ((A.)) of sets of O1, for which are infinite`, we have: Lim A. s 0 and for which not all

p(Lim A,.) = lim p(A.,). in

mo are also If jp(A,..) is finite, then according to 3.1.21 all p(Aw,) with m finite; thus we can assume all sp(A.,) to be finite. We set Lim A. = DAw, = A;

then A, = A + S (A.,-i - A,.); thus since A, A, - A2, .,>1

,

A,._I - A.,, .

are disjoint sets of n12, because of 3.1.11 we have:

p(Ai) = '(A) + E (p(A,.-I) w,> i

therefore p(A) = p(.4,) -

(p(A.,._,) - p(.4 w,)) = lim p(A.,).

We shall prove other theorems of this type in No. 5. 3.2.3. If p is totally additive and monotone increasing in the field $02, then for

every sequence ((A.)) of sets of X71, for which 8.4. s T1, we have: p(SAm) < M

If we set B. = :l, + A2 + sequence of sets of 0 and

wa

E "(A-). m

+ .4 m , then ((B.,)) is a monotone increasing Lim B. . Thus according to 3.2.2: p(SA,.)

lim p(B.,); as, from 3.1.41, p(B.,) 5 p(Al) + .. + p(A.,), it follows that F, ip(A.).

p(SAw.)

w

wt

We now give some conditions under which total additivity follows from additivity : 3.2.4. If p is additive in Xr1, if T1 is closed with respect to addition, and if for every monotone increasing sequence ((A,.)) of sets of 0 with Lim A,. t D1 we have

p(Lim A.,) = lim p(A,.), then p is totally additive in M. w)

.,

he a sequence of disjoint sets of 9N with SB,. a R. We set A.

Let

B, + B2 +

+ B. ; then ((A.)) is a monotone increasing sequence of set;:

of ? and Lim A. = SB... Thus according to the assumption, p(SB,.) = lim p(A _); wherein, because of 3.1.1, we have p(Av) = o(BI) + . 0

Thus p(SB,.) = M

v

+ p(B.,).

that is, p is totally additive in T1.

The theorems 3.2.2 and 3.2.4 together give: 3.2.41. If p is additive in the field X12, then in order that p be totally additive in X12, This condition is essential. Example: Let 14 be the system of all suheett of an infinite set E and ,.(A) the number of elements of A; if ((ai)) is a sequence of different elements of E and A. the set of all ai for i;ti m, then ((A.)) is monotone decreasing,.p(A,.) _ +oo, and, since Lim A. cow DA., - A, we have ..(Lim A.,) ,r 0.

[CHAP. I

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

16

it is necessary and sufficient that for every monotone increasing sequence ((A.)) of TZ with Lim Am a

l we have:

(Lim Am) = lim cp(Am). m

m

3.2.42. If rp is additive and monotone increasing in 9)t, if is closed with respect to addition, and if for every sequence ((Am)) of disjoint sets of Tl with SAm e 9)'t we

have cc (SAm) 5 E v(Am), then rp is totally additive in 9. M

Since A, +

m

. + Ak Q S A , we have ,o(A, + m

thus according to 3.1.1,,p(A,) +

+ A k) < V(SA m) ; m + cp(Ak) < cp(SAm), and hence E p(Am) 5 S

m

9(SA5). But according to the assumption we also have cp(SA,) S E r(Am), and thus (2) follows. 3.2.43. If ip is additive in the field 9)t and if there is a totally additive, finite set function 4, in 9t, such that always ¢(C..) -- 0 implies that p(C5) --> 0 also, then p is totally additive in 0.6 Let ((B.)) be a sequence of disjoint sets of M with SB,,, a V. We set SBm = B, M

B, + B2 +

m

+ R. = Am , B - Am = Cm ; by 3.1.11 we have #(C5) _

,I'(B) - ¢(A5); because of 3.2.2,t(A5) --> lt'(B). Thus 4,(B) - #(A5) -- 0, and hence ¢(Cm) 0. Therefore cp(Cm) -0 also. Since B = Am + Cm , and A m and C. are disjoint, we have p(B) _ cp(Am) + ,p(Cm), thus (p(A5) -. So(B); therefore cp(B,) + . + (p(Bm) --> cp(B), that is, Eso(Br) = v(B). Thus p is totally additive in Sgt. 3.2.5. If Sp is totally additive in Tl and c is a finite number 5,1 0, then cv is also totally additive in 9)1. 3.2.51. If p, and cp2 are totally additive in 97l, then in order that Sot +'p2 (or,p, - c2)

be also totally additive in 9'1, it is necessary and sufficient that there be no set A e SD2 for which p,(A) and v2(A) are infinite of different (or equal) sign, and that for not all A f S.D2 cp, + . (or
rp2) be infinite with the same sign.

Proof similar to that for 3.1.51. 3.2.52. If A e Tl and p, , Bpi are totally additive in 9)2, then in order that rpi + ,p2 (or gyp, - Sot) be also totally additive in T?, it is necessary and sufficient that there be no set A e TZ for which,pl(A) and "(A) are infinite of different (or equal) sign. This follows from 3:2'.51 because of 3.1.22. BIBLIOGRAPHY to Nos. 1 and 2: The notions of the set function and the totally additive set function are due to H. LERESO1JE, Ann. Re. Norm. (3) 27 (1910), p. 380. Subsequent to him: J. RADON, Sitzungsberichte Akad. Wiss. Wien 122 (1913), p. 1298; H. HAHN [1), p. 393; M. FRikcnET, Fund. math. 4 (1923), p. 338.

3. Extremes of totally additive set functions. We now prove a theorem which bears a certain analogy to the well-known theorem of the existence of a greatest and smallest value of a continuous point function on a self-compact set. 6 Of course, xm --> a means the same as lim x,,, = a.

BASIC PROP}RTIES

§3)

17

3.3.1. If p is totally additive in the non-empty a -field 7J?, then there is a set C e TZ and a set C' e 99)2, such that p(C) is the absolute maximum, p(C') the absolute minimum of 0. We prove the existence of C, for instance. If there is a set A e 7)? with p(A) -{- cc, then p(A) is the absolute maximum of 'p. Thus let us assume that o(X) <

+ 00 for all X e 0. We set g = sup p(X) (X e T?); then there is a sequence ((A,)) in T1 with (P(A;) -3 g; since - cc for all i; therefore, as we have assumed all co(A;) < + oo, all p(A;) are finite. We set A = S-4;, A -- A; = K; , and by P,; (j = r

2') we denote the 2m disjoint sets formed in every possible way by substituting some K. for the corresponding A; in A, A2 . . . Am , (that is, the 1, 2,

,

A,...Am_l.Km,K,.K2.A8...Am,

sets ,

K,-K2

K.); since A e 9?, we have K; a 9)2, and thus Pm, a 7)Z also.

Let B. be the sum of those Pm; (j = 1, 2,

, 2m) for which p(Pmrn) 0 (if there is no such Pm; , we have to set B. = A) ; then we have also B. a 772; as Am is a sum of certain of the Pm; , we get (p(Bm) Z O(A,,.). Since for nt" > m' either Pm.. f is c/ Bm' or and Bm are disjoint, we have

,p(B,)

co(B- + Bm+,) G ... <

-L ... + B. Fk),

and thus p(B- + Bm+, +

± B,,,,-k) ? v(A,,,). Let its set Cm Lim (Ben + Bm+, + ... + Bm+k) = B. + B,,,+, + ... + Bm+k + .. . k Then Cm e 9J also and, because of 3.2.2, ,p(Cm) = line ip(Rr -i- Bm,, + ± (P(X) k Since from the assumption < x for all X e 7)Z and may? ,

- -, (P(C. ,) is finite; thus setting C = DCm , we have C e JJ2 because of m 1.2.4 and ,p(C) = lim ,p(Cm) ? lim ,p(.1 ,) == g because of 3.2.21; therefore, since g m

m

was the supremum of all (p(X) (X e V): p(C) = g; that is, ,p((1) is the absolute maximum of ;p. 3.3.2. If ,p is totally additive in the a -field X1)2, then ,p is bounded either above or below.

Let g bethe supremum, g' the infimum of all tp(X) (X e 7l2 - A); by 3.3.1 there is a set C e 5172 and a set C' e 71?, such that o(C) = g, ,p(C') = g'; thus according

to 3.123 we cannot have simultaneously g = + co, g' = - oo . 3.321. If Sp is totally additive and finite in the a -field 972, then (p is bounded.

This follows immediately from 33.1. Let D? be any system of sets, £9 c 7)? a partial system of 0, and sp a set function

in 9; then the function in -0 which to every set B e'1) attaches the value co(B) is called the partial function of rp, restricted to 2;, and we designate this partial function by -01,p. For an arbiti ary set A we denote the system of all parts of A belonging to 912 by A19Z. If 9r is a field (or a-field), then 417)? is also a field (or u-field).

18

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

CHAP. I

If (p is additive (or totally additive) in the system iU2 and we set `ll - A 11111, then the partial function %1So is also additive (or totally additive) in 21. Applying the theorem 3.3.1 to MV, we obtain: 3.3.3. If p is totally additive in the o-field f2 and A11)2 is not empty, then in the set of values cp(X) (X e A 1971) there is a greatest one and a smallest one. BIBLIOGRAPHY: One owes theorem 3.3.1 to H. HAHN [11, p. 403; the simple proof given above is due to ff. HAIIN, Anzeiger Akad. Wiss. Wien 1928, No. S.

4. Positive-, negative-, absolute-function. Let 971 be any system of sets, e a set function in M. and A e M. We define: (4)

rp+(-4) = sup p(V) (X e A19N); 07(_A)

inf ce(X) (X e A1972).

Then io+, gyp` are set functions in ? which we denote as positive-function and negative-function of V. (4.1)

From (4) follows:

-4:-(.4) < v(X) <

+(.4)

for X e Alfi2,

and, in particular, (4.11)

-(p (A) S p(A) 5 p{(A).

Let A e 11)2 and p(A) = 0 (which according to 3.1.22 certainly is the case, if t e 9R and p is additive) ; then, since A Q A, we have: (4.12)

V+(A) ? 0,

(_4) z 0 for all A e M.

We now define: (4.2)

-{-

and we call p the absolute-function of c. Then we have also p(A) z 0 for all A e M. From (4.11) we get for every A e 971:

I a(A) I s O(A). From the definitions there results immediately:

(4.21)

3.4.1. In order that qp(A) be finite, it is necessary and suticient that both rp+(A) and 0-(A) be finite. 3.4.11. If rp is totally additive in the o-field 9)1, then to every A e 9)1 "there is a set

A' ,E AIM and a set A" e A 1971, such that p(A') = p+(A), (p(A") = -w-(A). This follows immediately from (4) and 3.3.3. 3.4.12. If y, is totally additive in the r-field 971, then at least one of the functions is bounded. :p+, This follows immediately from (4), (4.1.2), and 3.3.2. 3.4.13. If (p is totally additive in the a -field 11)2 and A e 9)2, then in order that p(A) be finite, it is necessary and sufficient that 9+(A) and lp'-(A) be finite. NECESSITY: This follows from 3.4.11 because of 3.1.21. SUFFICIENCY: This follows from (4.11).

P]

19

BASIC PROPERTIES

3.4.14. If fp is totally additive in the a -field 97? and A e 97?, then in order that ,p(A) be finite, it is necessary and sufficient that O(A) be finite. This follows from 3.4.13 and 3.4.1. 3.4.2. If 9)? is closed with respect to addition and fp is additive in 0, if A e 912, A e 91?, and A; a 91 (i = 1, 2, ) are disjoint subsets of A, then

I cn(A.) ( , m) with o(A.) Z 0 and A..' Let A., be the sum of the sets A; (i = 1, 2, m) with fp(A;) < 0. Then I fp(A,) I + the sum of the sets A; (i = 1, 2, . + I m(A.) I - tp(A.,) -- p(A') and, since from (4.1) 0 = p(A,m) 5 .p+(A), 0 k V (A".) -cc (A), we have I s(Al) I + ... -i- I fa(A,,,) I s ,p+(A) + (A) = O(A). Hence E I fp(Ai) I o(A). ,

3.421. If fp is additive in the field 9)?, them for every A e 9)? we have: p(A) = sup (I se(X) I + I sp(A - X) I) (X e A 191). We set g = sup (I p(X) I + I c,(A - X) I) (X e A 197?). Because of 3.4.2, we

have g 5 #(A). Thus we still have to prove: g z fp(A). In A 19 there is a sequence of sets X. with p(X,,,) -* ,p+(A), thus also with I f,(X,.) I --r ,p+(A) If fp+(A) = + co, then I fp(X.,) I --> + oo ; hence g = + c0, and the contention is proved. If .p+(A) < + -, thus finite, then because of (4) and since cc(A) _

,p(X) + 9(A - X) for every X e A1T?, it follows from yv(X.,) --+ v+(A) that lp(A-X,.) -+ -07(A), and thus I fp(A - X,) ---> fp-(A). But then we have: Ic(X.)I + I fv(A - X.) I --f w+(A) + cc (A) = o(A), whence g ;_ o(A) 3.4.22. If fp is totally additive in the a -field 9, then to every A e W2 there is a set A' e A 1971, such that I so(A') I + I fp(A - A') I = O(A).

Let A' be the set of the theorem 3.4.11; then v(A') = p+(A). If w+(A) _ + 00, thus also O(A) = + -c, then the contention is proved. If fp+(A) is finite, then because of (4) and since cp(A) = v(X) + p(A - X) for every X e A 1971, it follows from_fp(A') - fp+(A) that c(.4 - A') _ -V-(A), and thus I fp(A-A') I = (A); but then we have: I so(A') I + I p(A A') I = fv+(A) + o 7(A) = fp(A). 3.4.3. The set functions p+, j, fp are monotone increasing. Because of (4), an immediate consequence of A e 97?, B e i'?, and A a B is e+(A) S p+(B), c (A) 5 (B). Thus because of (4.2) also O(A) s fp(B). 3.4.31. If fp is additive in the ring 9)? and A e 9191', then fp+, ,p , fp are also additive.

We prove that p+ is- additive; analogously one proves that VP is additive; then because of 3.1.52, (4.12) it follows that also o is additive.-Let A e 97?, B e 1?, AB = A; we have to prove : fp+(A + B) = p+(A) + p+(B). Let X e (A + B)19)2; then X = AX + BX and AX e 912, BX e 97?; since AX, BX are disjoint, we

have fp(X) = p(AX) + P(BX) S p+(A) + fp+(B), thus also ,+(A + B) S fp+(A) + c+(B). Therefore we have only to prove that i+(A + B) k fp+(A) + fp+(B).

Because of (4), in 9 ? there are sequences of sets A. C A, B. Q B

with,p(A.,) - fp+(A), fp(B.,) - + a+(B); then A.B. = A. Thus jp(A,.

B.,)

T Because of 1.2.1 and 1.1.11, the assumptions on 9)1 are certainly satisfied, if 9)t is a (non-empty) field.

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

20

[CUAP. I

,p(4,) + p(B.); therefore cp(A. + B.,) -,p+(A) + v+(B) and, since A. + B. A + B, we have p+(A + B) Z fp+(A) + jp+(B). 3.4.32. If p is totally additive in the ring 212 and A e IV, then p+, P , 0 are also totally additive.

We prove that p+ is totally additive; analogously one proves that cP is totally additive; then because of 3.2.52, (4.12) it follows that also -p is totally additive.Let ((A,.)) be a sequence of disjoint sets of S1, A = SAM, A e 9; we have to

prove: p+(A) =

Let X e A 1D1; then X = SA..X, A.X a A,,,, and

thus- ,p(X) _ E p(A.X) 5 E

hence also ujp+(A). S E p+(A ). + A. C A, 3.4.31, 3.4.3, we have: On the other hand, because of A, + ,p+(Ai + ... + A.n) = p-"(A,) + ... + p+(A,.) 5 +(A); therefore also E V+(A,.) < rp+(A). Thus we have V+(A) = E (p+(A,.). 3.4.33. If p is additive, Ip finite and totally additive in the field 91, then 9 is also totally additive. This follows from 3.2.43 because of (4.21). 3.4.33 1. If N is additive and p+, rP are finite and totally additive in the field M, then,p is also totally additive.

If p+ and P are finite and totally additive, then because of (4.2), (4.12), and 3.2.52, 0 is also; therefore 3.4.331 follows from 3.4.33. 3.4.4. If Tp is additive and monotone increasing (or decreasing) in 9)1 and A e 1X12,

then v+=0=vs, 97 -0(orc =0= -'p, 'P+=0).

Because of 3.1.4 we have rp k 0; thus according to (4) Ir- 5 0; therefore by = 0. Hence as a result of (4.2) p+ _ (p and, since p is monotone in-

(4.12) p,

creasing, w+ _ (p by (4). 3.4.41. If p is additive in 9)1, if P = 0 (or rp+ = 0) and D2 is closed with respect to subtraction, then p is monotone increasing (or decreasing). From j = 0 it follows, because of (4), that ip (X) Z 0 for all X e 211. Thus the contention results from 3.1.4.

3.4.5.Ifp=91 +Ve,then tv+si++j+,w swi +V,p

+0

From V (A) = sup ( (X) + o2(X)) (X e A 19)'1) it follows that p+(A) 5 sup (X) + sup v(X) = Cpl (A) + Va (A); analogously the contention may be demonstrated for P ; it then follows for tp from (4.2).

3.4.51. If(p= vi -v2,then p':_9 v +q, ,(P S Pl

This-results from 3.4.5, since (- 02) + = tea and (- v2)- _ v+ 3.4.52. If w 5 4,, then p+ 5 iG+, cP ? r

+q

This follows immediately from (4). 3.4.6. If pisadditive in Oland TI is closed with respect to subtraction, if A e 11)2 and

at least one of the numbers p+(h,) and yo (A) is finite, then p(A) = p+(A) - 'p (A). At first assurae V(A) to be iinfinite; for instance, suppose V(A) + -; then

also 9+(A) = + oo, thus P -(A) is finite, and hence ,p(A) = p+(A) - (P-(A).Now let s(A) be finite,., According to (4) there is a sequence of sets X. e A 19)l, cp+(A). As a result of such that

21

3]

BASIC PROPERTIES

(4.3)

9(X..) + p(A -- X.) = p(A),

it follows from 9(X,,,) -* p+(A), according to (4), that p(A -

-07(A).

Thus from (4.3) we obtain: p+(A) - t-(A) = p(A). 3.4.61. If v is totally additive in the c;lteld Tl, then for every A e S2: -ptA)

p+(A) - 9;-(A).

This follows from 3.4.12 and 3.4.6. Because of 3.4.3 and 3.4.32 there results from 3.1.61: 3.4.611. Every set function totally additive in a a -field is the diferente between

two monotone increasing, totally additive set functions.

Because of 3.4.13 (and 3.3.21) there follows from 3.4.611: 3.4.612. Every finite set function totally additive in a a -field is the difference between two monotone increasing, totally additive, finite (and actually bounded) set functions. Among all possible representations of an additive set function p as the difference

between two monotone increasing set functions, the representation .p = 'P' - - is distinguished by the following property: 3.4.62. If A e , p is additive in 0 and v = p2 - Vs , where p2 and vs are additive and monotone increasing, then for every A e 0 we have:

(P+(A) 5 vi(A),

(A)

P2(A).

Let X e'"AlTl; since according to 3.1.4 jft(X) Z 0, c(X) ?_ 0, we have p(X) se (X)

vj(A), p(X) z -v2(X) z -V2(A), and thus p+(A)

"(A), -i (A)

ps(A). 3.4.7. If p is totally additive in the non-empty a -field 9)1 and p+ (or p-) is finite, then there is a set M+ a 92 (or Mr a 92); such that for every A e a writing A+ = AM+,

A- = A - M+ (or A- = AM-, A+ = A -- M-), we have: p+(A) = p+(A +) = p(A+),'P(A) = P -(A-) _ --p(A-); (A+) = 0,,P+(A-) = 0. According to 3.3.1, among all values .p(X) (X e !Ul) there is a greatest one, say

,p(M+). By (4) p(M+) is finite. We write A+ = AM+, Ar = A - A+ A - M+. Then p+(A-) = 0; for if p+(A-) > 0, then there would bean .X' e A-19 with p(X) > 0 and, since XM+ = A, we would have p(M+

+ X) =

P(M4)

+ p(X) > P(M+),

contradictory to the choice of M+. Furthermore, 07-(A4) - 0; for if .p (A+) > 0, then there would be an X e A+1912 with p(X) < 0 and, since X Q M+, we would have so(M+ - X)

tp(M+) - So(X) > p(M4), contradictory

to the choice of M+. From p+(A) = p+(A+) + p+(A') now it follows that So+(A) = p+(A+), and in the same way we obtain .p (A) = i (A-). Because of 3.4.61, it follows from y (A) = 0 that w(A+) = p+(A4) and from qi+(A-) - 0

we obtain ep(A-) - -yj (A'`). 3.4.71. If p is totally additive in the o-feld T and A e 0, then there is a deeomposition A = A+ + A- of A into two disjoint set A+ a 9)2, A- a V, such that p." (A-)

=0,9;(A+)=d.

[CHAP. I

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

22

Because of 3.4.12 this follows from 3.4.7. 3.4.8. If cp is totally additive and finite in the a -field fit and c > 0, then in Sl there are only finitely many disjoint sets with I ip(X) I

c.

Because of 3.4.14 0 is also finite, and thus by 3.4.32 and 3.3.21 g, is bounded: , X,., be disjoint sets of fit with g for all X e V. Let X1 , X2, , m). Because of (4.21)'(X;) z c also, and thus c (i - 1, 2, ,P(X1 + ... + X,.) mc. Hence inc S g, or m c .

.p(X)

I p(X;)

3.4.81. If p is totally additive and finite$ in the a-field fit, then in fit there are only eountably many disjoint sets with 9(X) , 0. This follows immediately from 3.4.8. BTBLioORAPHY: The notions of positive-, negative-, absolute-function are due to H. LnBasoun, Ann. Ec. Norm. (3) 27 (1910), p. 381. Furthermore Of. J. RADON, Sitsungsberichte Akad. Wise. Wien 122 (1913), p. 1299.-Theorem 3.4.71 is due to if. HAHN [11, p. 404; other proofs for it were given by R. FRANCK, Fund. math. 5 (1924), p. 262; W. StsRPIhSKI, Fund. math. 5 (1924), p. 262; 0. NIKODYM, Fund. math. 15 (1930), p. 176.

5. Theorems about limits. Referring to 3.2.2 and 3.2.21 we now prove a few

other theorems of this type: 3.5.1. 11.p is totally additive and monotone increasing in the o-field )t, then for every sequence ((A.,)) of sets of 9Jt we have: A

= Lim A., D. = A.-A.+,

A

m

; then D. J. DA+1 and, according

to §1 ,(1.32) and 1.1.2, we have A - SDA = LimD., ; thus because of D. e T?, M

A

A s SD?, and 3.2.2 we have ce(A) = lim (p(DA) ; but since .p is monotone increasing, 10

jp(D.,)

p(Am).

Thus so(A)

lim p(A.).

3.5.11. If tp is totally additive and monotone increasing in the o-field fit, then

for every sequence ((A.)) of sets of fit, for which 4 (SA.,) < + -,° we have: p(Lim A..) ? lim v(AA). A We set A = Lim A.. , S. = A. + A,.+I + in

; then S. Q SA+1 and, accord-

ing to §1 (1.32) and 1.1.2, we have A = DS., = Lim S.; because of 3.1.4 and 3.1.21, jo(S..) is finite; thus according to 3.2.21, we have 9(A) = lim tp(S A

therefore, since (v(SA) k p(A A), we get v(A)

urn cp(AA). A

This condition is essential here and in 3.4.8. Example: Let SDI be the system of all subsets of a non-countable set E and let a(A) be the number of elements of A. That this condition is essential (also if the v(A..) are finite), is shown by the example of footnote 5, p. 15, if there one considers the sequence of the sets {a,.1; for we then have

, (Lim la. l) - m(A) -° 0,sr(lam1) - 1.

BASIC PROPERTIES

131

23

3.$.2. If , is:totally additive in the o -field 5171, then for every sequence ((A.)) of acts of IN, left which ,p (SA m) is finite' and Lim A m exists, we have: p(Lim A m) ..

m

m

rim cp(A.). IZI

According to 3.4.13, ,p+(SAm) and iP (SAm) are finite. Because of 3.4.3, in

m

v+ (Lim A.) 6 litre yp+(A.); thus V+(Lim Am) = rim V+(_Am). In the game way we get m m r 0 (Lim A,.) - litre tj (A,.). Now from 3.4.61 it follows that p(Lim Am) r 3A.32, it follows from 3.5.1 and 3.5.11 that. rim ,p+(A.)

.

Jim p(Am).

.

'BIBLIOGRAPHY: H. HsUN fll, p. 396, 407.

6. Sequences of set tumctions. We shall use the following one-to-one mapping of the set of all (finite or infinite) real numbers a on the closed interval [-1, +1J: (6)

$(a)

a for - oo < a < -I- oo ; S(-I- °p) = 1; 8(- oo) . -1. I

(6) is called the bounding transformation. For any two real numbers a,, ag we now define: (6.1)

11 a, - a,11 _ I S(a,) - S(a,) 1;I0

if we ooneider.(6.1) as distance of a, , a2, then the set of all real numbers, including

+ oe and -- *, beeoeees.a metric space (§2, 3), which we denote as R1 at # as we have (6.11)

.

For

al-( 11 < fiat - a, .

Now we discuss sequences of set functions, referring to 3.1 .4 and 3,1.61. Let ((,p,)) be a sequence of iset functions in SR. If for every A e M the sequence ((,o,(A))) is convergent", then it is called convergent in 9; if te(A) rim ,p,(A) for every A e $12, then we also say: ((gyp,)) converges in D2 to ,p.

=

Furthermore, we say the sequence ((,p,)) converges uniformly to the set function a in D2 if to every 8 > 0 there is a r., such that I ,p,(A) - p(A) < 8 for all. e2andAP, 9 re." 3.6.1. I f As. s e t f u n c t i o n s sv, (r - 1, 2,

) are totally additive in the field $1 and the sequence ((es,)) conceive, uniformly to a function p which does not have infinite values of different sign, then co is also totally additive. _' This notation is due to H. Haha (2j, p. ITS. +I That is to say: -there is a real (finite or isSnite) number a with lint V,(A) - a. also p. 121, footnote 10. It Cf. also p. 124.

Cf.

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

24

[CHAP. I

Let ((Ak)) be a sequence of disjoint sets of 9)t, A = SAk, and A e 0; we have to k

+ Ak - At ; then because of k + cp(A,); thus the contention is equivalent to 3.1.61: r(B,) _ co(Al) + II 4,(A) - ,.(A) 11 + ;p(I3k) --> p(A). Now we have II cp(A) - 1P(Bk) II prove: jo(A) = E 4p(Ak). Let us set Al +

IP(Bk) II Because of the uniform convergence II ip,(A) - se,(Bk) II + II of ((,p,)) there is a p*, such that I I 4p(A) I I < 3, I I p, (Bk) ,(BA) I I <

for all k; since (p,. is totally additive, by 3.2.2 we have lim 9,.(B,) is

for k z k*. Therefore

thus there is a k*, such that (I ,p,.(Bk) - 4p,.(A) II <

we have II p(A) - ,p(Bk) II < e fork z k*, that is, 4(Bk) --> co(A). 3.6.2. If A e 9)2 and ((gyp,)) is a monotone increasing sequence of totally additive and monotone increasing set functions in 9)2, then op = lim rp, is also totally additive.

Let ((A,)) be a sequence of disjoint sets of 9)1, A = SAk , and A e SD'l; we have k

to prove:,p(A) = E ,p(A,). Since w, is totally additive, we have E 4p,(A,,) _ k k cp,(A); thus, as 4(Ak) z tp,(Ak), we get E.p(Ak) p,(A) and hence also k 4p(Ak) S 4p(A). Since 4,(Ak) z 4p(A). It remains to be proved that k

k

161

F rp,(Ak) = jp,(A), because of 3.1.4 we have E 1P,(Ak)

,p,(A) S ,p(A). Thus

k-l

k nt

E v(Ak) <
x..1

k

3.6.21. If 9)t is closed with respect to subtraction, if (((p,)) is a monotone decreasing

sequence of totally additive and monotone increasing set functions in 9'l, and if" co for all X e 9)1, then se = limp, is also totally additive. is Of course, here one can substitute "jp,,(X) < +oo" for ",r1 (X) < (since for Furthermore one vo > 1 one can omit the terms 9ft , ... ,,p,e_, in the sequence can generalize the condition in the following way: for every X e fl1 there shall exist a first index vx with ,p,z(X) < -i- oo. For if ((X,)) is a sequence of sets of 0 with SXk e'1)2 and if we set SXk - A, then there is an index VA with sp,_, (A) < + m ; thus according to 3.1.2 we

haveS,,A (X) < +- for all X e A10; this makes the application of 3.6.21 possible for A1171 and v $ PA. If t1Jl is a a-field, then the last condition is not a real improvement over the original one. For then there cannot be any sequence ((Xk)) with Xke9R, vxk since, because SXk s A e 91, we always have n,A (Xk) < -1- oo (k = 1, 2, ... ). s

That the condition is essential, is shown by the following example: Let Tt be the system of all subsets of an infinite set E. Let 5'1(X) be the number of elements of X e 1l (thus

r1(X) !. +'o if X is infinite) and ,p,(X) - 1 e,(X). Then we have ,p(X) = lim ip,(X) m 0 if X is finite, and w(X) - +oc if X is infinite; thuso is not totally additive.

25

BASIC PROPERTIES

§31

By 3.1.4 we have p, z 0; thus the functions p, are finite. v, - p,(v = 1, 2, - . ) is a monotone increasing sequence of functions which, according to 3.2.52, are totally additive and, since vi - gyp, z 0, are monotone increasing as a result of 3.1.4. Thus, because of 3.6.2, lim (VI - sp,) - tot - p is totally additive; hence, r

by 3.2.52, p = - (,pi - p) is also totally additive. The theorems 3.6.2 and 3.6.21 are, essentially, special cases of the general theorem:

3.6.22. If t is closed with respect to subtraction, if ((gyp,)) is a monotone increasing (or decreasing) sequence of totally additive set functions in V, and if VI(X) > -- x (or < + co) for all X e 9R, then, = lim ip, is also totally additive.

Since p,(X) z o,(X) > - , 0,-(p = 1, 2,

) is finite as a result of (4.12).

Therefore according to 3.4.6 we have g*, _
(p = lim a, = lira gyp; - lim (p is also totally additive. r

v

D

A very general condition, under which the total additivity of 'r = lim cp, follows from the total additivity of the -p, , will be given in 5.8.2. 3.6.3. If ((gyp,)) is a sequence of set functions in M and ep = lim jo, exists, then V

+(A) S lim

(A) 49 lim ,p-(A) for all A e 812.

(A ),

V

Y

In ((p,)) there is a partial sequence (( . )), such that 4 (A) -+ lim gyp; (A). To every z < .p+(A) there is an X C A, such that X e STZ, So(X) > z. Since v*, such that z < to,;(X)

w i(A).

Thus

z S Jim p +1,,(A) = lim
M

3.6.31. If (((p.)) is a sequence of set functions in fit, if A e 9 with p,(A) = 0,1s and if Sp = lim 9, exists, then c(A) 1im p,(A) for all A e U. This follows immediately from 3.6.3 and (4.2). 3.6.32. If ((p,)) is a monotone increasing (or decreasing) sequence of set functions in iD and p = lim pp,,, then also the sequence ((0+,)) (or ((cpy ))) is monoto-e increasing D

and we have: tp+(A) = lim 4p+, (A) (or ,,-(A) = lim co (A)) for all A e V. D

V

According to 3.4.52 ((4)) is monotone increasing.

Let z < lim gyp, (A); then

there is a v*, such that 0+,.(A) > z; thus there is also an X C A, such that X e V and p...(X) > z. Then, since ((So,)) is monotone increasing, we have also ,p(X) > z, and hence ip+(A) > z; therefore V+(A) 9 Jim 4; (A). Thus, because r

of 3.6.3, we get: w+(A) = lim gyp, (A). Y

1+ If the So, are additive, then, because of 3.1.22, the condition p,(A) F 0 is satisfied :,utornatically.

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

26

[CHAP. I

3.6.321. If 9t is closed with respect to subtraction, ((p,)) is a monotone increasing (or decreasing) sequence of additive set functions in 9It, and Sp = lim is,, then the

sequence ((V )) (or ((gyp+))) is moiwtone decreasing and, for every A e SR for which

0(A) < + - (or > - - )16, we have: (A)

lim w. (A) (or +(A) = lim (P: (-4))

is monotone decreasing. If ip (.4) _ + -, then According to 3.4.52 the contention follows immediately. from 3.6.3. Thus let 0-(A) be finite. We

set t - A 19; then according to 3.1.22 and 3.1.6 the partial function

I1(p

(§3, 3) is additive; therefore by 3.4.6 we have: p(A) _ p+(A) - 07-(A). Thus p+(A) is finite and, because of 3.6.32, jp, (A) is also finite; therefore according to 3.4.6 we have: p.(A) =gyp; (A) - p- ,(A). Since p,(A) -+ ,p(A) and, by 3.6.32, 0+,(A) - (P+(A), it follows that V. (A) -+ v 7-(A) also. 3.6.322. If ((,o.)) is a monotone increasing (or decreasing) sequence of totally

additive set functions in the o field 9111 and 0 - lim gyp, , then p-(A) = lim 0-(A) (or

w+(A) = line N; (A)) for all A s M.

If ip- (A) _ + oo, then the contention follows from 3.6.3. Thus let (p -(A) be

finite; then we have '(A) > - o o, and hence (p,(A) > - eo for almost all v, and so we can assume this to be true for all v. Then because of (4), 3.3.3, and 3.1.2, (o. (A) is also finite for all v; thus as a result of 3.4.3 ce (X) is also finite for

Now we define a sequence ((A,)) of sets of fit by induction according to the following rules: corresponding to 3.4.71, let A = At + Al with Al -Ai = A, g'(A;) = 0, rpi (Ai) = 0; and, if A;_t is known, let A;_, = A-, + A; with A+,-A-, = A, w7(A;) = 0, cp; (A-) = 0. Then ((A-,)) is monoevery X e A19)1.

tone decreasing and we prove by induction that

p-(A -- A-) = 0. For P = I this follows immediately from 3.4.71 because of 3.4.7 and 3.4.31. Let us assume (6.2) for v - 1. Since A - A; _ (A - A;_1) + (A-_, - A;) _ (6.2)

op-(A) _ --,p.(A-.),

(A - A,-,) + A,, we have V-(A - A;) = qp-(A - A 1) + 07,(A;); therein,

according to the definition of A: , we have (p-(A;) = 0. By our assumption, we have (p-_,(A - A;-,) - 0, and thus, since by 3.4.52 ((07)) is monotone decreasing, (p-(A - A-_,) = 0 also; hence ip; (A - A-) = 0. Thus the second equation (6.2) has been proved. Therefore we have 4o (A) = O (A-), and the is This condition is essential, as the following example shows. Let A be the set of the elements ak(k = 1, 2, ... ) and let W be the or-field of all subsets of A. If X is a finite subset .; if X is an infinite of A, then let -,o, (X) be the number of elements ak in X with k

subset of A, then let w,(X) _ +-. The sequence ((p.)) is monotone increasing and V(X) = lim cp,(X) - 0 if X is finite, or - + - if X is infinite; and we,have so. (A) _ -1- cc

,o(A)=0.

,

21

ZERO-SETS AND COMPLETE FIELDS

§41

first equation (6.2) means: 4: (A,,) _ -,p,(A -V) ; but this results immediately from,p,(A;) _ (A;) --- 4-(A, ), since gyp; (A;) = 0; thus also the first equation of (6.2) is proved.-Then since a ,-(A) is finite, y,,(A;) is also finite. Now we set B = DA-,; then, because of 1.2.4, we have B e SDl and w,(A,) = 4,,(A- - B)

+ O.A. Since Lim (A- - B) = D(A' - B) = A, and since, because of the finiteness of 4p,(Ai) and 3.1.21, 9,(Ai - B) is also finite, by 3.2.21 we have lim 4p1(A-. - B) = 0. Thus, as ((so,)) is monotone increasing: lim `p,(Ay - B) r

y

0.

Therefore, since Tp,(B) --+,p(B), we have lim p,(A,) k 4(B), that is -4,(B) V

thus according to (6.2): -4p(B) z lim jo; (A), and hence, since B Q A: g (A) k lim a.(A). But by 3.6.3 we have 07(A) S lim vP, (A);

thus c(A) = lim 0-(A). 3.6.323. If J)1 is closed with respect to subtraction, ((,p,)) is a monotone increasing (or decreasing) sequence of additive set functions in 91, and ap = lim po, , r

then we have vp(A) = Um o,(-t) for all A e 912.

't'his follows immediately from 3.6.32 and 3.6.321, if 40(A) < + co. But if 4o(A) _ + co and thus lim p,(A) = + oo, then because of (4.21) we have also p(A)

+ m, lim cp,(A) _ + oo, and hence again (p(A) = lim o,(A). V

14. Zero-sets and complete fields 1. Zero-sets. Let rp be a set function in A2, let A e 91 and p(A) = 018 A set A e 931 is called a zero-set for o", symbolically written: A =, A, if for all sets X Q A belonging to TZ we have p(X) = 0. Of course, A =, A. If A e 93? and A is not a zero-set for cp, we write: A ,, A. From the definitions of the positive-, negative-, and absolute-function (§3, 4) there ..follows immediately: 4.1.1. In order that A =, A, it is necessary and sufficient that O(A) = 0.

4.1.11. In order that A =, A, it is necessary and sufficient that P '(A)

w(A) - 0. 4.1.12. A -, A is equivalent to A =, A; furthermore A =, A is equivalent to both A =.{. A and A = o- A simultaneously. If A =, A, then c(A) = 0; for a monotone yr we also have the converse: 4.1.13. If v is monotone, then in order that A -, A, it is necessary and sufficient

.that P(A) = 0. 4.1.2. If 0 = cci -l- cps (or qp = 91 A

then A =,1 A, A = A imply that

A also.

is Because of 3.1.22, this certainly is the case, if j' is additive in the non-empty field V. 17 The words "for i." can be dropped, if there is no question.

[CHAP. I

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

28

4.1.3. If A =, A and B e A 19J , then also B =,, A. 4.1.31. If cp is additive in the field 1, then the system of all zero-sets for (P is a field.

Let A e 9W, B e 9Ji, A =, A, B =,, A. Since A -- B e l and A- B Q A, we have A - B = v A according to 4.1.3. Thus we only have to prove that A + B ==, A also. Let X e P, X Q A + B. Then we have X = AX + (B - A)X; the terms of the right side are disjoint sets of 9n, hence P(X) = ,p(AX) -{- ,p((B - A) X).

Since AX C A and (B - A)X Q B, we havep(AX) = 0

and p((B - A)X) = 0; thus
_, A. 4.1.32. If Sp is totally additive in the u-field T?, then the syctcm of all zero-sets for (p

is aafield. Because of 4.1.31 we only have to prove: If A. e M, A m = A, and A = SAm ,

then also A =,, A. We set B. = A, +

0

+ A.,; then ((Bm)) is a monotone

increasing sequence of sets of Dl with Lim B. = A and, by 4.1.31, we have Bm =, 0

A; thus because of 4.1.1 rp(Bm) = 0; then, by 3.4.32 and 3.2.2, we have P(A) = 0,

and hence according to 4.1.1: A =, A.

2. Neglecting zero-sets. Two sets A e STZ, B e 9 are called disjoint for rp

if AB =, A. If A and B are disjoint, then they are also disjoint for gyp.

A

system TZ' C 9Th of sets (or a sequence ((Ak)) of sets of 9)2) is called disjoint for fP if every two sets of D1' (or of ((Ak))) are disjoint for gyp. , m), B e 91, and if 4.2.1. If p is additive in the field 912, if Ai a fit (i = 1, 2, - -}- Am) -{A also. , m), then B . A (i = 1 , 2,

+ Am) = BAI + This follows from 4.1.31, since B(AI + 4.2.11. If V is totally additive in the o-field 0, if Ai a 0 (i = 1, 2, and if B Ai =, A (i = 1, 2,

+ BAm . ), B

), then B SAi =, A also.

This follows from 4.1.32. 4.2.2. In order that A and B be disjoint for gyp, it is necessary and sufficient that they are disjoint for 0, or that they are disjoint both for.p+ and for V7. This follows from 4.1.12. 4.2.21. If cp is additive in the field 9)1, if A e 9)t, B e 9)2, and A, B are disjoint for gyp, then we have:

p(A + B) - ,.(A) + o(B), (o+(A + B) = p+(A) + ,+(B),

v 7(A + B) - 9`(A) +

(B), .p(A + B) - AA(A) + .p(B)..

This follows from §3 (1.2) ; for since AB =, A, we have p(AB) = 0. Similarly for rp+, jp , o bedaubs of 3.4.31.

4.2.211. If rp is additive in the field .'t, if Ai a 9)? (i = 1, 2, are disjoint for p, then we have:

,

m), and the Ai

29

ZERO-SETS AND COMPLJ= FIELDS

§4)

,'(A1 + ... + Am) _ p(A1) + ....{... jo(Am), (Q+(A, + ... + Am) - yo+(A,) + ... + v+(A,.),

-(A,+ ... +As.) =io (A,) + ... +a (Am),

+ .. + q'(A..).

co(A1 + ... + Am) This follows by induction from 4.2.21, because of 4.2.1.

4.2.22. If y, is totally additive in the field SD t, if ((A k)) is a sequence of sets of D1, disjoint for,p, and SAk a m, then we have: k

cc(Ak), c+(SAk) _ E w+(A,),

,p(kA,)

sv (SAk) _

E

E

+ A,,; then ((Bk)) is a monotone increasing sequence We set Bit = A, + of sets of )1 with Lim Bk = SAk . Because of 4.2.211 we have ,,(B,) - p(A3) + k

k

+ ,p(A,) ; thus according to 3.2.2 p(SAI) - lim rp(Bt) laxly for p+,

, (p

P(A&).

k

A

Simi-

because of 3.4.32.

If A e V, B e fit, A - B =. A, then we write: A a;, B; in words: A Is a subset If A C B, then also A C:. B. 4.2.3. In order that A C. B, it is necessary and sufficient that A C; B, or that

of B, neglecting a zero-set for rp.

both AC.,BandAC,B. This follows from 4.1.12. 4.2.31. If v is additive in the field 1131, then for any three sets A, B, C of X71, from A - C C (A - B) + (B - C), 4.131, and 4.1.3, from A - B B - C =, A it follows that also A - C =c A.

4.2.32. If w is additive in the field ?, if A1, A, , B1, B2 are sets of 1171, and Al C,, B1,

1 3 6 , then we have also A, + A, C, B, + B2, A1A, C. BLB2 , A, - B, Q,

B, -- A2. Because of 4.1.31 and 4.1.3, this follows from (A, + A,) - (B, + B2) C

(A, - B,) + (A, - B,), A1A, - B,B2 _ (AiAs - B,) + (A,A, - B,) C (A, -- B1) + (A, - B,), (A, - B,) - (B, - A2) ((A, -- B,) - B1) + (A, - B,) At C (A1 - B1) + (A, - B,) .

4.2.33. If o is totally additive in the afield J1 and if A. e 1,D1, B. a 1Sl1, A. Cv

B,., then also SAm C SBm and DAm Ce DBm. m

,»

m

M.

Because of 4.1.32,4.1-3, and 1.2.4, this follows from SAm -- SBm in

m

S(Am - Bm) in

and (cf. §1 (1.21)) DAm - DBm = S(DAi - Bm) Q S(A,, - Bm). m

m

m+

w

{

4.2.34. If p is totally additive in the a field SJ.1l and if Am e 171, A. a 1, A

then also Lim A. Qw Lim B. and Lim q, C. Lim B.. fn

m

m

11

Bm,

[CHAP. I

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

30

B.xcause of §1 (1.32), this follows from 4.2.33. 4.2.35. If p is totally additive in the o --field lR, if Am M, Bm a .12, A

B,,, and

if Lim A,,, Lim B. exist, then we have also Lim A. Q, Lim B..

,

+

m

m

This follows from 4.2.34. If A e 91, Be 971, A C, B, and B a, A, then we write A =, B, or, in words: A and

Thus for B - A the symbol A =, A means: consistent with our notation introduced in No.1. If .4 is a zero-set for po, A e 9i1, B e 912, but if it is not true that A =, B, then we write: A ; , B. 4.2.4. In order that A =, B, it is necessary and sufficient that A =; B, or that both .4 =,,+ B and A =_ B.

B differ only as to zero-sets for gyp.

This follows immediately from 4.2.3.

For an additive p in a field 0, the relation =, is an equality relation: we always have A =, A, and both the propositions A =, B, B =, A are equivalent; that the relation =, is also transitive, is shown by the theorem: 4.2.41. If p is additive in the field 0, then for any three sets A, B, C of n2, from

A =, B, B -, C it follows that also A =, C. This is an immediate consequence of 4.2.31. The theorems 4.2.32, 4.2.33, 4.2.34, and 4.2.35 imply: 4.2.42. If p is additive in the field T2, if A,, A2, B1, B2 are sets of 9R, and

.41 _ B, , A2 -, B2, then we have also A, + A2 = o B, + B, , A,A2 =, B,B2,

A1-A2=,B,- B2.

4.2.43. If v is totally additive in the a--field fit and if Am a 9N, B. a 9)2, A. Bm , then we have also SAm =, SB,,, DA., _, DBm, Lim A. _, Lim Bm, Lim Am m

m

m

m

m

m

m

Lim B,, and, if Lim Am, Lim B. exist, also Lim A, =, Lim B.. M m

m

m

4.2.44. If w is additive in the field M, if A e 9)l, B e 9)t, and A =, B, then p(A)

,o(B), p+(A) = o+(B),

(A, = V(B), O(:1) = O(B)

From A = AB + (A - B), B = AB + (B - A) it follows that (p(A) _

,p(AB) + p(A - B), ,p(B) = to(AB) + tip(B - A) ; thus since fp(A - B) = 0 and p(B - A) = 0, we have: rp(A) = p(B) _ p(AB). Similarly for (p+, , ip because of 3.4.31.

We now show that the decomposition A = A+ + A-, given in 3.4.71, is unique except for zero-sets: 4.2.45. If v is totally additive in the a -field 91, if A e 9)2, and if A = A+ + Aand A = A* + A** are two deeonmpositions of A into disjoint sets of 912, such that +(_4) _ p+(A**) = 0, 0-(A) =' o7(A*) = 0, then A* A+ and A** =, A-.

Since A* - A+ Q A- and A+ - A* a A**, we have p+(A* - A+) = 0 and 1P+(.1T-A*)=0; sinceA*-A+c;A*andA+-A*cA+, we have4, (A*-A+)

= 0 and p (.4+ - A *) = 0; thus, according to 4.1.11, we have A * - A + _, A and .4+ - :i* =, A, that is, A* =, A+. Similarly we obtain A** =,A-.

31

ZERO-SETS AND COMPLETE FIELDS

§4l

If (p is totally additive in the a-field fit, if Am e P1, A e 931, Lim A. =,, A, and m

Lim A. =o A, then we write: Lim, A. = A. From Lim Am =,A it follows m that Lim, Am = A; conversely from Lime A. = A it follows that Lim A. m m m A if and only if Lim A. exists. 4.2.5. If 9 is totally additire in the a -field 971 and if A m e 9931, A e 931, A' a 9)'l,

then from Lim, Am = A and A =,, A' it follows that also Limo A. = A'. in

m

This results from 4.2.41. From 4.2.43 and 4.2.41 we get: 9'1 and if A. a 93'1, Am a 911, 4.2.51. If * is totally additive in the a-field "n A e 9)1, then from Lim, Am = A and Am =,. A it follows that also Lim A-' = A. m

of

Let 972 be a field, p an additive set function in 9)1. Then, as we saw, the relation =, is an equality relation defined in 971. Thereby fit is decomposed into disjoint partial systems, such that for every two sets A, B of 9)1 we have A =,, B if they belong to the same partial system, and A 76, B if they belong to different partial systems. We designate these partial systems of 971 by small Greek letters a, f, . and we call every set A e 9971 belonging to a a representative of a. If A, A' are two representatives of a and B, B' two representatives

of f, then according to 4.2.42 we have: A + B =, A' + B'; thus we are able to give the definition: If A and B are representatives of a and l4, respectively,

then a + S shall be the partial system, a representative of which is A + B. Analogously a Q, a - f can be defined, according to 4.2.42, and, if cP is totally additive in the a-field 0, then Sam , Dam , Liln a., Lim am , Lim am can also m

m

m

m

m

be defined, according to 4.2.43.

Now let p be a finite, totally additive set function in the a-field 0; then by 3.4.14 rp is also finite. We show that then the set of all the partial systems a, S, , into which 9911 is decomposed, becomes a metric space M (§2, 3) by the

following definition of the distance: if A and B are representatives of a and respectively, and o is the absolute-function of gyp, then the distance shall be (2)

a/3 = o(A -- B) + o(B - A).

From 4.2.42, 4.2.44 it follows immediately that the value of aft defined by (2) does not change, if for A and B we substitute other representatives of a and p. Furthermore condition 1 (§2, 3) obviously is satisfied. Thus we only have to prove that condition 2m is also satisfied; that is, we have to prove the triangle inequality: (2.1)

ay S a#+7$.

Let A, B, C be representatives of a, $, 'Y. From

A - Cc(A - B) + (B-C),C-Ac(C-B) +(B-A)

ADDITIVE ARID TOTALLY ADDITIVE SET FUNCTIONS

32

[CHAP. I

it follows that

O(C - B) + o(B - A),

,?(A - C) 6 O(A - B) + p(B - C), i3(C - A)

and hence by addition we get (2.1). The metric space M just defined will be called the metric space associated with SR by cp. If E M, X E SDZ, and X is a representative of (;, then we say that is the point associated with X by gyp. 4.2.6. Let cp be totally additive and finite in the a --field 0, let A. a SR, A e SDZ, and let M be the metric space associated with SDZ by gyp; then from Lim, A. = A it folm

lows, if A. is a representative of am and A is a representative of a, that in M we have: lien am = a. M

Let Lim A. = A', Lim A. = A"; then A' _, A, A" _, A. We set m

in

SA, = An., DA, = AZ ; then ((A')) is monotone decreasing, ((A")) mono-

.tm tm tone increasing, and, according to §1 (1.32) and 1.1.2, we have: A' = Lim A;., m

A" = Lim A". Hence by 3.4.32, 3.2.21, and 3.2.2: p(A,,) -,p(AJ, m

0, .p(A,n) -+ O(A") ; thus, since A' Q A,' , A" -2 A", we get: O(A;. - A') o(A" - Am) --+ 0. As Am C; A'", A m a A."', it follows that ,p(A m - A') --.* 0, A, we get: qp(A. - A) 0, -p(A" - Am) -- 0; thus, since A' =, A, A"

am = a. -p(A - A.,) - 0. Hence, because of (2), ama --' 0; that is, Tim m

4.2.61. If V is totally additive and finite in the o-field 0, then the metric space M associated with DZ by 'p is complete (§2, 3).

Let ((am)) be a Cauchy sequence (12, 3) of M; we have to prove: there exists

an a e M, such that lim am = a. There is an m; , such that am;am < 1 for 0

mi ; and we can choose the sequence ((m,)) monotone increasing.

m

A. is a representative of am , we have -p(A, - Am;) <

,

Then, if

o(Am, - Am) <

for m 9 m; . We set A = S Am, , A;' = D Am, ; then A; - Amt c its

itti

S (Ami+, - Am;), Am; - A;' Q S

iz{

j i

o(Ami+, - Am;) <

,'(Ami - Am;+,) <

(Amt

Ami+,).

Thus because of

, and 3.2.3, we get: O(A; -

(A mt - A;) < 21

Am) <

We set Um A m; = A', Lim A m, = A"; then, by §1 (1.3) and (1.32), we have AA" A' Q A; , and hence ,p(A' - A") rp(A; - A;') = ,p(A; -Am) + p(Amr A;') < 1: for all i. Thus p(A' - A") = 0, that is, A' =, A"; hence Liin, Am, = A'. We now denote the point of M, a representative of which is A', by a; then, according to 4.2.6, we have 21'

,

1

-

Eal

lieu amt = a; thus because of 2.3.2 lim am = a. i

m

ZERO-SETS AND COMPLETE FIELDS

§41

33

BIBLIOGW'nr to Nos. 1 and 2: The concept " _," is due to M. Fsttcuwr, EneeiWment math. 22 (1922), p. 126; Fund. math. 4 (1923), p. 353; we owe the notation to O. ,NISoDYnc,

Fund. math. 15 (1930), p. 137. The metric space M associated with 91 by v and theorem 4.2.61 are due to O. NIKODYM, C. R. I. Congres math. d. pays slaves (Warsaw 1929), p. 307, and loc. cit.

3. Zero-sets in the wider sense. Again let rp be a set function in fit, A e fit, and The set A is called a zero-set for fp in the wider sense, symbolically written: A (=,) A, if it is a subset of a zero-set for 'p, that is, if there is a set A' a fit, such that A' =, A, A Q A'. As contrasted with the zero-sets for rp, the zerosets for fp in the wider sense do not have to belong to M. If A e fit, then A (_,) A and A =, A are equivalent. From the definition we get immediately: .p(A) = 018.

4.3.1. If A(=,)A and B C A, then also B(=,)A. 4.3.11. If v is additive in the field fit (or totally additive in the a -field 931), then the system of all zero-sets for p in the wider sense is a field (or a a -field).

This results immediately from 4.1.31 and 4.1.32.

If A - B (=,) A, then we write: A (a,) B. If A e fit, B e 0, A - B e fit, then A(Q,)B and A C, B are equivalent. 4.3.2. If 0 is additive in the field fit, then from A(c,)B, B(C,)C it follows that also A(C,)C. The proof is analogous to that of 4.2.31. 4.3.21. If (p is additive in the field fit, then from AI(Q,)BI , A2(c,)B2 it follows

that also AI + A2(C:,)BI + B1, A1A2(c,)B1B., AI - B2(c,)B1 - A2. The proof is analogous to that of 4.2.32. 4.3.22. If Sp is totally additive in the a -field 171, then from Am(C;,)B, it follows that also SAm(c,)SBm , DAm(C,)DBm . ff1

.1

in

m

The proof is analogous to that of 4.2.33.

If A(C,)B and B(C,)A, then we write: A(=,)B. Thus for B = A the symbol A(=,) A means: A is a zero-set for 1p in the wider sense, consistent with our notation introduced above. If A e fit, B e 5102, A - B e fit, B - A e fit, then A(= )B and A =, B are equivalent. If So is additive in the field fit, then the

relation (=,) is an equality relation; in particular, because of 4.3.2, we have: 4.3.3. If go is additive in the field 91, then from A(-,)B, B(=,)C it follows that also A(=,)C. From 4.3.21, 4.3.22 we obtain: 4.3.31. If 9 is additive in the field 971, then from A1(=,)BI , A2(=,)B2 it follows

that also Al + A2(=,)B1 + B2, A1A,(=,)B1B2 , Al - A,(=.)B1 - B2. 4.3.32. If 9 is totally additive in the a -field fit, then from A.(=,)Bm it follows that alto SAm(=,)SBm , DAm(-,)DB . 4.3.33. If V is additive in the field fit, then the system of all the sets A for which there is a B e l't with A (=,)B is afield V° -,2 U. 11 Cf. footnote 16, p. 27.

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

34

[CHAP. I

If A e 9N, then A (_ c)A ; thus also A e 972°, and hence 991°Q M. Let AI E 971°, A2(=,,)B2 . A2 a 91f°; then there is a B1 a 9)I and a B2 a 971, such that

Then because of 4.3.31 we have Al + A2(=.)B1 + B2 , Al - A2(=.)B2 - B2 ; since B1 + B2 a 992, B1 - B2 a 9)?, we have A, + A2 e 9)2°, Al - As a 92°; thus M° is a field. 4.3.331. If (p is totally additive in the afield )I, then the system of all the sets A

for which there is aBe9)?with A(=0)Bis aafield971°Q911. According to 4.3.33, 971° is a field Q 99I. Now let A. a 9)2° (m = 1, 2,

);

then there is a B. a SDI, such that Am(= o)B,,, . Thus by 4.3.32 we have SBm and, since SBm a 97I, we have SA m e 97I°; therefore 9)2° is a o-field. m

m

m

4. Complete fields. Let cp again be a set function in 971, At 97I, and 'p(A) = Then the system 971 is called complete for rp if )I contains all zero-sets for

0.19

,p in the wider sense, that is, if from A (_,,) A it follows that A e V. If fit is complete for p, then A (= m) A and A = A are equivalent; or in other words: then every zero-set for Sp in the wider sense is also a zero-set for (p. Now we assume p to be additive in the non-empty field 972. Let 971° again be

the field of 4.3.33; then we have: 4.4.1. If rp is additive in the field 9)2, then in order that 971 be complete for (p, it is necessary and sufficient that 9)2° = V. NECESSITY: Let At SDI°; then there is a B e 971, such that A (_ ,) B. So we have

A-

A

B-

A

= (B + (A - B)) - (B - A), then also A e SD2, and thus 9n° c 9; and,

since according to 4.3.33 971°Q 97I, we get 971° = TI. SUFFICIENCY: Let 9)2° = SDI.

If A(=,) A, then, since A e 9)1, we have A e 972°, thus A e )I; that is, SD2 is complete for gyp.

We now shall prove that the set function p additive (or totally additive) in the field (or the a-field) 9 ? can be extended to a set function additive (or totally additive) in the field (or the a-field) 992° of 4.3.33 (or 4.3.331), such that 9)1° is complete for . 4.4.2. If A e 9)I°, B e 97I, B' e 971, and then p(B) = sp(B'). According to 4.3.3 we have B(=,,)B'; thus, since B e 97I and B' e 992, we have

B = B' also, and hence by 4.2.44: So(B) = p(B7. Therefore we can define a function rp° in 97I° by: (4) p°(A) = p(B) if A e 971°, Be 992, A(= )B. If A e971, then, since it follows from (4) that So (A) = tp(A); thus 9)lLp° = rp, that is: is an extension of rp on 9)1°. 4.4.21. If rp is additive in the field TI, then is additive in 97I°. 4.4.211. If rp is totally additive in the a -field 9)2, then P is totally additive in 812°.

The proofs for both these theorems are quite analogous. First we prove 4.4.211.

Let ((Am)) be a sequence of disjoint sets of 991°; there is a B. t 9)?, such

19 Cf. footnote 16, p. 27.

§41

ZERO-SETS AND COMPLETE FIELDS

35

that A,.(=,)Bm . Because of 4.3.32 we have SAm(=,)SBm, and thus accord(A m) - p(B,.), o°(SA.) _ ,p(SBm). By 4.3.31 we have BmBm, (_,) m M A,.A.,; thus BmB.,,(=,)A form m' and, since B,B,, a 91, also BmBm, =, A; hence according to 4.2.22: so(SBm) = F, V(B,.). Therefore we get: ye°(,SAm) = ing to (4) :

,p(SB.) = Lr 5'(Bm) = m4 !Am). w w Then, to prove 4.4.21, one only has to consider finitely many disjoint sets A..(m = 1, 2, - , mo) of l72° instead of the sequence ((Am)), and to apply the theorems 4.3.31 and 4.2.211 instead of the theorems 4.3.32 and 4.2.22. 4.4.22. In order that the set A e 932° be a zero-set for p, it is necessary and suffi-

cient that A(=,) A. Necessrrr: Let A =,. A; since A e 9)2°, there is a B e 2, such that .A (==,)B. Then we have A - B(_,) A; hence there is a C e 9J2, such that A - B C; C and

C =, A, and thus also C(--,) A. Setting B + C = A', we have A' e 91 and A C; A'; thus it is enough to prove that A' =, A. Because of 4.3.31, from

B(=,)A and C(=,)A it follows that B + C = A'(=,)A. Now let X e 0 and X C A'; because of 4.3.31 from A'(=,)A and X(=,)X it follows that A'X = X (=,)AX ; thus according to (4) : cp(X) = sa°(AX) ; since AX a D2°, it follows from

A =, A that p°(AX) = 0; thus p(X) = 0 for X A A'19, that is, A' =, A. SUFFICIENOY : Let A (=,) A; because of 4.3.1, for X C A we have X(=,) A, and

thus according to (4) : sp (X) = (s(A) = 0; hence A =,. A. 4.4.23. 9)2° is complete for p*.

We have to prove: if A =,. A and B C A, then B e W. According to 4.4.22, we have A(=,) A; thus because of 4.3.1 also B(=,) A; hence, since A e ?, also

BeW.

By 4.3.33, 4.4.21, and 4.4.23 the following theorem is proved: 4.4.24. If (p is additive in the field 911, then there is a field an' 2 0 and an additive set function 9' in W, such that (p = 9Y2lso and 9)2' is complete for ra'.

Analogously the following theorem is proved by 4.3.331, 4.4.211, and 4.4.23: 4.4.241. If (p is totally additive in the or-field 9)1, then Acre is a o-field fit' Z 1112 and a totally additive set function ,p' in 9)?', such that 0 - T?l9o' and 9)1' is complete for so

Because of both these theorems, for any additive (or totally additive) met function v in a field (or in a o-field) )2 one can always at once assume that if is complete for (p.

Restricting ourselves now to monotone set, functions, we still prove that the field 9W° of 4.3.33 (or 4.3.331) is the smallest field X02' on which ss can be extended to an additive set function so', such that X12' is complete for gyp', and that this ex-

tension of p on SJ2° is possible only in one manner. Thus let S)2° be the field of 4.3.33 and rp° the set function defined by (4). Then we obtain: 4.4.25. Let fit be a field and W a field fit; let rp and ,p be additive and monotone increasing in 971 and 9)2', respectively; let 9)2' be complete for gyp', and 9)21,' = p;

then we have I' Q 92° and 9)2°is = so

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

36

CHAP. I

X (=,) A implies X (_,') A; for, since X (=,) A, there is a Y e X12, such that ,V C; Y, ,p(Y) = 0; then also (Y) = 0; thus according to 4.1.13 Y =,- A, and hence X(=,')A. Now let B e Jl2°; then there is a COW, such that B - C(_,)A,

C - B(=,) A; thus also B complete for

,

C - B (=,') A and, since 971'

is

B - C e 1712', C - BOW. As T?' ? 912, we have C e 971'; thus from

B = (C + (B -- C)) - (C - B) it follows, since ID?' is a field, that B e W. Hence 1))2°.-Now again let we have proved: if B c W, then also B e 971', that is: 9)2' B e 12°; then there is a C e $t, such that B(=9,,)C; then also B(=,-)C and, since Be 9)2', Cc W, also B =,' C. Therefore, according to 4.2.44, v(B) _ -P (C) ; thus, since I21p = (o, ,p'(B) _ 'o(C) and since by (4) also (B) = cp(C), we get (B) = (B) for all B e X92°; that is, I02°lo = (p'. BIBLIooatPHY to Nos. 3 and 4: M. FPACSET, Bull. Soc. math. de France 43 (1915), p. 262;

H. HAm., Annali Scuola norm. di Pisa (2) 2 (1933), p. 429; B. JEasEN, Mat. Tidaskrift 11 1934, p. 82.

§5.

Regular and singular parts of totally additive set functions

1. Singular sets. Let 9)1 be a non-empty a-field and let 5 be a non-empty a-field it 02 with the following property: if S e 5, then 51971 C e.20 The sets belonging to 5 shall be called singular sets. If a subset of a singular set belongs to 9)2, then it is also singular; in particular, A e 5. Let (p be totally additive in IA2. The set R e 9)2 is called regular (for (p) if cp(X) = 0 (or, what amounts to the same thing: X -, A) for all X c R16. We designate the system of all sets regular for

If R, e 9?, R2 c 9?, then, since R, - R2 e R,1971, also R, - R2 a 9?. Thus we have only to prove: if R,,, e 9t (m = 1, 2, . ) and R = SR,., then also R e 9t. m

Let X e R1:5; then X = SXRm; since XRm a 915, we have XR, =, 4; thus according to 4.1.32 X =, A also, and hence R e T. Since 92 and 5 are a-fields, M19? and MIS are also (non-empty) Because of 3.3.3, in the set of all values of rp in M19? or M1C there is a greatest one and a smallest one. Thus we can define four set functions in 1)'2 Let M e 972.

or-fields. by21:

20 As to the notation SI 9 , cf. §3, 3. " If there is no question of ambiguity, the index v can be dropped in the following formulas.

37

REGULAR AND SINGULAR PARTS

§51

*(111) = max
min'p(X) (X e M19Z(9))

a**(M) = max cp(X) (X E MIS), ,***(M)

min io(X) (X eMIS).

(1)

Because of §3 (4.11) and 3.4.11, these definitions are equivalent to: «mt,111)

= max p+(X) (X e M19t(',)), ,e*(M) = Max

(X) (X e

ap*(M) = max N+(X) (X E MIS), S**(M) = max So (X) (Xe MICA). As an immediate consequence of the definitions we obtain: 5.1.2. The set functions a*, 0*, a**, 3** are monotone increasing. 5.1.21. If v is monotone increasing (or decreasing), then 3* = i3** = 0 (or a* _

a** = 0). This follows from (1.1) and 3.4.4. 5.1.3. In order that .&f e 9t, it is necessary and sufficient that a**(M) = S**(M)

=0. NECESSITY: From M e 92, X e M1( it follows that co(X) = 0. SUFFICIENCY: From a**(M) = p**(M) = 0 it follows for all X e M1S according to (1.1) that 'p +(X) = 0, 07(X) = 0; thus, in accordance with 4.1.11, X A; hence M e T. 5.1.31. If M e Cam, then a*(M) = ,B*(M) = 0. For from M e Cam, X e MIN it follows that X =* A, and thus'p(X) = 0.

5.1.32. In order that M e 9t, it is necessary and, if 'p(M) is finite, also sufficient

that a*(M) = w+(M), ,6*(M) =

(IYI).

NECESSITY: If M e 9?, then X e i1119t is equivalent to X e MI D1; thus the proposi-

tion follows from (1 and §3(4). SUFFICIENCY: According to (1.1), there is an M* e M19t, such that'p+(M*) = a*0M) = P+(M). Let X e M15; then'p+(X) = So+(XM*) + rp+(X - M*). From'p+(M*) = p+(M) it follows that 'p+(M -M*) = 0, and thus also w+(X - M*) = 0. Since XM* e T e, we have XM* =0 A,

and so p+(XM*) = 0. Therefore we have v+(X) = 0 for all X E M1C. In a similar manner one proves that'- (X) = 0 for all X e M1(. Thus we have

X =,.AforallXeM1S;that is:Me9t.

$**()lf) = ,p`(1l1). 5.1.33. If M e Cs, then a**(M) = This follows from (1) and §3(4), since, due to the fact that M e Cam, both the systems M1( and M1V't coincide. 5.1.34. For every M e ffl there are two disjoint sets M* e MIN, M** e MI S, such that:

a*(M) = p+(M*), ,6*(M) = io^(M*), a**(M) =

O"(M) - So (M**).

According to (1.1), there is an M' a M1e and an M" e 11110, such that a**(M) =

,o+(M'), 6**(M) = 'p (111"). We set: M** = M' + M"; then M** e M1( also "This condition to essential, Example: Let fit be the system of all subsets of an infinite set M and Iet,,(X) be the number of elements of X. Furthermore, by a, designate a fixed element of M, and let 6 consist of A and {ail, Then we have a*(M) m w+(M) + -, tZ*(M) - (M) - 0; but'it is not true that M e 8t, because e(la,)) - 1.

ADDITIVE AND TOTALLY ADDITIVE BET FUNCTIONS

38

and ,p+(M**) z p+( if') = a**0M),

[CHAP. I

z P -(M") = $**(M). Thus

because of (1.1): a**(M) = So+(.11**), 13**(111) = lp (M**). Similarly one gets (M). Since MM** e 92 an M E MIT, such that a*(31) _ P+(M), 0*(M)

we have MM** = A. Thus setting M* = M - M** we have M*

hence, according to 4.2.44, a*(M) =

4(31*),

+{lb1) =

M, and

#*(M) _ V (M)

(P (M*), and M*, M** are disjoint. 5.1.3 5. The set functions a*, 8*, a**, 13** are totally additive in 91t.

Let ((Mk)) be a sequence of disjoint sets of TJ and let

We prove this for a*.

M = SAlk ; we have to prove: a*(M) = Ea*(Mk). According to (1.1), there is an Mk E Mk19t, such that a*(Mk) = V+(Mk). We set S11fk= M*; then, since k

the Mk are disjoint, by 3.4.32 we get: g+(M*) _ F_w+(A1k) k Thus, since by 5.1.11 M* a M192, we obtain from (1.1): a*(M)

Ea*(A11k). k

Ea*(Mk). k Therefore we still have to prove: a*(M) S Ea*(Mk). Let X e M192; then x = k

SXMk, where XMk e MOT; thus according to (1.1) we have p}(XMk)

a*(Mk).

k

Hence because of 3.4.32: rp+(X) = Ep+(Xdfk) k

we get also a*(M)

Fla*(Mk); and so by (1.1), k

Ea*(,11k). k

Let Al a M; since according to (1.1) both the numbers a*(M) and a**(M) are

values of p+, both the numbers 0*(M) and 6**(M) are values of 07. Thus according to 3.4.12 either both the numbers a*(M), a**(M) or both the numbers $*(%f), ,6**(Ilf) are finite for all ill a alt. (1.2)

p*(Alf) = a*(11f) - (i*(Af),

Therefore we can define:

0**(31) = a**(M) --

O"(M).

5.1.4. The set functions 0*, p** are totally additive in fit, and there are.no two sets A e fit, B e fit for which p*(A), rp**(B) are infinite of different sign. This follows from the definition, 5.1.35, and 3.2.52. 5.1.41. For every M e 92(jo) we have rp*(M) = (p(M), ***(M) = 0; for every M e C5 we have c*(M) = 0, cp**(3f) _ 'P(M). This follows from 5.1.3, 5.1.32, 3.4.61 and from 5.1.31, 5.1.33., 3.4.61. 5.1.411. fR(?) = fit, 92 (ip**) = 92 (,p) . This follows from 5.1.41. 5.1.42. For every M e fit there are two disjoint sets M* e M191(, ), M** a M1r,

(Al*), such that 0*(M) = pw**(Df) _ p(M**). This follows from 5.1.34, because of (1.2) and 3.4.61.

5.1.43. If rp is monotone increasing (or decreasing), then 0* = a*, rp** = a**

(or P* = _#*, IP** = -0**). This follows from 5.1.21.

Denote the positive-, negative-, and absolute-function of ip again by P+, , -p; then according to 5.1.43 we get:

39

REGULAR AND SINGULAR PARTS

§51

a.+;

(So+)* = a.+,

(1.3)

('p )* = aw (o)*

(g)** = aa

We designate the positive-, negative-, and absolute-function of (P*, 4** by

*- * **+

*+ 5.1.44. cc

**-" ** then we obtain:

= ('P_)* = 49'. = (,p+) * = a,, , Let M e t and let M = M + M- be the decomposition of 3.4.71. Because of 5.1.35 we have a* (M) = a* (M+) + a* (M)), a"+(M) = aw+(M+) + a,p+(M-) Here, according to (1.1) and (1), we have aw*(M-) = 0, a,,+(111-) = 0; thus o*(M) = a,*,(M+), a*+(M) = a*+(M+). Since (M4') = 0, we have cc(X) _ (P+(X) for all X e M+1 TZ; hence M+1 9?(,p) = M+1 96pi). Thus by (1.1) a*+(M+); therefore a*+(M) and, according to and (1): (1.3), (rp+)* - a,*..-Because of (1.2) and 3.4.62, we have o*+(M) .5 a,,(M); thus we only have to prove that fp*+(M) z a,(M) also. Sinceao(M) = a.(M' ) and i4;(M+) = 0, we have v*(M+) = ao(M) = a*(M); thus indeed c*+(M) (P*(M+)

9P

= cr (M).

5.1.441. **+ _

a,*.* so**- _ (,)** _ v*.

From (1) and (1.1) it follows immediately that a,*P* = am+. Thus, according

to (1.3), (Sc+)** = ao. Then the proof continues in the same manner as that of 5.1.44. 5.1.45. ;P*= W* _ (,v+)* + (.P-)*; c** _ (,D)** _ (m+)** + (w-)**. Let M e 1 and let M = M+ + M- be the decomposition of 3.4.71. According to 5.1.35 we have a*(M) = af(M+) + a'(M-). For all X e M+1D2 we have O(X) _ W '(X), for all X e M-193 we have o(X) = rp (X) ; thus, since as a result of 5.1.1 9t(#) = 9t(c), by (1) and (1.1) we have a*o(M+) = a*(M+), ai(M--) _

8;(M-). Therefore $(M) = a.(M+) + (3 (M_); since a,(M) = ao(M+) + and 0, i9 (M+) = 0, there follows that a*(M) = ao(M) + 3,*,(M); hence, according to (1.3) and 5.1.44: ()* = (v+)* + W-)* = (so*)+ + (cc*)- _ * Let M e fil; a representation M = R + 8, where R e 9?(,p), Se 25, RS = A,

«:(M-), 0:(M) _ O.(M) +

shall be called a singular decomposition of At (for ip) u

S.I.S. If M = R + S (R a 92, S e 5) is a singular decomposition of M and if

R'eM19t(orS'eMI ), then R' For since R' - R Q S, we have R' - R e TS, and thus R' - R =, A.

5.1.51. If M = R + S (R e 9i,SsS)andM=R'+8'(R's92,S'e6)

are singular decompositions of M, then R = o R', S =, 8'.

For according to 5.1.5 we have R'cwR,RcwH';S'C;w8,ScoS'. 5.1.52. If At = R + S (R e 92, S e G) is a singular decomposition of At, then in 5.1.34 we can set M* = R, M** = S. Because of 5.1.5 we have rp+(X) S yo+(R), co'(X) S tp (R) for X e MIT; "Such a decomposition need not exist. Example: Let 9 l consist of all subsets of a noncountable set E and let 4'(X) be the number of elements of X e 9)t (thus - + oo if X is infinite); let f$ be the o-field of the countable subsets of E. Then A is the only set of R(P). Thus there is no singular decomposition of E (for p).

40

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

[CHAP. I

thus by (1.1) : a*(M) = gs+(R), 8*(M) = w (R). And similarly we get a**(Al) = lp+(S), #**(M) = (S)

5.1.53. If there is a set S e M1S, such that S Q,, S for all S e HIS, then

M = (M - S) + S is a singular decomposition of M. We have only to prove: M - S e R. Let X e (M - S)1S; since according to the assumption X - S =, A, we have X =, A; thus M -- S e 9Z. 5.1.54. If o** is ftnite2`, then there is an S** e e, such that M = (M - S**) + MS** is a singular decomposition for every M e V.

Because of 5.1.4, 3.4.32, and 3.3.1, among all the values y,**(X) (X e S) there is a greatest one, say ,p*s(S**).

For X e S we have X - S** =, A ;for other-

wise, according to 4.1.1, we would have O(X - S**) > 0, thus o(S** + I) = g,(S**) + {p(X - S**) > o(S**), contradictory to the choice of S**, because of 5.1.41. Therefore we have X a, S** for all X e C5; thus also S . r-, S**M for every M e Pt and for every S e M1(. Hence the proposition follows from 5.1.53. 5.1.541. If M e SD2 and cp**(M) is finiteU, then there is a singular decomposition of M.

We obtain this from 3.1.21, by applying 5.1.54 to the function p restricted to M1SDl.

5.1.55. If M = R + S (R a 92, S e C) is a singular decomposition of M, then .F*(M) = 'P(R), w**(M) = v(s) For, because of 5.1.52, (1.2), and 3.4.61, we have rp*(M) = +(R) - jp (R) _ p(R).

5.1.56. 'P = c* + Let M e. If p**(M) is finite, then, according to 5.1.541, there is & singular decomposition M = R + S. and, by 5.1.55, p(R) _ pp*(M), p(S) = ip**(M); w**.

thus p(M) = p(R) + sp(S) = p*(M) + cp**(M). If now p**(M) is infinite, then, according to (1.2) and (1.1), also either a**(M) = + co or 6**(M) = + oo ; say a**(M) _ + co . Then, by (1), there is an X e MISS with p(X) + co; then, because of 3.1.2, we also have p(M) = + 00 . Furthermore, according to (1.2), 0**(M) = + ao ; thus, by 5.1.4, p*(M) 0 - ao . Hence tp(M)

V*(M) + P**(M) = + 00. + **+ 5.1.57.

*_

*+

**_

_

.

jp+

As a result of 5.1.56 we have = (,p+)* + (gyp+)**. Thus the proposition follows from 5.1.44 and 5.1.441 (or 5.1.45). 2. Regular and singular set functions. Let ,p be a totally additive set function in the a-field T?. We call `o regular (with respect to S [cf. No. 1]) if from X e C it follows that cp(X) = 0. Since X e e implies that X101 C S, we can here

substitute X -, A for p(X) = 0. We call p singular (with respect to e) if from X e 92(,p) [cf. No. 11 it follows that p(X) = 0, or, which amounts to the same thing, that X =, A. In other words: rp is called singular if for every set *' That thin condition is essential, is shown by the example of the preceding footnote.

REGULAR AND SINGULAR PARTS

§5]

41

M e 1D2 with cp(M) r 0 (or with M 5,4-. A) there is a set S e MF with P(S) 0 0 (or with S , A). If p is regular (or singular), then crp (c i' 0) is also. ,p is regular in SR, if and only if R(.p) - D't.

Let A e 0l; we set A lit = W. Then A is a regular set if and only if the restricted function sf1(p (§3, 3) is regular. If A is a singular set, then 2(lip is singularxe 5.2.1. (p* is regular, 0** is singular. This follows immediately from 5.1.4 and 5.1.41. 5.2.11.

*+, p*

40* are regular; p**+, p**-, 4P** are singular.

Because of 5.1.44, 5.1.441, and 5.1.45, this follows from 5.2.1, applied to 5.2.12. In order that to be regular (or singular), it is necessary and sufficient that (P** = 0 (or p* = 0).

Nssrry: If v is regular, then p(X) = 0 for all X e.25; thus according to

(Y}1' a**(M) = 0, l3**(M) = 0 for all M e V. Hence, by (1.2), 4P **(M) = 0 SUFFICIENCY: This follows from 5.1.56 and 5.2.1. 52.121. In order that 'p be regular (or aingular).tt is necessary and sufficient that

for all M e 971.

= e (or p =

0**).

NECESSITY: This follows from 5.1.56 and 5.2.12. SUPFICIIdNCY: This follows from 5.2.1. 52.13. If p and 4' are totally additive in the a -field D2, if rq, and if 1n is regular, then 4' is also regular.

Because of 4.1.1, since X =, A for all X e (, it follows that also X =,y A for all X e S. 5.2.131. If p and 4, are totally additive in the afield D2, if gyp, if p** is finite" and cp is singular, then 4' is-also singular. .et M e 9Z(4,); we have to drove: 4'(M) = 0. According to 5.1.541 we have M = M* + M** with M* a 9?(4p), M** e G, M*M** = A; then 0,(M) _ 4(M*) + 4'(M**). Since so is singular, we have M* =, A, thus also M* =,y A, and hence 4,(M*) 4G(M**)

0. Since M e 82(44) and M** a M1c, we have M** =,u A, hence 0. Therefore 4'(M) = 0 also.

5.2.14. If p = v, + cps (or = ip, - w,), where vn and tp:.are regular in the afield !R, then cp is also regular.

For every X e " we have 0,(X) _ (X) = 0, and thus also P(X) = 0. 5.2.141. If 'p = vn + Vp2 (or = vn - ), where V, and GPs are singular in the afield 9.Dt, then 'p is also singular.

Let M e 82(v) ; we have to prove: rp(M) = 0. First we prove that 4 *(M) and 'p; *(M) are finite. If, for instance, 1**(M) were infinite, then, according ° That the converse is not generally true, is shown by the example of footnote 23, p.

39, for A - E. "This condition is essential: Let S be the system of all countable sets of Dl and let

w(A) - 0, io(X) - + eo for all the other sets X e Dl; then T(v) - IA),.p is singular, and for every set function sk totally additive in 9 we have q,.

42

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

CHAP. I

to (1.2) and (1.1), we would have either a, * (M) = + ao or 9,.,* (M) _ +00 and, by (1), there would be an X e 11110 with g,(X) = t co ; but then we would have also p(X) = f co, contradictory to The assumption M e 9('P).-According to 5.1.541, we have M = M* + M** with M* s (go ), M** e Cam, M*M** = A,

andM=M'+M" with M' e ¶R(), M"e 25,M'M"=A. Then M=M*M'+ M*M' = M** + M", (M - M*M'), where M*M' e 91( M thus M - M*M' e S. Since Cpl and ep2 are singular, it follows from M*M' e that rpl(M*M') = 0, go2(M*M') = 0, hence also p(M*M') = 0. Since M e 91(,p), it follows from M - M*M' e C that ep(M - M*M') = 0. Thus we have g(M)) = ep(111*M') + go(M - M*M') = 0. 5.2.15: In order that p be regular (or singular), it is necessary and sufficient that cp+ and go be regular (or singular). NECESSITY: If gp is regular, then because of 5.2.121 So = ep*, thus ep+ and the contention follows from 5.2.11. SUFFICIENCY: Because of 5.2.14 and 5.2.141, this follows from 3.4.61.

5.2.16. In order that ep be regular (or singular), it is necessary and. sufficient that,p be regular (or singular). This follows immediately from the definitions of the regular or singular set

functions, since, according to 4.1.12, X =,, A is equivalent to X =i A and, by 5.1.1, 9'i(g) = T(ep). 5.2.17. Every regular (or singular) set function is the difference between two monotone increasing regular (or singular) set functions. If 0 is regular (or singular), then because of 5.2.15 ep+ and go are also.

Thus

the proposition follows from 3.4.61 and 3.4.3. 5.2.2. Every totally additive set function in the a -field 91 is the sum of a regular and a singular set function. This follows from 5.1.56 and 5.2.1. 5.2.21. If gp = 91 + pi, where pi and g2 are totally additive in the a -field 9)"t and Sol is regular, then gp**+

, 40**

5_ P2

, **

s 402

.

Let Al e D2; according to 5.1.441, we have go**+(M) = a**(M); thus, by (1.1), there is an S e MIS, such that go**+(M) = sp+(S). Since gyp, is regular, we have 4pl(X) = 0 for all X e 519)1; thus Sp+(S) = cps (S), and hence {p**+(M) = w2 (S) 5 ,p2+(M). In the same manner we get:,p**- S gZ . Because of §3 (4.2),

it follows that

S 92-

5.2.22. If the set function ep is totally additive and27 finite in the a-field x!11, then g = p* + rp** is the only decomposition of ep into a regular and a singular term.

Let go = pi + V.-, where go' is regular and go. is singular. Since 0 is finite,

we have go = -got + go; thus, because of 5.1.56: o2 = (-gpl + p)* + (-,pl + gyp) **. Hence, asp, is regular, by (1), (1.2) : gPs = (-,pl + ep)* + ep**. Thus, by 5.2.21, g, *+ < tP**+. According to 5.2.121, we have v2 = go *, and thus

37 That this condition is essential, is shown by the example of footnote, 23, p. 39. There, according to (1), we have for every M e TZ: a* - 0* - d** = 0, a** - gyp; thus by (1.2): V* - 0, (p** = v. Now we get a decomposition of (pinto a regular and a singular term, V = ,y, + ,P**, which is different from _ .e* + ip** and in which c, (M) - 0 for every countable set M and ec, (M) = + - for every non-countable set M of R.

REGULAR AND SINGULAR PARTS

151

43

*+; therefore we have also + ; So**+. On the other hand, because of 5.2.21, we have v**+ ;9 (Ps+; thus v**+ = yea+. Similarly one proves that (p**- = Then from ip = p* + 0** _ ws . Hence, because of 3.4.61, we have vz = gyp**. since jp is finite, that 0, = w* also. V, + is it follows, Because of 5.2.22 we call p* and rp** the regular and singular part of 0, respectively. -I- V: , where (A and .pa are totally additive in 5.2.23. If rp is finite and P = the a -field 9)2, then (p* = rpi + (Ps , 0p** = Bpi + gyp! . are also finite. Because of 5.1.56 we have Since p is finite, cp, and ,p; = rp,* -}- ,p* *(i = 1, 2) ; thus gyp?, rpr* (i = 1, 2) are also finite and we have rp =

(4 -f- 92*) + (V * + qft *). According to 5.2.1, 5.2.14, and 5.2,141, Vi + Sc * is singular. Thus the proposition follows from 5.2.22. +02** is regular and 3. Continuous and purely discontinuous set functions. Let 914 be a e-field: then we can choose the system of all the countable sets of 9)2 as the a-field (e of the singular sets (No. 1). If ,p is totally additive in 9)2, then the regular sets R e 9)2 for rp are characterized by the following property: for every countable set X e RITZ we have (p(X) = 0 (or, what amounts to the same thing: X =, A).

in this way, we call the regular (or singular) set functions Choosing continuous (or purely discontinuous). Thus a totally additive set function qp in the a-field 9f1 is called continuous if for every countable set X e 9>2 we have ,p(X) = 0 (or X =, A); it is called purely discontinuous if for every set M e 9)2 with cp(M) $ 0 (or with M 0, A) there is a countable set X e M 1912 with 9p(X) 0 0

(or with X 0, A). 5.3.1. Every continuous (or purely discontinuous) set function is the difference between two monotone increasing continuous (or purely discontinuous) set functions. This follows from 5.2.17.

Let the system a of the singular sets again signify the system of the countable sets of 972 and, according to 5.1.56, let us decompose (p into jo = go* + ce**; then we call go* the continuity-unction, ,** the discontinuity-function of jo. According to 5.2.1, the continuity-function is continuous and the discontinuityfunction is purely discontinuous. 5.3.2. Every set function totally additive in the a-field 9? can be represented as the sum of a continuous and a purely discontinuous set function. If gp is f niO, then there is only one such representation. This follows from 5.2.2 and 5.2.22.

Let E be a fixed set, let 9.12 be a a-field consisting of subsets of E, and let go be totally additive in 972. An element a e E is called a discontinuity-element of rp if { a } e 912 and rp({ a }) $ 0 (or, which means the same: if { a) , E, A). Every 28 The example of the last footnote shows that this condition is essential. There, since + .p**, we get

`p, is regular, by 5.2.121 we have Bpi o wj; therefore, although we have So s so* s 0 $ Bpi + (+P**)* s soi -i-,o* - to,.

29 That this condition is essential, is shown by footnote 2?, p. 42.

44

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

(CHAP. I

element a e E which is not a discontinuity-element of gyp, is called a continuityIf a is a discontinuity-element of 'p, then { al e 5, where again C is the system of all countable sets of W. 5.3.3. In order that a be a discontinuity-element (or a continuity-element) of re, ip is necessary and sufficient that a be a discontinuity-element (or a continuityelement of (p.

element) of gyp.

For ip(1a}) 5 0 is equivalent to rp({ a }) P6 0. 5.3.31. In order that a be a discontinuity-element (or a continuity-element) of (p, it is necessary and sufficient that a be a discontinuity-element of #P+ or of V (or a continuity-element of rp+ and of rp`).

This follows from 5.3.3, since 0({a}) 0 0 is equivalent to the validity of one of the relations: (p+({a}) 0 0, fp-({a}) 0 0. 5.3.4. If rp** is fnite80, then the set of all discontinuity-elements of 4, is countable.

Let a be a discontinuity-element of cp; then (a({ a }) 0 0; thus, according to 5.1.41, also p**({a!) 0 0. Hence, because of 5.1.4, the proposition follows from 3.4.81 if there one substitutes p** for gyp. 5.3.41. If M e V? and rp**(M) is finiye80, then the set of all the discontinuityelnrtents of p which belong to Al is countable. Because of 3.1.21, we obtain this by applying 5.3.4 to the function rp restricted

to Mlt.

5.3.5. For every a e E suppose that {a} e 9)?; then in order that M E 9)? be regular, it is necessary and sufficient that M contain no discontinuity-element. NECESSITY: Let M be regular; then for every S e M1( we have 4*(S) = 0; thus also for every a e M we have ,p({ a }) = 0. SUFFICIENCY: For every a s M

we have y,({ a }) = 0; then for every S e M1(, since S is countable and 0 is totally additive, (p(S) = 0. Similarly we obtain: 5.3.51. For every a e E suppose that (a} a 9)?; then in order that rp be continuous, it is necessary and sufficient that every a e B be a continuity-element of (p.

5.3.52. For every a e E suppose that {a} a M; then in older that v be purely discontinuous, it is necessary and sufficient that every set M e A2 with 4(M) 0 0 contain at least one discontinuity-element. NECESSITY: Let V be purely discontinuous; then every M e 9)l with 4o(M) , 0

contains an S e Mie with c(S) 0 0. Let S consist of the countably many elements a.,(m = 1, 2, ) ; since c(S) = (p({ a1 }) + gyp({ a2 }) + .. 96 0,

there is at least one p({ a., : 0. SUFFICIENCY: This is obvious. 5.3.53. For every a e E suppose that { a) a T?; if jp** is finite81 and S** signifies the set of all discontinuity-elements of so, then M = (M - S**) + MS** is a singular decomposition for every M e W.

According to 5.3.4, we have S** e 6; according to 53.5, M - S** Now, in particular, let E be a metric space (*2, 3). Instead of discontinuityor continuity-element we then say discontinuity-point or continuity-point. 10 That this condition is essential, is shown by the example of footnote 23, p. 39. " That this condition is essential, is shown by the example of footnote 23, p. 39.

REGULAR AND SINGULAR PARTS

§51

45

5.3.6. In order that the point a e E be a continuity-point of p, it is sufficient and, if n is finiteE2, also necessary that for every p > 0 there be a o > 0, such that (§2, 3), we have I p(X) I < p. f o r every a d X e X 1 2 which is Q Surs'ICIENcY: If (a) ... a V, then certainly a is a continuity-point. Thus let

(a) t 9; then, according to the assumption, we have I p((a}) I < p for every

p > 0, that is, p({a}) = 0. NECESSITY: Let a be a continuity-point of gyp. If the condition were not satisfied, then there would be a p > 0 and for every m a set X. a 9)2 which is Salim, such that I gp(Xm) I z p. Now if {a} ti e 02, then Lim X. = A; for otherwise we would have Lun X. = (a I. Thus, because m

m

of §1 (1.32) and 1.2.4, {a} e 912, contradictory to the assumption. But since Lim X. = A, it follows, because of 3.5.2, that lim v(Xm) = 0, contradictory M

"S

If, however, {a} e W2, then setting X*, = Xm + {a}, we also have X.' a 9)2; and, since a is a continuity-point,,p({ a}) = 0 and we get p(X4,) to I V(Xm) I == p.

,p(Xm), so that I y (X;) I z p. But this is impossible, since now Lim X,;, = (a) and because of 3.5.2 lim cp(X,) = p({a}) = 0. IZI

Always substituting (p for rp in this proof, we obtain:

5.3.61. In the theorem 5.3.6, we can substitute O(X) < p for the condition I ,c(X) I < p. BIBLIOGRAPHY. The notion of the continuous set functions is due to J. RADON, Sitzungs-

bericbte Akad. Wise. Wien 122 (1913), p. 1320, as well as theorem 5.8.6, for which another proof was given by W. BIBaPIfvsal, Ann. Soc. Polon. Math. 7 (1928), p. 76; cf. also E. SzPILRAJN, Fund. math. 22 (1934), p. 309. Besides, as to No. 3, we mention especially H. HAHN (1], p. 408; also R. S. PHILLIPS, Bull. Amer. Math. Soc. 46 (1940), p. 274; A. SOBCZYK and P. C. HAi[xsR, Duke Math. Journ. 11 (1944), p. 839.

4. Atoms. Again let 9)2 be a o-field and let p be totally additive in W't. The set A e Wt is called a (p-atom if for all X e A IV either X = A or X =, A.

Every zero-set for v is also a c-atom; every set { a } a W2 is a v-atom. 5.4.1. If A' is a 9-atom, then (y(A) = I (p(A) I ; if furthermore 4p(A)

0, then

(A) = 0; if so(A) 5 0, then cc (A) = 0, cc (A) = 14p(A) I. This follows immediately from the definitions ofq, and 0 and from

'P+(A) = c(A),

§3 (4.2). 5.4.11. If A is a ip-atom, then in order that A =, A, it is necessary and sufficient

that p(A) = 0. This follows from 4.1.1 and 5.4.1. 5.4.12. If 'A e 0)2, then in order that A be a 4,-atom, it is necessary and, if ce(A)

is finite", also sufficient that for all X e A1D2 either p(X) = 0 or cc(X) = p(A). u This condition is essential: Let M e E be a set which is not closed, let 9 be the o-field of all sets of E, and for every X e Jt let cp(X) be the number of points of XM. Then every

point a e Me - M is a continuity-point of v, but for every a > 0 we have p(S«) so. u This condition is essential. Example: Let v(A) - 0 and v(X) - + eo for all the other X e Dt. ForA(* A) a D2 and for all X(* A) a AlW? we then have: (X) - V(A); but for X * A we have4p(A - X) - + ao, thus X:o A.

[CHAP.

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

46

NECESSITY: This follows immediately from the definition.

I

SUFFICIENCY:

Let p(A) z 0, for instance. Then for every X e Al 1l we have 07(X) = 0 and either rp+(X) = 0 or rp+(X) = rp(A); thus either O(X) = 0 or p(X) = ,p(A). If p(X) = 0, then X =, A; if p(X) _ p(A), thus also p(X) = o(A), then p(A - X) = 0, and hence X =,, A. 5.4.2. If A is a cp-atom and B =, A, then B is also a c*-atom. Let X e B 19Q; then X = XA -}- (X - A). Since A is a to-atom, either XA =,A

or XA

A, and in the second case, according to 4.2.41, also XA =, B.

Since X - A C B - A, we have X - A =, A. Thus, because of 4.2.42, we have either X =, A or X =, B. 5.4.21. If A is a cp-atom and B s 912, B Q, A, then B is also a .p-atom.

Since B a A + B and since, because of A + B =, A and 5.4.2, A + B is a so-atom, we can assume from the first that B Q A. Now let X e B11fl; then also X e A19.fl, and thus either X =, A or X =, A. But, as X C B CA, in the second case also X =, B. 5.4.22. If A is o f-atom and B e A 19, then either B =, A and A- B =, A or

B =, A and A - B =, A. We have either B =, A or B =, A. If B =_, A, then A - B =, A. If B =, A,

then A - B =, A. 5.4.23. If A and B are 49-atoms, then either A =, B or AB =, A. We have A = AB + (A - B), B = AB + (B - A). If here AB

, A, then, according to 5.4.22, A - B =, A, B -- A =, A, that is, A =, B. 5.4.24. In order that A be a 9-atom, it is necessary and sufficient that A be a 0-atom.

This follows from 4.1.12 and 4.2.4. 5.4.2 5. If A is a p-atom, then A is also a jp+-a1om and a jp -atwn. This follows from 4.1.12 and 4.2.4. The converse of 5.4.2 5 is not true. Example: Let AB = A, IN = { A, A, B,

A + B 1, and cp(A) = 0, rp(A) = 1, 49(B) = -1, rp(A + B) = 0. Then A + B

is a rp+-atom and a j -atom, but not a c-atom.

Now let Q5(So) be the system of all the sets of 0 which are a sum of countably many 4p-atoms. Because of 5.4.21 we have: if S e then S141J1 C C(,p); and since CS(V) is a we can choose 6(,p) as the system (S of the singular

sets (No. 1). But we have to take into consideration that now the system Z(,p) of the singular sets depends on the choice of the function 'p. 5.4.3. Every set S t S(,p) is a sum of eountably many disjoint 4p-atoms. According to the definition of CU(,p), we have S - SA , where every A. is a 0

(p-atom. We set BI = A (Ai + -}- A,.-4)(m > 1); then S = A SB., where the B. are disjoint and, according to 5.4.21, rp-atoms.

5.4.31. In order that R e %((p), it is necessary and sufficient that for every p-atom

X Q R we have; w(X) = 0 (or X -, A). NECESSITY: Since every fp-atom belongs to (5(c), this follows front the defini-

REGULAR AND SINGULAR PARTS

§51

47

Lion of fi(,p) (No. 1). SUFFICIENCY: Let X e RI S(,p); by 5.4.3, we have X = where the X,,, are disjoint gyp-atoms. Since pp(Xm) = 0, we have also m

,P(X) = EM P(X,,.) = 0. 5.4.32. Z(O) = This follows immediately from 5.4.24 and 4.1.12.

For the set function p totally additive in the a-field 9?2, we form the functions 0* and p** according to (1.2) (again by CS meaning the system C5(,p) of all the sets of 932 which area sum of countably many s*-atoms). Similarly we form the functions (ip+)*, (,p+)** (or (gyp-)*, (gyp-)**, or (rp)*, (gyp)**), now by C5 meaning

the system (,R+) (or (((P-) or e4) = Ca(w)). Then we obtain: _ (,p-)*; ,p**' = (P+)**, V** _ (lp)** 5.4.4. ip*+ = (So+)*, ca The proof is quite analogous to that of 5.1.44 and 5.1.441 if we take into consideration that, since p(X) = p+(X) for all X e M+19J , we have M+1 lt(cp+) = M+15)t(,p) and M+]C(,p+) = M+1G(,p). 5.4.41. ,p* = ( r p ) * = (w+)* + (c)*; ** = (0)** = (W+)** + (w )** Referring to 5.4.32, the proof is quite analogous to that of 5.1.45. _ 5.4.42. V+ = ,p*+ + P**+, c = 4P*- + + 4*s. The proof is analogous to that of 5.1.57. 5.4.5. If p** is finite" and c > 0, then every system ?1 of tp-atoms X with I p(X) P**-,

*

c, no two of which differ only as to zero-sets for gyp, is finite.

According to 3.4.14, ,p** is also finite; thus, because of 5.1.4, 3.4.32, and 3.3.21,

V** is bounded:.V**(X) S g for all X e D1. Let Xi e $( (i = 1, 2, according to 5.4.23, the Xi are disjoint for p; thus, since by 5.4.42 ** X, are disjoint also for p**.

, m);

w, the Hence, according to 4.2.211, W*=' (X1 + - - - + X,,,) _

,p**(X,) + . + **(Xm). But because of §3 (4.21) and 5.1.41, we have "(Xi) J sv**(X,) I = (p(X:) I ? c; thus ,p**(X, + ... + X,,,) mc, hence me s p and m :_5 5.4.51. If ip** is finite, then every system W of atoms X, no two of which differ only as to zero-sets for rp, is countable. This follows from 5.4.5, because in 81 at most one X =, A. 5.4.52. If M e SY and iv**(M) is finite, then every system of p-atoms X C. M, no two of which differ only as to zero-sets for gyp, is countable.

Because of 3.1.21, we obtain this by applying 5.4.51 to the function jo restricted to M1TI. 5.4.53. If p** is finite, then there is a countable system 81' of disjoint
48

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

[CHAP. I

able. A system of p-atoms which contains exactly one p-atom of every such class provides the system 2C' required. Again (as in No. 1) we call a decomposition M = M* + M** (M* M** = A) singular if M* a 9Z((p), M** e 6(,p); then we obtain: 5.4.54. If o** is finite, if $C is the system of 5.4.53, and if S** is the sum of all

X e 8I', then M = (M -- S**) + MS** is a singular decomposition for every M e fit. Since, by 5.4.53, ?l' is countable, we have S** e e(,p). Let S e M1(; then S = SAM , where the A. are (p-atoms. According to the definition of £i', there

is an A;, e `f', such that A' =, A.; then we have SA;, _, S. Thus, since in SAW. a S**, we have S Q, S**, and hence also S Q, MS**. Therefore the proposition follows from 5.1.53. 5.4.55. If M e TZ and P**(M) is finite, then in M19) there are countably many ), such that M - SA , L. disjoint 4o-atoms A. (m = 1, 2, Because of 3.1.21, we obtain this by applying 5.4.54 to the function so restricted to MI TI.

5. Atomless and atomic set functions. Let ,p be a totally additive set function in the o--field 9R, let S(V) be again the system of all the sets of I which are a sum of countably many 9-atoms (No. 4), and choose e(go) again as the system of the singular sets (No. 1). 9?((p) designates the system of the regular sets ( (No. 1), defined on the basis of S((p). Then we call V atomless if for every .p-atom A we have: g(A) = 0 (orA =, A); we call ,p atomic if for every set R e T (,p) we have: ,p(R) = 0 (or R =, A). 5.5.1. If ,p is atomless, then X =, A for all X e 6(,p). This follows from 5.4.3.

Hence the proposition: ",p is atomless (or atomic)" means the same as: "So is regular (or singular) which respect to e(,p)" (No. 1). That here %p) depends on ,p, has to be taken into consideration. For instance, 5.2.1 implies that ,p* is regular with respect to e(,p) and that .p** is singular with respect to "(,p). Yet both the following theorems: ",p* is atomless, (p** is atomic" mean: ,p* is regular with respect to'(so*), g** is singular with respect to C(g"). 5.5.11. ,p* is atomless.

Let A be a g*-atom. According to 5.1.34 there is an A* e A 1 R't(go), such that ,p*(A) = ,p(A*). Let X e A*1Tt; because of 5.4.21, A* is a Sp*-atom; thus either X =,,. A or X =,. A*. Since from 5.1.41 ,p(M) = o*(M) for all M e A*1W , there results that either X =, A or X =, A*; that is, A* is a gp-atom. Thus because A* a 9'i(`o), we have ,p(A*) = 0; hence also ,p*(A) = ,p(A*) = 0. Since this is true for every ,p*-atom A, go is atomless. 5.5.111. ,p** is atomic.

§5]

REGULAR AND SINGULAR PARTS

49

Let Re 9Z(V**) ; we have to prove that4**(R) = 0. According to 5.1.34 there is an R** e R1,!a(4p), such that p**(R) = 4p(R**). Thus we have to prove that ,p(R**) = 0. If cp(R**) 54 0, then, since R** e Z(.p), there would be a cc-atom A Q R** with 4p(A) 34 0. For all X e A112 we have either X =p A or X _,, A. But since from R** e e5(cc), according to 5.1.41, So(M) = 0**(i1I) for all Ale R**19Yl,

there results either: X =,,.. A or X =,,.. A; that is, A is also a 4,**-atom. Thus since A e RITZ, we have j,**(A) = 0, contradictory to cc**(.A) = 4(A) 96 0. 5.5.112. p*{, p* , p* are atomless; p***, 9**-, 4p** are atomic. The proof is analogous to that of 5.2.11. 5.5.12. In order that rp be atomless (or atomic), it is necessary and sufficient that ,p** = 0 (or rp* = 0). This follows from 5.2.12 for 6 = Cu(sp).

5.5.121. In order that rp be atomless (or atomic), it is necessary and eufficient that p = rp* (or rp = 0**). This follows from 5.2.121 for e = C5(,p). 5.5.13. If 4, = Bpi + v2, where rp, and 402 are totally additive in the a -field 9)1, if gyp, is atomless, and A is a yr-atom, then there is an A' e A19.1, such that A' =,, A

and A' According to 3.2.52 p is also totally additive in M. Let ¶1 be the system of all the zero-sets for rp which are c A; according to 4.1.32 % is a a-field. Thus, because of 3.3.1, among all the values -pi(X) (X e 91) there is a greatest one, say sp,(N). Now let 9' be the system of all the sets of 91 which have no element in common with N. Then 91' is also a u-field and hence, because of 3.3.1, among all the values o1(X)(X e 91') there is a greatest one, say 7,,(N'). W-

set A' - A - (N + N') ; then A' =, A.-In order to show that A' =,, A, we prove first: there is no X e A'1V with 0 < 01(X) < 01(A'). Indeed, since N e %, we have,p(N) = 0; thus, as sp = sp, + ,pz , we have ,pi(N) finite and hence, according to 3.4.14,.1(N) is also finite. Thus from 0 <
it would follow, since A' + N =A - N', that,pt(N) <1(X+ .N) < ,p1(A - N'). Because of the first part of this inequality we would have X + N -' e 91,

that is, X + N b A; thus, since A is a o-atom, X + N = A; and hence A - (X + N) e 9'. Because of the second part of the inequality ,pl(X + N) would be finite; thus according to 3.1.11:

01(A - (X + N)) = q1(A) - gpi(X'. J,- N). If here q1(A) is finite, then from q,(X + N) <
,p1(X + N) > 1(A) - sp1(A - N') = ql(N'); hence p1(A - (X + N)) > o1(N'), contradictory to the definition of N'. If, however, p1(A) = + co, then also Sd1(A - (X + N)) = + w and, since 1(N') is finite (which results from a consideration similar to that above for sp1(N)), we would get q,(A - (X + N)) > 'MN') also in this case. So we have proved the contention that there is no X E A'1SJ,il with 0 < 4p1(X) < pl(A'); in other words: for every X e A.'llD? we have either ,pi(X) = 0 or 01(X) = ,p1(A').-We now prove that A' is awl-atom. If 41(A') is finite, it follows from 5.4.12 that A' is a pi-atom, and hence from 5.4.24 that

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

50

CHAP. I

A' is also a vi-atom. Now let 01(A') = + co ; then, as we have shown, for every

X e A'1E either o,(X) = Oor p1(X) = + b. Since A'e A1l, according to 5.4.21 A' is a,p-atom; thus for every X s A'1WDI either X =,A or X =,A". Inthe first case we have p(X) = 0; thus since p = q n + V2, we have ft(X) finite; hence according to 3.4.14 0,(X) < + co; therefore 01(X) = 0; that is, X =,, A. In

the second case we have p(A' - X) = 0; thus p,(A' - X) is finite; hence ,pl(A' - X) < -1- - and therefore, since A'- X e A'ISTZ, we have o1(A' - X) = 0, that is, X =,, A'. Thus A' is a p,-atom.-But since V, is atomless, it follows

that A' =,, A. 5.5.131. If p = p, + p: , where p, and p2 are totally additive in the v-field V, if p, is atomless, and if A is a p-atom, then in A 171 there is a pr-atom A' with

pr(A') = p(A).

According to 3.2.52 p is also totally additive in

M.

Let A' be the set of 5.5.13.

Since A' =,, A, we have p(X) = p,(X) for all X eA'19. Thus, as A' is a p-atom according to 5.4.21, A' is also a n-atom and we have p(A') = pr(A'). Therefore, since A' =, A, we have also p(A) = pr(A). 5.5.14. If p = .pt + v2 (or p = vi - pr), where p1 and pr are atomlesa set functions in the a -field iDl, then p is also atomless.

According to 3.2.52 p is also totally additive. Let A be a p-atom. Because of 5.5.131, in A lit there is a pr-atom A', such that pr(A') = O(A). Since pr is atomless, we have pr(A') = 0; thus also p(A) = 0; that is, p is atomless. 5.5.141. If p = p1 + pr (or p = p, - pz), where p, and pr are atomic set functions in the a -field SDl, then p is also atomic.

Let p = p1 + pr ; according to 3.2.52 p is also totally additive. Because of 5.4.3 we have to prove: if M e 1t and p(M) 5 0, then in MI P there is a p-atom A , A. We discern three cases. tat case: There is a B e MIT?, such that B 0,, A and B =,, A. Then, since 9P1 is atomic, there is a q,-atom A ,, A in B1 12. As A Q B, we have also A = A; that is, for all X e A 1 !R we have p(X) = VI(X). Thus A is also a p-atom and A , A. 2nd case: There is a B e M1 D1, such that B = A and B ,, A. This case can be treated analogously.

3rd case: For every X e Ml9R we have either X = A, X =,, A or X -* A,

X P6 A. Then every pi-atom B e M11B is also a pr-atom and conversely; for if B is, for instance, a pi-atom, then for every XsBlT1 either X =,, A or B - X =,, A; thus either X = A or B - X =,, A also. But every such a ,pi- and pr-atom is also a p-atom; for, according to 4.1.2, X =,, A, X =,, A imply X =, A, and B - X = A, B - X =,, A imply B - X =, A. Now we have

still to prove that for not all of these pi-, pr-, and p-atoms we have p(B) = 0, that is, pr(B) = -pI(B). If this were the case, then according to 5.4.3 we would also have p,(X) _ -p1(X) for all sets X of M1 (4,1)(= M1((p,)). Thus from (1) we would get a' *(M) = 6p; (M), A*,*(M) = a** (M); hence from (1.2): pi *(M) = -pi *(M). Therefore, because of 5.5.121, we would also have pr(M) = --p1(M) ; hencep(M) = 0, contradictory to the assumption. 5.5.1.5. In order that p be atomless (or atomic), it is necessary and suf cient that p+ and ,p be atomless (or atomic).

§51

REGULAR AND SINGULAR PARTS

51

The proof is analogous to that of 5.2.15. 5.5.16 In order that p be atomless (or atomic), it is necessary and sufficient that o be atomless (or atomic). NEcESsITY: If cp is atomless (or atomic), then according to 5.5.15 rp+ and io are also; thus, because of 5.5.14 (or 5.5.141), 0 is also. SUFFICIENCY: If 0 is atomless, then according to 5.4.24 (p is also.-Now let 0 be atomic and M e Because of 5.4.32, we have M e 92((p) also; thus O(M) = 0, hence p(M) = 0 also;

that is, p is atomic. 5.5.17. Every atomless (or atomic) set function is the difference between two monotone increasing atomless (or atomic) set functions. This follows from 5.5.15, 3.4.61, and 3.4.3. 5.5.2. Every totally additive set function in the o-field 992 is the sum of an atomless and an atomic set function. Because of 5.5.11 and 5.5.111, this follows from 5.1.56. 5.5.21. If rp = + pc , where rpi and p2 are totally additive in the a -field 9.)2 and (pi i8 atomlesa, then rp**+ S p+2, p**- -5 rPs ,,p**

pz .

Let M e M. According for 3.4.71 there is a decomposition M = M+ + M(with M+ e MlSDJ, M- a M15)2, M+M- = A), such that c+(M-) = 0, j (M+") = 0 Then because of 5.1.441 and (1.1), we also have ,p**+(M-) = 0, c**-(M+) = 0;

thus p*"+(M) = p**+(M+) = cp**(M+). According to 5.1.34 there is an S e M+1CS((p), such that cc**(M+) = p(S); therefore we have rp**+(M) = cp(S). By 5.5.13

From 5.4.3 we get S = SAM , where the A. are disjoint cp-atoms.

there is an A. a A.15Dt, such that A' = A and A., =,A.. We set S' = SAC, ; M then we also have S' = A, S' =, S; hence rp(S) = cc(S') and thus rp**+(M) = p(S'). Since S' =,, A, we have cp(S') = "(S'). Therefore ip**+(M) = (S') 9(M). 5.5.22. If the set function p is totally additive and finite" in the a -field SDt, then = p* + Sp** is the only decomposition of p into an atomless and an atomic term. We prove this in a similar way to 5.2.22, applying 5.5.21, and 5.5.121. BIBLIOGRAPHY to Nos. 1 and 2: The notion of the p-atom (but not this expression) and an essential part of the results of No. 1 and 2 are due to M. FRICHET, Enseignement math. 22 (1922), p. 126; Fund. math. 4 (1923), p. 355.

6. Intermediate values. We here prove an "intermediate value theorem" [5.6.22] for totally additive set functions p, which is analogous to the intermediate value theorem for continuous point functions (namely, analogous to the well-known theorem that a continuous point function f(x) on a connected set assumes every value between any two of its values). As in Nos. 4 and 5, let cc be totally additive in the a-field fl, let S(O) be the u That this condition is essential, is shown by footnote 27, p. 42, if one still understands from footnote 36, p. 52, that pi is atomless.

AND TOTALLY ADDITIVE SET FUNCTIONS

52

[CI kp. I

,system of all the sets of 901 which are a sum of countably many p-atoms, and let be the system of the regular sets defined on the basis of (1p). 5.6.1. If .411 e 9l(,p), M 0, A, and "' cp(M) is finite, then for every e > 0 there is an X e M 1711 with 0< 53(X) 5 e. According to 3.4.14, O(M) is also finiteand,because of 5.4.32, M-E l((p). Since

therefore M is riot a rp-atom, according to 5.4.12 there is an 11f' e MIT with 0 < p(11') < o(M). Hence from p(M) = 0(M') + 53(M - M') we get either 0 < o(11') S Arp(M) or 0 < . (11f -- M') s- i rp(M). Thus there is an M1 e M1 0, such that 0 < rp(d11) 5 21(p(M). Similarly there is an 1112 e M119)l, such that 0 < ;,(i112) 5 rp(M1), and so on. We thus obtain a sequence of sets Mk a t11 15D2, such that 0 < rp(Mk) 5 l,1.p(M). From rp(Mk) --> 0 the proposition follows. 5.6.1.1. If M e 9? (,p) and 5,(M) is finite, then for every e > 0 there is a decomposi+ Ali of Al into finitely many disjoint sets of V, such that 1111 + tion Al rp(,il;) s e(i. = 1, 2, , k).

For all X e M11Y1 we set: y(X) = sup p(Y) (Y e X19)2, ,p(Y) S e). Then 0 < -y(X) S e and, because of 5.6.1, -y(X) -= 0 if and only if X =, A. Now we define a sequence of disjoint sets X, of 1111971 with ib(X.) S e according to the , X, following rule: Let X1 be any set of Mlnt with 0 5 rp(X1) 5 e; if X1, are determined, then we set Al - (X 1 + . + X.) = Y, and choose X,+1 e Y, IT? according to Ay(l ,) 5 rp(X.-,-1) 5 e. We set M' = M - SX, ; then (from 2rp(X,+1). But since the X. are disjoint, Al' C Y,) 1ve get: y(M') < y(Y,) we have E p(X,) = ,p(SX,) 5 O(M); thus since,g(M) is finite, E fo(X,) is also

finite.

Hence ip(X,) -+ 0 and therefore y(M') = 0; thus dl' =, A. Since

i'(X,) is finite, there is a k, such that M'; them since
}

rp(X,) S e. We set Mk = S X, +

So(.ill') and M' =, A, we also have rp(Mk) 5 e.

Now we still set Al, = X,(v = 1, , k. provide us with the desired decomposition.

1). Then the sets M1 ,

,

Mk

N:wr we prove the "intermediate value theorem", first for monotone increasi'r1g 1p and then in 5.6.22 for the general case. 5.6.2 If c is totally additive and monotone increasing in the o ,field 971, V M e %,(,P) and cp(!til) is finite, then for every y satisfying the inequality 0 5 y 5 So(M) there is an h e _?4f19)2 with w(X) = y.

For y = 0 the contention is true, since So(A) = 0, A e M 19R. Thus we can assume y > 0. Since p is monotone increasing, we have rp = p; thus according u This condition or the weaker one of 5.6.21 is essential here and in all the other theorems of No. 6. Example: Let E be a non-countable set and 9J1 the system of all subsets of E. Let '(X) = 0 for every countable set X ESDD, jv(X) - +m for every non-countable set X est. Then S(yo) consists of all countable sets of 971, and llt(,p) consists of all sets of V. Indeed, no non-countable set M e Dt is a p-atom. For every non-countable set M is a sum of two disjoint, non-countable sets M1 and A (of., for instance, F. Hausdorff 121, p. 71); hence ,p(M,) - ,p(M) - +oo,a(M - M,) = p(M,) - +-, and thus M, A and Al, 5ee M.

53

REGULAR AND SINGULAR PARTS

§5]

+ Mk, of M into finitely to 5.6.11 there is a decomposition 3f = M11 + We set Mo = A. , ki). many disjoint sets of 9, such that v(M;) 5 1 (i = 1,

+ MI) Then there is an index h1 (0 5 hi S kl - 1), such that p (Mo + < y 5 ip(Mo -+- ... + Mk, + 11MA,+1] - P(M10 + - + 111k) + 1. According to 5.6.11 there is a decomposition Mkl+1 - Mi + .. + Mk, of M11, +1 into finitely many disjoint sets of V, such that ip(M=) 5 1 (i = 1,

, Q.

We set Mo = A. Then there is an index h, (0 5 h2 5 kg - 1), such that

,p(Mol + .....}_ Mk -}- 1i1o -}-..

M% + Mo +

MA,) < y 5 So(Mo -+-...

-}- lblk + Mn2f1) 5 w(Mi }- ... {. Mh1 + Mo + ... + Mk,) -{- J. Con, M;,, for every P, such tinuing in this way we obtain sets Mo = A, M; , , ... are disthat the sets joint and M0,...,M;,,Mo,...,M,,,,...,Mop,

A,

i0

A,

A,

i0

Al

so(SM,-{-SMi-+-...-}. SM{)
i0

1

0

...,MAY,

As

A,

10

1-0

v

J.

Thus if we now set X = S 8 M{ , then p(X) = y and X e M1 M. . i-o 5.6.21. The theorem 5.6.2 remains true, if in it for the condition "s(M) is finite" we substitute: M = Lim M., where ((M.)) is a monotone increasing sequence of ri

sets of fit and

is finite.

oo also. Let P(M) = + oo, and then, according to 3.22, lim For y = + oo, X = M satisfies the proposition. Thus we assume y < + oo. y. Therefore, because of 5.6.2, there is Then there is an n, such that an X e with V(X) = y, and, since M. M, we also have X e M1112. 5.622. If V is totally additive in the o f eld 1, if M and if M = Lim M,,, where

is a monotone increasing sequence of sets of T? and

then for every y satisfying the inequality - rp (M)

is finite,

y 5 p+(M) there is an

X e M1SD2 with V(X) = Y. According to 3.4.13

and V -(M.) are also finite. Let M = M+ + Mbe the decomposition of 3.4.71. For every A e M{'1T2 and B e M-19)2 we have lp+(A) = cp(A) and lp (B) = -,p(B) ; therefore from M+ a 91(ip) and ,11- e 9 ?(,p) it follows that M e (,p+) and M- e ftp) also. Thus, since ip+(M+) = P+(M) and V -(M-) = V-(M), from 5.6.21 applied to ,p+ (or top) there results that for

every y satisfying the inequality 0 5 y 5 w+(M) (or - lp (M) S y 5 0) there is an X e M+1171 (or an X e M-1ft with p+(X) = y (or (p (X) = -y). But since qp(X) = ip+(X) for X e M+1x'.02 (or p(X) _ -97(X) for X e 111-19)1), we obtain jp(X) = y. BIBLIOGRAPHY; W. SIERPIIi8EI, Fund. math. 8 (1922), p. 240; M. FRaCHET, Fund. math. 4 (1923), p. 364; G. FICRTENHOLZ, Fund. math. 7 (1925), p. 296; H. HARK, Anzeiger Akad. Wise. Wien 1928, No. 12; Bulletin Calcutta Math. Soc. 20 (1930), p. 227; G. POPROUG$NSO, Fund. math. 12 (1928), p. 254; 14 (1929), p. 221.

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

54

CHAP. I

7. 4-continuous set functions. Let 4, be a totally additive set function in the s-field M. Because of 4., 1-32 and 4.1.3 we can choose the system of all zerosets for ¢ as the o-field Z of the singular sets (No. 1). If ,p is also a totally additive set function in 9)l, then the sets R e Dt regular37 for ip are characterized by the property: if X e RITZ and X =# A, then o(X) = 0 (or, what amounts

to the same thing, X =, A). Choosing Z in this way we call the regular (or singular) set functions 4-continuous (or purely 4-discontinuous). {Cf. the terms of No. 3.1 Thus the totally additive set function' in 9 is called 4-continuous if X =, A implies to(X) = 0 (or X =, A) ; it is called purely ,y-discontinuous if for every set M e 9)2

with 4o(M) 96 0 (or M 0, A) there is a set X e MIT? with X =* A and ,(X) , 0 (or X 0, A). Every totally additive set function ip in'the o-field 02 is u-continuous. If ye+,

,

and 0 again signify the positive-, negative-, and absolute-function

of (p (§3, 4), then we have: 5.7.1. 0, rp+, and 'p- are `o-continuous; qp, rp+, and rp are ip-continuous.

For according to 4.1.12, X =, A implies X =; A, X =,+ A, X =,.- A; and similarly X = o A implies X =, A, X =,+ A, X =, A. If X is also totally additive in 9)'1, then we have: 5.7.11. If jp is t4-continuous and 9j X, then 9 is also X-continuous. For if X =,,'A, then X = # A also. z, then v is also purely X-dis5.7.111. If p is purely "iscontinuous and continuous.

ForifX =#A,thenX =x Aalso. 5.7.12. In order that p be ¢-continuous (or purely. 4-discontinuous), it is necessary and sufficient that 4p be i-continuous (or purely i-discontinuous). We obtain this from 5.7.11 (or 5.7.111), substituting at first 0 for 4', y4 for X, and then J' for 4, 4 for X. 5.7.121. If jo is purely *discontinuous, then cc is also purely 4+-discontinuous and purely ,y--discontinuous. Since ;' 4+ = y4±, by 5.7.111 v is purely 4,'-discontinuous, and similarly purely 4. -discontinuous. 5.7.13. If tp is 4-continuous and 4 is X-continuous, then 0 is also X-continuous.

For if X =x A, then X =,p A also. 5.7.131. If tp is purely 4-discontinuous and X is 4.-continuous, then 0 is purely X-discontinuous.

Let M e 0 and p(M) 5-6 0. There is an X e Ml9 with X =0 A and sp(X) Pe 0.

Since X = x A also, the proposition follows. 5.7.2. If rp and 4, are totally additive in the o-field x111, then in order that tc be

i4-continuous, it is sufficient and, if (p is finiti8, also necessary that for every sequence ((Mk)) of sets of x11 with (Mk) -- 0 we have also O(Mk) -+ 0. "r If we also want to express the dependence on 4,, then instead of regular or singular set we say more explicitly y.- regular or ,.-singular set. 19 This condition is essential here and in 5.7.21. 5.7.22, 5.7.24. Example: Let E be the

REGULAR AND SINGULAR PAITS

Is]

SUFFIcIENCr: Let X -+ A; that is, j(X) - 0. If we now get M, - X, then we have RMk) - 0, and hence ,p(Mk) -+ 0 also, that is, o (X) - 0. NEcES$rry:

If the condition is not satisfied, then there is a c > 0 and a sequence ((M,r)) belonging to W2, such that (Mk) - 0 and tp(Mk) k c for all k. Taking a subSetting N. - k8 Mk, we sequence, if necessary, we can assume:

have N, e TI, p(N.) ? c, and from 3.2.3: J(N,) <

Z

I

.

Still setting M =

DN., by 3.2.21 we get: o(M) Z c(>0), (M) = 0; that is, go is not J,-continuous. 5.7.21. In 5.7.2 one can substitute ,p(Mk) -- 0 for ,p(Mk) - 0. SUFFICIENCY: Substitute v for pin the proof of 5.7.2. NECESSITY : ,p(Mk) - 0

implies that yo(Mk) - 0. 5.7.22. If p and . are totally additive in the a-field Dt, then in order that v be ,P-continuous, it is sufficient and, if p is finite, also necessary that for every e > 0 there be an q > 0, such that, for all X e 9R, J(X) < q implies (X) < e (or I ,p(X)

< e). Sumcc$Ncy: If X =* A, then (X) < q; thus p(X) < e (or I (p(X) I < e) for A (or c(X) = 0). NECESSITY: every 'e > 0; that is,,p(X) = 0 and hence X If the condition is not satiified, then there is an e > 0 and a sequence ((X,,)) 0, ,p(Xk) is e. Thus the contention follows belonging to 9, such that from 5.7.2.

5.7.23. If p is f-continuous in the a -field O1, then A -# B (A a D1, B e 91`t) implies:

,p(A) _ p(B), o(A) _ CA(B),,+(A) - p+(B),'v (A) = qp-(B). Since A =,. B implies A =, B, the proposition follows from 4.2.44.

Let M be the metric space associated with 9Y by ¢ (44, 2). If a e M and if A and B are representatives of a, then we have the equations of 5.7.23. Thus in M we can define the point function f associated with ,p in the following way: if a e M and A is a representative of a, then f(a) - W(A). 5,724. If sp is t#-continuous and finite, then the point function f associated with ,p is uniformly continuous in M. Let are M, 19 a M.

then a#

J(A

If A and B are representatives of a and 0, respectively,

B) + -;(B - A) - ;((A - B) + (B - A)).

Since A

AB + (A - B) and B - AB + (B - A), we have f(a) - f(5) _ p(A - B) -I 'p(A - B) I + I ,p(B - A) I tp(A - B) + ro(B -- A); thus If(a) - f(d) 1

,p(B -- A) - p((A - B) + (B - A)). Hence from 5.7.22 there results: for sequence (a, , a2

0(la*j)

ak ,

(k - 1,

} of points and let )I consist of all subsets of B. Let G(A) - 0,

0, p(lakl) - k (k - 1,

from this-let 1' and 0 be defined as totally additive in

;(larl) -+O,w (lab)) -++co.

and besides--starting

Then, is #-continuous. But

56

[CHAP. I

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

;;very e > 0 there is an n > 0, such that ao = t ((A - B) + (B - A)) < Implies i f(a) - f(!g) I < e. 5.7.25. Every 4.-continuous (or purely 4-discontinuous) set function is the difference between two monotone increasing 4.-continuous (or purely 4.-discontinuous) :et funetion.a. This follows from 5.2.17.

Further on, using the system of zero-sets for 4, as the system G of singular nts, we decompose the totally additive function (p into rp = (p* + rp** according to 5.1.50. Then we call rp* the ,k-continuity-function and y,** the ip-discontinuity function of c. [Cf. the corresponding expressions of No. 3.] According

to 5.2.1, the 4.-continuity-function is 4.-continuous and the 14-discontinuityfunction is purely ,y-discontinuous, and we get from 52.2 and 5.2.22: 5.7.3. Every totally additive set function rp in the a -field x>z can be represented as the sum of a ,.-continuous and a purely fir-discontinuous set function. If .p is finiten, then there is only one such representation. Binuooa&euy: H. Lnnssaan, Ann. Ec. Norm. (3) 27 (1910), p. 381; J. RADON, Sitzungsberichte Akad. Wise. Wlbn 122 (1913), p. 1318; H. HAHN [11, P. 416; 0. NIHODYM, C. R. I.

Congr6s math. pays slaves (Warsaw 1929), p. 309; Fund. math. 15 (1930), p. 166; (to him the expression i.-continuous is due, instead of absolutely continuous or totally continuous with respect to i.);B. JassuN, Mat. Tidsskrift B 1938, p. 17; C. E. RieuART, Duke Math. Journ. 10 (1943), p. 653.

8. Sequences of 0-continuous set functions. Let p and cp, (i = 1, 2, ) be totally additive set functions in the o-field M. Referring to 5.7.22 we call the sequence ((,pi)) equi-4.-continuous if for every e > 0 there is an 71 > 0, such that for all X e 0 from yi'(X) < rt it follows that 4pi(X) < e (or I ,pi(X) I < e) for all i. 5.8.1. If 4, is totally additive ands0 finite in the v-field M, if ((,p.)) is an equi-4#continuous sequence of totally additive functions in 0, if SP(X) = lim c;(X) exists for all X e TI, and if Sp does not attain infinite values of different sign in xff, then w is totally additive and 4.-continuous.

Let X = SX, (X, e M) where the X, are disjoint. Weset Y., = S X, ;then ,2v$

since Lim Y. = A, from 3.2.21 it follows that lim 4'

0. Thus for every e > 0 there is an m, , such that I ,p;(Ym) f < e form ? m. and all i. Then we also have a form 9 m. , that is, lim p(Y.,) = 0. Since according to 3.1.61 p is adV(Y,,,)

ditive, we have -AX) _ E p(X,) + e( Y,. ). Thus p(X) = lim E p(X,), and hence m t41

rp is totally additive.-Let X -# A. Then pi(X) = 0; thus also p(X) = 0, that is, rp is *-continuous. " Cf. footnote 27, p. 42. +^ That this condition is essential, is shown by the example of footnote 13, p. 24, if one sets 4.(X) - w,(X)-

57

REGULAR AND SINGULAR PAINTS

§51

5.8.11. If >G is totally additive and finite in the c-field 912, if ((Pi)) is a sequence

of finite, totally additive, and #-continuous functions in 0, and if, in the metric space M associated with 0 by ' (§4, 2), those points t for whose representatives X e 0 a finite limit p(X) = Urn pi(X) exists form a set A of the second category t

(§2, 1), then thc+ sequence ((,pi)) is equi-4 continuous.

Let fi(I;) be the point function associated with rpi (No. 7) in M. If e > 0 is arbitrarily given, we designate by Ali the set of all the 1: e M for which Since according to 5.7.24 fi is . I fj(E) - fi(t) 1 < a for j = i, + 1, continuous, Mi is closed in M. Since A

SMi and A is a set of the second catei

gory, at least one Mi, say M. , fails to be nowhere dense. As a result of 2.1.5,

Mi. has an inner point *; that is, there is a p > 0, such that the sphere of the space M (with the centre r* and the radius p) FE., Q Mi.. Then for all e for j i' i*; in other words: if X* e 912 is arepreE e Ec., we have: I fj(E) - fi.() 1 sentative of r*, then for every X e 1 with (X - X*) + (X* - X) < p and for all j i* we have: 1 pi(X) - p,-(X) 15 e. Since all cpi are ,y-continuous

and finite, according to 5.7.22 we can choose p so small that (pi(Y) < e for If we now set X' = X* -1- Y, A:" , s* and all Y e 912 with (Y) < p. i = 2, 1,

= X* - Y, then we have (X' - X*) + RX* - X') _ RY - X*) <

and (X" -- X*) + J(X* - X") = (YX*) < p ; and hence: 1 VAXO - 'Pi-W) 1

e,1-Pj(X") - -W) 1

e

for j 2 i*; that is, I p1(X*) - pi.(X*) '+- soi(Y - X*) - -PAY --- X*) 15 C, (8)

I vj(X*) -

pj(YX*) + p,.(YX*) 1

e for j z i*.

e for j i*. Furthermore Since t* e EE., , we have I pj(X*) - pi.(X*) 1 e; thus alsol ic,.(Y - X*) I < e and I pi.(YX*) I < e. p was so chosen Hence it follows from (8) that rpj(Y - X*) I < 3e and I co j(YX*) I < 3e for j ' i*; thus also I pi(Y) I < 6e for j i*. And since p was so chosen that , i*, we have I pi(Y) I < 6e for all i; that is, ((ipi)) I ipi(Y) I < e for i = 1, 2, is equi4-continuous. 5.8.12. If the conditions of 5.8.11 are satisfied and the set A of the second category

is dense in M, then. for all X e 9J a finite limit p(X) = lim cc,(X) exists and ip is i

totally additive and 4;-continuous.

Let fi(b) be the point function associated with v s. First we prove that at every point r e M the sequence ((f )) is equi-continuous'. According to 5.8.11 ((,pi)) is

e > 0 there is a p > 0, such that

" The sequence ((fi)) of point functions, defined on the space M, is called equi-continuous in the point to M if for every e > 0 there is & neighborhood Ut, such that 11 I. (t') - fi(t) 11 < e [cf. §3 (6.1)] for ail t' a OF and for all i. If the fi are finite, then according to 13 (6.11) we can here substitute Ifi(t') - (t) < e for 11f,(t') - fi(t) 11 < e.

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

58

for all Y e SDl from (Y) < pit follows that I ppi(Y) I <

CHAP. I

for all t. Let

M,

V a M, and ff' < p. If X and X' are representatives of and f', respectively, in 9, then (X - X') + (X' - X) < p; thus I tp;(X - X') I < 2 and

:pr(%' - X) < 2 for all i. Thus, since :pi(X') = :pi(X) + ,;(X' - X) -pt(X - X'), we have I sp:(X') - :pt(X) I < e; hence also

I ft(f') - f:(f) I < e for ff' < p and for all i;

(8.1)

that is, ((f{)) is equi-continuous at f.-We then prove that him f{(f) exists and t is finite at every point f e M. Since A is dense in M, according to 2.1.52 there is a point fo e A with ffo < p. Since him f,%) exists and is finite, there is an index 4 , such that for i z io and j io we have I f:%) - f j(fo) I < e. Thus, using (8.1) also, we get

f+(f) - f j(f) I s I f.(f) -1;(fo) (-{- I J:(fo) - .f j(fo) I + I f j(f) - fj(f) I < 3o for i z io and j io ; that is, ((fi(t))) converges at every point f e M. We now set f = him f:. Since f(fa) is finite, and since from (8.1) we get I f(fn) - f(f) I e, {

M) is also finite.

Hence p(X) = lim :pt(X) exists and is finite for every X e .Q.:

Now from 5.8.1 it follows that 4p is totally additive and ,'-continuous. 5.8.13. If ' is totally additive and finite in the 47-field l)2, if (((pt)) is a sequence of finite, totally additive, and ¢-continuous functions in $t, and if for every X s D'1 there exists a finite" limit :p(X) = him :p;(X), then Sp is also totally additive and 4-continuous.

Since according to 4.2.61 the metric space M associated with 9 by ,P is complete, from 2.3.31 it results that M is of the second category in itself. Thus the proposition follows from 5.8.11 and 5.8.1. 5.8.14. If yt is totally additive and finite in the c field SD'l, if ((:pt)) is a sequence of finite, totally additive, and #-continuous functions in D2, and if, in the metric space M associated with hl by st', those points f for whose representatives X e D? the

sequence pi(X) (i - 1, 2,

) is bounded form a set A of the second category,

then there is a p > 0 and a finite c, such that for all Y e 9)2 with , (Y) < p we have:

I pi(Y) I < c for all i. Let f,(f) be the point function associated with -pt in M. We designate by M. the sets of all f e M for which I ft(f) I s n for all i. Since according to 5.7.24 the f; are continuous, M. is closed in M. Since A Q SM and A is of the second cate-

gory, at least one M,,, say M.., fails to be nowhere dense. According to 2.1.5 M,,. contains an inner point that is, there is a p > 0, such that the sphere

Et., a M,,.. For every f e FF., we then have:

I ft(f) I 43 That this condition in essential, is shown by footnote 43, p. 60.

;j n" for all i; in

59

REGULAR AND SINGULAR PARTS

§51

other words: if X* a W is a representative of E*, then for every X e T1 with

(X -X*) + a(X* - X) < p and for all i we have: I

qo,(X)

n*.

Now let Y e l)1 and (Y) < p.

We set X' = X* + Y and X" - X* - Y; then (X' - X*) + +G(X* - X') < p and ,'(X" - X*) + 1 (X* - X") < p. Thus I

pi(X')

I

n* for all i; that is, 5 n*, pi(X") 1 I ,p:(X*) + pi(Y - X*) ( s n*, I p.(X*) - (pi(YX*) 16 n* for all i. n* for all i; hence we get: But since ¢* a Et., , we here have I pi(X*) 1 cp,(Y - X*) 1

2n*, I p,(YX*) 1 5 2n* for all i.

qa,(Y) I

Therefore

,p,(Y - X*) I + I q,(YX*) I s 4n* for all i.

5.8.15. If the conditions of 5.8.14 are satisfied and if the set A of second category is dense in M, then the sequence tp,(X) (i = 1, 2, ) is bounded for every X e V.

According to 5.8.14 there is a p > 0 and a finite c, such that for all Y e 9R with ,{,(Y) < p we have: I ,pr(Y) I < c for all i. Let X e W' `t and let E e M be the point associated with X. Since A is dense in M, there is a E' e A with if' < p. If X' e SD2 is a representative of E', then the sequence O,(X') (i = 1, 2, ) is

bounded; that is, there is a finite c', such that 1 ,p,(X') I < c' for all i. Since

if' < p, we have (X - X') + ; (X' - X) < p; thus (e,(X - X') I < c, I -p,(X' - X) I < c for all i. Hence from ,pi(X) = qo;(X') + 9,(X - X') (P,(X' - X) we get J p;!X) I < c' + 2c for all i.

5.8.16. If >' is totally additive and finite in the o-field A2, if ((p,)) is a sequence of totally additive and J,-continuous functions in IN, and if for every X e T't the sequence,p,(X) (i = 1, 2, ) is bounded, then there is a finite 1, such that cp,(X )

< lforallXd0 and for alt i. Since 4, is finite, according to 3.4.14 j' is also finite. As a result of 3.3.1 there

is a C e TI, such that y (C) is an absolute maximum of j. For every X e T? we then have RX - C) = 0; thus, since the pi are *-continuous, cp;(X - C) = 0 Therefore it suffices to prove the contention for all X e C19'1.-According to 5.4.55, in CM there a r e countably many disjoint #-atoms A,(v = 1, 2, . ), such that setting B = C -- SA, we have B e Dt(#). Since by 4.2.61 and 2.3.31 M also.

is of the second category in itself, according to 5.8.14 there is a p > 0 and a finite c, such that for all Y e T2 with (Y) < p we have: I pi(Y) I < c for all i. Since is finite and the A, are disjoint, (A,) is finite; thus there is a v*,

such that E (A,) < p. Hence, if we set Be = S A,, we have: i`(Bo) < p. According to 5.6.11 there is a decomposition B = B, + + Bt of B into disjoint sets, such that (B,,) < p (X = 1, 2,

, k). Then we have C = Al + + A,. + Bo + B, + + Bk, where A, , , A,. are 4-atoms and (B)) < p = 0, 1, . - , k). Now let X e ClV; then X = XAI+ . -}- XA,. +

XBo +

+ XB1 . Since y (XB),) < p (X - 0, 1,

,

k), we have I pi(XB,,) I

ADDITIVE AND TOTALLY ADDITIVE SET FUNCTIONS

60

[CHAP. I

, k and for all i. According to the supposition, the se< c for X = 0, 1, , v*; thus there is a quence vi (A,) (i = 1, 2, ) is bounded for v = 1, 2, finite c', such that I vi(A,) < c' for v = 1, 2, - , v* and for all i. Because of 5.4.12 we get pi(XA.) = 0 or = pi(A,) ; thus we also have 1 vi(XA,) < c' + for v - 1, 2, - , v* and for all i. Hence from rpi(_X) = 1Pi(XAi) + - + ipi(XBf) there results: qp:(X A,.) + o,(XBo) + -

I pi(X) I < v*c' + (k + 1)c for all X e C1)2 and for all i.

Now let a o-field T't be given together with a sequence ((,pi)) of finite and totally additive functions in 9N.

According to 3.4.14 Ipi is also finite; thus because

of 3.3.21 there is a finite ci > 0, such that ; (X) S c; for all X e 9)2. If we set 1

¢(X) = E 21 rp,(.Y), then 4, is finite and, according to 3.2.52, 3.6.2, totally additive in W. Since 4'(X) = 0 implies pi(X) = 0, the vi are i-continuous. Now from 5.8.13 and 5.8.16 we obtain the theorems: 5.8.2. If ((,p,)) is a sequence of finite and totally additive functions in the e -field 9)2, and if for every X e $t there exists a finite" limit ip(X) = lim pi(X), then p is i

also totally additive.u

5.8.21. If ((,pi)) is a sequence of totally additive functions in the a-field 0, and if for every X e 92 the sequence rpi(X) (i = 1, 2, - - -) is bounded, then there is a finite 1, such that 19i(X) I < l for all X e s92 and for all i. BIBLIOGRAPHY: S. SASS, Trans. Amer. Math. Soc. 35 (1933), p. 965.

The theorems 5.8.2

and 5.8.21 are due to 0. Nlsouru, C. It. Paris 192 (1931), p. 727; Monatshefte f. Math. u. Phys. 40 (1933), p. 418, 427; the demonstration of these theorems given above is due to S. SASS, toe. Cit.

a This condition is essential. For if In, attains positive and negative values and if vi # i-vi , then according to 3.1.23 .p = lim fi certainly is not additive. 44 As to the particular case of monotone sequences ((,pj)) ef. also 3.6.22.

CRAPTER II

MEASURE

§6. Measure functions 1. Measurable sets. Let any set E be given and let (Y be the system of all subsets .A of E. We call a set function ,p(A), defined in (9, a measure function in L (or on E) if it satisfies the following conditions: 1.) p(A) = 0. 2.) p(A) is monotone increasing.

3.) rp(SAt) 5 Ek v(Ak) k From 1.) and 2.) we obtain: (1)


0forallA e(2.

If not both cp(A) and cp(B) are infinite, then we have (1.1)

p(A) - p(B) s tp(A - B).

For if jo(B) = + co, then (1.1) is trivial; thus assume rp(B) to be finite. Since B + (A -- B), from 2.) and 3.) we get p(A) .19 cp(B -}- (A - B)) S p(B) + o(A - B). Thus, since jp(B) is finite, we obtain (1.1). A

6.1.1. Every monotone increasing, totally additive set function (p in (I is a measure function. Because of 3.1.22, 1.) is satisfied, and 2.) is assumed. Since 12 is a o-field, according to 3.2.3, 3.) is also satisfied. 6.1.11. If the measure function V is additive in (9, then 0 ie also totally additive

in L. This follows from 3.2.42.

However, a measure function is not additive in general; but the notion of measure functions in (2 is much wider than the notion of monotone increasing and totally additive set functions in (,I. This is shown by the following example of a non-additive measure function in (2: Let B be the sum of countably many ), each of which contains more than one element. disjoint sets E. (m = 1, 2, For every A e (2 let op(A) be the number of those E. which have a non-empty intersection with A.--Other examples of non-additive measure functions, in particular those from which the general theory of the measure functions started, will be found in §8. Although a measure function ip is not additive in (2 in general, we shall define a o-field it Q (2 on which (p is totally additive. The sets M of this a-field will be called (p.-measurable, and the value p(M) will be called the So-measure of M. We define: The set M e (2 is called 9-measurable if M, together with every set A e (2, satisfies the equation: (1.2)

p(A) = p(AM) + cp(A - M). 61

MEASURE

62

[CHAP. II

Because of 6.1.1, from (1.2) we obtain immediately: 6.1.12. If rp is a monotone increasing, totally additive set function in (, then every set of (9 is rp-measurable.

In the example given above, A, E,,, (m = 1, 2, - ), and every sum of these E. is 9-measurable. Every other set B e (9, however, is not p-measurable. For

if .9 designates an E. for which E,,,B 54 A and E. - B 0 A, then, for this A and for M = B, (1.2) is not satisfied, since p(A) = p(AB) = V(A - B) = 1. We now discuss the gyp-measurable sets in general. 6.1.2. In order that M be gyp-measurable, it is sufficient that (1.2) be satisfied for all A e (I with finite sp(A).

For if sp(A) is infinite, then (1.2) is always satisfied, since then Sp(A) ,p(AM) + p(A - Al), while from A = AM + (A - M) and 3.) it follows that w(A) 6 r(AM) + P(A - M). 6.1.21. A and E are (p-measurable.'

For M = A we have AM = A and A - M = A. For M = E we have AM = A and A - M = A. Hence, because of 1.), in both cases (1.2) is satisfied for every A e ( . 6.1.22. If p(M') _ + - for all the non-empty M' Q M, then M is sp-measurable. This follows immediately from 6.1.2. 6.1.23. If M is ce-measurable, then E - M is also.

For since A (E - M) = A - M and A - (E - M) = AM, (1.2) is transformed into itself, if we substitute E - M for M. 6.1.24. If M, and M2 are gyp-measurable, then M, + M2 is also.

Weset,11,+M2=N,AM,=A,,A(312-M,)=A2,andA-N=As. Since M, is(p-measurable, we have [setting M = M, in (1.2)]:

,p(A) - p(Al) + (p(A2 + A3). Since M2 is (-measurable, we obtain from (1.2) substituting A2 + A, for A, and (1.21)

M2 for Al:

p(A2 + As) = p(A2) + p(Al). Now substituting A, + As for A, and M, for M in (1.2), we get: (1.23) p(A1 + A2) = p(Al) + 0(A2)Putting (1.22) in (1.21) and taking (1.23) into consideration, we obtain: 'P(A) _ v(A, + A2) + 4r(A,). Since here A, + A2 = AN and A, = A - N, the contention is proved. 6.1.25. If M, and M2 are ce-measurable, then M, - M2 is also. According to 6.1.23 E - M, is p-measurable; by 6.1.24 (E - M,) + M2 is also; thus according to 6.1.23 M, - M2 is also. We now prove, generalizing (1.2): (1.22)

6.1.3. If M, , M2, we have: (p(A(M, +

, M, are disjoint gyp-measurable sets, then for every A e ($

+ M1)) _ p(AM,) + ... + r(AMI).

' For the example given on p. 72, A and E are the only ip-measurable sets.

63

MEASURE FUNCTIONS

§6]

We prove this by induction. For 1 = I it is trivial. Suppose, we have

for I> 1 c (A(M, + ... + M,-,)) = .p(AMI) + ... + co(AMI-,). + MI) for A, and M, + In (1.2) we substitute A(M, + + M,-, for M (1.3)

(which is admissible according to 6.1.24), and we obtain:

,p(A(M, + ... + MI) _ p(A(Mi + ... + M,-1)) + p(AM,). Because of (1.3), this is the contention. For A = E there results from 6.1.3: 6.1.31. If M, , M2, , M, are disjoint rp-measurable sets, then

ee(M, + ... + M:) = 'P(MI) + ... + 'P(MI)

-

6.1.32. If ((Mk)) is a sequence of disjoint 9-measurable sets, then for every A e Er

we have: v(S AMk) = E ,p(AMk). k

k

According to 2.) we have cp(SAMk) k

p(AM, + + AM,), and hence + p(AM,). Thus by letting l --p :

v(AM,) + z E p(AMk). But according to 3.) we have So(SAMk)

by 6.1.3: cp(SAMk) ,p(SAMk) k k

k

k

F,(AMk) k

also.

For A = E there results from 6.1.32: 6.1.321. If ((Mk)) is a sequence of disjoint rp-measurable sets, then p(Wk) k

k

V(Mk).

6.1.322. If ((Mk)) is a sequence of disjoint c measurable sets and if At Q; Mk then cp(SAk) = E p(Ak). k

,

k

In 6.1.32 one has only to set: A = SAk . k

6.1.4. If all Mk(k = 1, 2,

) are p-measurable, then M = SMk is also. k

Furthermore we can assume the Mk to be disjoint, as we can replace them by the sets Mk - (M, + + Mb_,), which are also rp-measurable according to According to 6.1.2 it suffices to prove (1.2) for all A with a finite 'P(A).

6.1.24 and 6.1.25. We set: A(M, + .. + M,)

A,, k>t SAMk = B,, and 'P(A

SAMk = B. According to 6.1.32 we have w(B) k

Mk) and p(BI) k

E ,p(AMf). Since q,(A) is finite, by 2.) and (1) cp(B) is also finite, and hence

k>t ,p(B,) --> 0.

By 6.1.24 M, + .

+ M, is se-measurable, and thus because of

(1.2) we have:

cv(A) - p(Az) + p((A -- B) + B,).

(1.4)

According to 6.1.3, we here have p(A1) = rp(AMI) + + rp(AMI), and hence .p(Al) -- l..+ rp(AMk) = .p(B). Because of 2.) and 3.) we have 9P(A -- B) k

[CHAP. H

MEASURE

64

,p((A - B) + B1) S p(A - B) + co(Bj). Thus, since tp(B1) - 0, we get tp((A - B) + Bt) -- tp(A - B). Hence from (1.4) we obtain by letting l -- ao : to(A) = p(B) + p(A - B), and, since B = SA11k and A - B = A -- k M,, this is (1.2) for M = SMk. Thus SMk is (p-measurable. 1:

k

From 6.1.25 and 6.1.4 we obtain: 6.1.41. The tp-measurable sets form a u-field. From 6.1.41 and 6.1.321 we obtain:

6.1.42. The measure function V is totally additive in the u-field of the 0measur-able sets.

6.1.5. If (p(M) = 0, then M is c-measurable.

tp(A) S tp(AM) + p(A -- M). Because of 2.) and 3.) we have tp(A - M) But because of 2.) and (1), tp(M) = 0 implies that tp(AM) = 0 also. Thus from

the inequality there results: V(A) _ p(A.M) + p(A - IV); that is, M is tp-measurable.

From 6.1.5 we obtain immediately because of (1) and 2.) : 11, then M' is tp-measurable. 6.1.51. If ep(11) = 0 and M' According to §4,4 we can formulate this also in the following way: 6.1.511. The or-field of the (p-measurable sets is complete for tp.

6.1.512. If p(M) = 0, then for every A s C we have: tp.(A - M) = sp(A). This follows from 6.1.5 and (1.2), since p(AM) = 0 because of 2.) and (1). 6.1.52. If rr(A) is f nice and if there is a o-measurable M C; A with c(M) = p(A), then A is also gyp-measurable.

Since M is tp-measurable, we have tp(A) - c(M) + p(.4 - 111) according to Hence tc(A - M) = 0. Thus because of 6.1.5 A -- M is tp-measurable, and therefore M + (A - M) = .4 is also sp-measurable as a result of 6.1.24. 6.1.53. If M C A Q M', if M is p-meae-arable, and c(11' -- 111) = 0, then A (1.2).

and M' are also tp-measurable and tp(.4) = c(21I) =

By 6.1.5 Al' - 111 is tp-measurable. and hence by 6.1.24M' = M + (M' - M) is also ip-measurable; then by 6.1.31 p(111') = y-(M).

Since A -- M M' - M,

by 6.1.51 A -- Eli is tp-measurable and we have p(A - M) = 0. Thus since A = M -1- (A - M), by 6.1.24 A is also 9-measurable and, according to 6.1.31, we have tp(A) = tp(11)(=jp(M')). Btattooa.+ruy: The theory of measure functions and of cp-measurability is due to C. C.1ttATHAonoxr. Na.chrichten Gen. d. Wiss. Goi;tingen 1914, p. 404; [11, Chap. V(and p. 359);

cf. also Sitzungabericl,te Bayer. Akad. d. Wise. 1938,p.27. Subsequent to him: H. HAHN 111, p. 424.-As to particular concepts of measure, from which the general theory of measure started, cf. the bibliography to 18, 2.

2. Outer and inner 4 -measure. Let tp be a measure function in. (9. We designate by M the tp-measurable sets and for every A e (Y we set: 2 This. condition is essential. Example : Let M be sq,-measurable set with v(M) -+a* and let _\ be a non-c-measurable net with NM - A (cf. the example above). Furthermore set A = M ± N. Then because of 2.), sa(A) - 4. also; but A is not, -measurable, sine, if it were, according to 6.1.25, A - M - N would also be p-measurable.

§6]

65

MEASURE FUNCTIONS

V> (A) = inf p(M),

(2)

Vx(A) = sup pp(M); MC A

M?A

we call Sox(A) the outer gyp-measure of A and cx(A) the inner p-measure of A.' We always have So(A) S rpx(A).

px(A)

(2.1)

A, such that rp(M) = -,X (A).

6.2.1. To .every A e Cr there is a gyp-measurable M

According to (2) there is a sequence of (p-measurable sets M; Q A, such that ,o(Mi) -* cp'(A). Set M = DM;. Then because of 6.1.41 and 1.2.4, M is also

gyp-measurable and from A cM a M. we get ox (A) 5 p(M) ; p(Mi). Thus from 9(M;) - lox (A) it follows that v(M) = ,x(A). Analogously one proves: 6.2.11. To every A e ( there is a go-measurable M c A, such that cp(M) = c x(A) . 6.2.2. The outer measure i8 a measure function. It is clear that wx has the properties 1.) and 2.) of a measure function. Thus we have only to prove that 3.) is also satisfied for cox. According to 6.2.1 there is a 9-measurable Mk Q Ak, such that .p(M,) _ px(Aj). Then by 6.1.4 SMk is k

a V-measurable set Q SAk ; thus cpx(SAk) S yv(SM,). k

k

k

we have cp(SMk)

But according to 3.)

E pox(A,). Hence cX(SAk)

k

k

k

6.2.21. The inner gyp-measure has the following properties: 1.) 2.) rpx(A) is monotone increasing. 3.) If the At are disjoint, then

F,,px(Ak). k

(A) = 0.

,px(SA,) ? Ek cax(Ak) k

That px has the properties 1.) and 2.) is clear. According to 6.2.11, there is a c-measurable Mk C A,;, such that gp(M,.) = px(A0). Then, because of 6.1.4, SMk is a 9-measurable set c SAk ; thus wx(SAk) z p(SMJ. But since the Mk k

k

k

k

are disjoint and immeasurable, we have ip(SM,) = E p(M,) = E px(Ak) k

according to 6.1.321.

k

Hence s(SA,) r E cx(As) k

k

k

6.2.3. If A is c-measurable, then VX(A) _ ,p(A) = svX(A).

This follows from (2), since now A itself belongs to the v-measurable sets

M;2 AorMCA.

OX possesses the following converse property : 62.31. If v(A) is finite` and px(A) = ye(A), then A is ip-measurable. According to 62.11 there is a c-measurable M C A with p(M) = 1 (A). This the proposition follows from 6.1.52.

From 623 and 62.31 we obtain, because of (2.1): ' Because of 6.1.21, the definitions (2) have meaning for every A e 2. 4 Forte the analogue is not true; cf. No. 4.

' That this condition is essential, is shown by the example of the set A of footnote 2, p. 64.

[CHAP. 11

MEASURE

66

6.2.32. In order that A be ip-measurable, it is necessary and, if p(A) is finite', also sufficient that spx(A) = x(A). 6.2.4. If M is ,p-measurable, then for every A e L:

V(M) - px(MA) + px(M - A) = ,px((.11A) + csx(1I - A). According to 6.2.11 there is a,p-measurable A

MA, such that c(A) = csx (Af-4).

Then because of 6.1.25 M - A is also ,p-measurable and M - d Q M - A; ,p(M - A). But since A is -p-measurable, we have ,p(M) = ,x(MA) + cpx(M - A). On the other v (A) -1- qp(M - A), and thus c(M) hence ,px(M - A)

hand, by 6.2.1, there is a 0-measurable A a 111 - A, such that gp(A) _ 1ox(M - A). Then M - A is also ,p-measurable and M - A a MA; hence ,px(MA) z ,p(M - A). Since M a A + (M - A), we have ce(M) 5 ,p(A) + (p(M - A) as a result of 2.) and 3.), and thus

w(M) - px(M - A) + vx(MA) also. Hence ,p(M) = ox(MA) + ,px(M - A). If we here substitute B = E - A for A, then, since MB = M - A and M - B = MA, we obtain also w(M) = ,px(M.4) + px(M - A). Analogously to 6.1.32 we have: 6.2.41. If ((Mk)) is a sequence of disjoint ,p-measurable sets, then for every A e 12 ,px(AMk). we have: gox(SAMk) _ k

AM, M. According to 6.2.1 there is a cD-measurable A We set SMk such that ,P(A) _ ,px(AM), and substituting AM for A we can assume: A Q M.

We now set At, = AMk ; then we have A = SAk and, since these sets are ,p-meask

urable, from 6.1.42 it follows that E o(Ak) = ,p(A) (_ ,px(_id1)).

Since

k

AMk c Ak , we get from 6.2.2 and 6.2.3: ,px(AMk) 5 ,,x(,41.) = sp(Ak), and hence

px(AMk) ; ox(AM). But as SAMk = AM, by 6.2.2 we have also k

iox(AMk) ? ,x (4M). From 6.2.41 we obtain, in the same way as 6.1.322 was derived from 6.1.32: If ((Mk)) is a sequence of disjoint p-measurable sets and if Ak C Mk, then 5v (SAk) _ E cp'(A&). k

k

If we here set M, = M, M2 = E- M, Al = AM, and A2 = A - M, then we obtain: 6.2.412. If M is ,p-measurable, then for every A e

we have

,ox(A) = p'(_1 l1) + p'(A - r11 j

(2.2)

In other words: Every ,p-measurable set is alsopx Analogously one proves: 6.2.42. If ((Mk) is a sequence of disjoint gyp-measurable sc(s, then for every A e (i we have: ,px(SAMA) _ E jpx(4Mk). -rneasurable.

k

MEASURE FUNCTIONS

§6}

67

6.2.421. If ((Mk)) is a sequence of disjoint rp-measurable, sets and if AA: Q M>I5 , Ox(Ak).

then(SAs) _ k

6.2.422. If M is tp-measurable, then for every A e ($ we have:

cx(A) ' cex(AM) + ccx(A - M).

(2.3)

In other words: Every rp-measurable set is also px-measurable. We now prove the converses of the theorems 6.2.412 and 6.2.422.

6.2.43. If M is `px-measurable and ife .px(M) is finite, then M is p-measurable also.

Since px(M) is finite, there is a gyp-measurable set M, Z M with a finite we have according to (1.2): ip(M,)(- cx(Ml)). As M is ,px(M,) - ,x(M,M) + eX(M, - M), and hence px-measurable,

4,X(M1 - M) = to(MJ - 00P.

(2.4)

Since M, is rp-measurable, because of 6.2.4 we have: yc(M,) = (px(M) + AM, - M). Putting (2.4) into this equation, we get V(MI) _ p(MI) -(cx(M) - ,px(M)), and hence(M) - ox (M) = 0. Thus, according to 6.2.32, M is cp-measurable. 6.2.431. If E = SEi , where the E. are `o-measurable sets with finite i

then

every `px-measurable set M is c-measurable also.

According to 6.2.412 E; is `px-measurable also. Thus, because of 6.2.2 and 6.1.41, E;M is `px-measurable s,lso, and pX(EiM) is finite. Hence by 6.2.43

the E;M are tp-measurable, and thus M = SE;M is also ip-measurable as a i

result of 6.1.4.

Analogously one proves both the following theorems': 6.2.44. If M is jpx-measurable and if there is a tp-measurable set M, finite,p(M,), then M is ,p-measurable also. 6.2.441. If E = SE; , where the E, are (p-measurable sets with finite i

M with a then

every c-measurable set M is ,p-measurable also. .6.2.5. For every monotone increasing sequence ((Ak)) of sees of Cr we have: ,px (Lim AA,) = lim cx(Ak). k

k

According to 6.2.1 there is a So-measurable Mk Z Ak with cp(Mk) = Sox(Ar,). We set Nk = D Mi . Then, because of 6.1.41 and 1.2.4, Nk is also cs-measurable

ilk

' That this condition is essential, is shown by the following example: Let E be an infinite set. Furthermore let p(A) be the number of elements of A if this number exceeds 1, let p(A) a 2 if A consists of a single element, and let p (A) = 0. Then no set M other than A and E isp-measurable. For let a, e Al and at a F. - M, and set A - +a, , a,{ ; then (1.2) is not satisfied. Thus px(M) - + cc for every M # A, and a 0 for Al - A; that is, every M 44 is rX-measurable.

For the proof of 6.2.441 one has to make use of the fact that the intersection of two px-measurable sets is also .px-measurable. Butt hat the system of the,px-measurable sets form a field, and hence according to 1.2.1 a ring, follows from the analogies of 6.1.24 and 6.1.25 for the px-measurable sets (which can be proved in the same way as there).

(CHAP. u

MEASURE

68

A,& - A and and At c Nk C Mk ; thus po(Ni) = lpx(A.) also. We set Lim k Lim Nk = N; then N is gyp-measurable and N

Since ((Nk)) is monotone

A.

k

increasing, we get from 3.2.2: `p(N) = lime ,p(Nk) = Jim SX(A,), and hence from k

k

A C N we obtain also px(A)

lim ceX(A,).

On the other hand, since A

Ak,

k

we have px(A) z jpx(AL.), and thus Vx(A)

lime X(Ak) also. k

Analogously one proves: 6.2.51. For every monotone decreasing sequence ((Ak)) of sets of 11 we have, if

not all y <(A,) = + cc8: 4Px (Lim AA;) - lime vx(Ak). k k

BIBLIOan+YHY: The same as in No. 1; in addition: A. ROSENTHAL, Nachrichten Gee. d. bliss. Gottingen 1916, p. 305; F. RIDDeR, Acts math. 73 (1941), p. 131. According to F. HAUSDORFF [11, p. 419, 6.2.5 is not true for monotone decreasing sequences ((Al,)) and 6.2.51 does not hold for monotone increasing ones.

3. Measure-covers and measure-kernels. A 9-measurable set Ax

A is

called a measure-cover of A (for rp) if for every 0-measurable M we have: (3)

c(AXM) _ 9,x(AM).

A gyp-measurable set A X a A is called a meaatue-kernel of A (for (p) if for every (p-measurable M we have: rpx(AM). o(AxM) If in particular the set A is gyp-measurable, then A is both a measure-cover and a measure-kernel of itself. 6.3.1. In order that the p-measurable set B 2 A be a measure-cover of A, it is necessary and, if px(A) is finite', also sufficient that cp(B) = (px(A). (3.1)

NECESSITY: This results, if in

(3) we set M = E.

SUFFICIEwcY: Let

,p(B) = iox(A) be finite and M so-measurable. According to (1.2) and 6.2.412

we have: 4p(B) = q,(BM) + cp(B - M), rpx(A) = cpx(AM) + cX(A - M), and hence

p(BM) + co(B - M) = px(AM) + ,X_(A - M). rpx(AM), cp(B - M) cpX(A - M). Since B Q A, we here have p(BM) (3.2)

Both these inequalities are compatible with (3.2) only if we have the equality sign in both cases. Thus we have p(BM) = ipx(AM), as (3) requires. Analogously one proves: s That this condition is essential, is shown by footnote 5, p. 15, as for the function p considered there we have wx - p, according to 6.1.1 and 6.1.12. s This condition is essential. Example: Let A and M be two disjoint p-measurable sets with p(A) - p(M) = + m (for instance, cf. the example at the beginning of No. 1). We set

B = A + M. Then because of 2.) and (2.1) we have p(B) - p(A) - ,X(A) - +ao; but q,(BM) - ,'(M) = + o and pX(AM) - pX(A) - 0. Thus B is not a measure-cover of A.

69

MEASURE FUNCTIONS

161

6.3.11. In order that the rp-measurable set B Q A be a measure-kernel of A, it is necessary and, if tox(A) is finite10, also sufficient that cp(B) = ,px(A).

From the definition we obtain immediately: 6.3.2. If B is a measure-cover (measure-kernel) of A, then every immeasurable

set C with A C; C c B (with A a C a B) is also a measure-cover (measurekernel) of A.

6.3.21. If B is a measure-cover (measure-kernel) of A, if C is cp-measurable,

if C =, B, and if C .2 A (C G A), then C is also a measure-cover (measurekernel) of A.

Since C =, B (§4, 2), we have cp(C - B) = 0 and 4p(B - C) = 0. Thus as C = B + (C - B) - (B - C), for every immeasurable M we have: cp(CM) _ w(BM) + cp((C - B)M) - cp((B - C)M) = p(BM). Hence in (3) we can substitute C for Ax = B. 6.3.22. If there is a measure-cover (a measure-kernel) B of A and if M is a immeasurable set

A (or c A), then there is also a measure--cover Q M (a measure-

kernel LD M) of A.

According to 6.3.2, BM (or B + M) fulfils that which we desire. 6.3.23. If B is a measure-cover (a measure-kernel) of A and if N is ,p-measurable, then BN is a measure-cover (a measure-kernel) of AN..

This follows immediately from the definition. 6.3.3. In order that the w-measurable set B D A (B

A.) be a measure-cover

(a measure-kernel) of A, it is necessary and sufficient that

wx(B - A) = 0

(wx(,l - B) = 0).

NECESSITY: If cpx(B - .4) > 0, then there is a w-measurable M Q B - A with,p(M) > 0. Then AM = A and hence wx(AM) = 0, but BM - M and thus cp(BM) _ cp(M) > 0, so that for B (3) is not satisfied. SUFFICIENCY: If B is not a measure-cover of A, then there is a immeasurable If, such that wx(MA) < p(MB). Thus, according to the definition of wx, there is a So-measurable C Q MA, such that ip(C) <
Therefore MB - C is a w-measurable set with ip(MB - C) > 0, and, since

MB - CCB - A, we havecpx(B-A)>0.

6.3.31. The propositions "B is a measure-cover of A" and "E - B is a measure-

kerml of E - A" are equivalent. For since B - A = (E - A) - (E -- B), both statements are equivalent to the proposition .px(B - A) = 0 according to 6.3.3. 6.3.32. If A is a measure-cover of A k, then SA,x is a measure-cover of SAk . k

Let M be a yrmeasurable set C SAC` -SAk k

have M = SMk. Since Mk c Ak k

.

k

If we set Mk = Ak M, then we

k

At, we have cp(Mk) = 0 according to

6.3.3, and hence also V (M) = 0. As this is true for every co-measurable set 10 This condition is essential, as may be shown by interchanging the roles of A and B in the preceding footnote.

[CHAP. II

MEASURE

70

If C; SA,< - SAk, we have lpx(SAk - SAk) = 0 and the contention follows k

k

k

k

from 6.3.3.

For measure-kernels an analogous theorem is not true (even for finitely many terms); this is shown by 7.3.2. But we have: 6.3.33. If ((Mk)) is a sequence of disjoint p-measurable sets, if At Q ML , and Akx is a measure-kernel of Ak , then SAkx is a measure-kernel of SAk . k

k

Let M be a (p-measurable set C SAk - SAkx ; we have M = SJL1Tk . k

k

Since

k

Al, - Akx , and because of 6.3.3 ,px(4k - Akx) = 0; thus lp(MMk) = 0. Therefore also p(M) = 0, hence the Alk are disjoint, we have 11MIk

,px(SAk - SAkx) = 0, and the contention follows from 6.3.3. k

k

6.3.34. If B is a measure-cover (a measure-kernel) of A and if M is p-measurable, then B - M is a measure-cover (a measure-kernel) of A - M.

According to 6.3.3 /px(B - A) = 0. Thus, because of the properties 1.) and

2.) of 6.2.21, we have also lpx((B - A) - M) _ px((B - M) - (A - M)) = 0. Hence, again by 6.3.3, B - M is a measure-cover of A - M. 6.3.4. If /px(A) (or px(A)) is finite, then there is a measure-cover (a measurekernel) of A. This follows from 6.2.1 (or 6.2.11) and 6.3.1 (or 6.3.11).

6.3.41. If A = SAk , where all px(A#,) are finite, then there is a measurek

cover of A.

This follows from 6.3.4 and 6.3.32. 6.3.411. If A = SAk, where all lpx(Ak) are finite, then there is a measure-kernel k

of A. There is a lp-measurable Mk Ak with a finite p(M,). We set Nk - Mk + Mk-1); then the Nk are disjoint pp-measurable sets. Furthermore (M, + we set AN,, = Bk ; then A = SBk and Bk Nk. Since lp(N,) is finite,,px(B,) is k

also finite, and according to 6.3.4 there is a measure-kernel Bkx of Bk. because of 6.3.33 SBkx is a measure-kernel of SBk(= A). k

Then

k

6.3.42. If E - A - SBk , where all cpx(Bk) are finite, then there is a measurek

corer and a measure-kernel of A.

For by 6.3.41 and 6.3.411 this is true for B -- A, and hence because of 6.3.31

also for A. We shall now prove a supplement to 6.1.41. For this purpose we give the following definitions" :

Let 9 be a system of sets, and to every complex (n, , (k = 1, 2,

n2 ,

,

nk)

) of natural numbers let a set M,,,,,,...,,k E 9)1 be attached; then the

11 Cf. H. Hahn [2), P. 339.

71

MEASURE FUNCTIONS

§6l

constitute a Suslin scheme e5 in D1. The set Dlnlni"'nknk+l ni of sets C5 (for k < 1) is called a successor of Ma,,.,.. nk Now to every sequence Y = ((nk)) of natural numbers we form the intersection

M. = M,,, Mn, n,'

M,,,,...,k

; then the set A (S) = SM. (where the

summation extends over all sequences r of natural numbers) is called the nucleus of S. Every such nucleus A(() is also called an analytic set over T1. The Suslin scheme e is called monotone if we always have M,,,,1...n* h+1 c Mnln,...,k .

If e5 is a Suslin scheme in l'l, then the successors of the fixed set M,.,n,...nk E S

form also a Suslin scheme in Y'l; we designate it by and its nucleus by If the nucleus A ((Z) of every Suslin scheme t in alt belongs to x] '',12 then TZ is called a Suslin system of sets." 6.3.5. If ip is finite, then the rp-measurable sets form a Suslin system of sets. We have to prove: if the sets . of the Suslin scheme S are 4p-measur-

able, then its nucleus A = A ((Z) is also. We can assume the scheme S to be monotone"; then we have According to 6.3.4, for A and Q

A,.,,.,...,k there are measure-covers Ax and AX,,.,...,,. Since A,,,.,...,. M,1n,...,k , there is also a measure-cover r. M,.,,,...ak of 6.3.22. Hence forthwith we can assume: Ax,,,,...,,, Q M,,,,,...,k

.

C;

according to Now we set:

Ax _ SA* - B, A*,n,...,k - SA*,a,...,k, = B,,,,...,k , and C = B -{n S

B,,,.,...,k , where

n1 as ... nk)

(n, , n2,

, nk)

the summation extends over all complexes

of natural numbers. Since A - SA, and A * is a measure-cover

of A., by 6.3.32 SA* is a measure-cover of A, and hence p(B) = 0.

One sees

in the same way that (e(B,.,,,...,k) = 0. Hence 4p(C) = 0 also, and thus ,p(Ax - C) = ,(A x) _ fox(A). We now prove: Ax - C C A. Let a e Ax - C; then a e A x, but a - e C, and hence also a i., e B (= Ax - SA *) . Thus a e SA * ; n

that is, there is an n, , such that a e A*, . But since a

n

a C, we have also

e B,, (= A*, - SA*, ,), and hence a e SA that is, there is an n2, such n n that a e A*,,, . Continuing in this way we obtain a sequence ((nk)) of natural numbers, such that a e A*,,,...,k for all k. Thus since AA *, n1"'^k M,,,1...,, , we have also a, e for all k, and hence a e A. Therefore, if a s A x - C, then ad A also; that is, Ax - C C A, as contended. Since furthermore Ax - C is gyp-measurable and sp(Ax - C) _ 4px(A), we obtain (A) = rpx(A), a

and hence by 6.2.32: A is 4p-measurable. 1t In other words: If every analytic set over 0 belongs to V. 1i Cf. H. Hahn 121, p. 543. 14 According to H. Hahn 121, theorem 40.1.2.

[c31 AP. II

MEASURE

72

6.3.51. If E = ,SE1 , where all gPX(E,) are finite, then the (p-measurable sets form a Suslin system of sets.

According to 6.3.4 there is a measure-cover E; of E6 ; hence E = SEx and all c

,p(Ex) = ((E,) are finite. Again let the sets of the Suslin scheme Z be *-measurable and let A = A (C) be the nucleus of Cam. If M is any set and we

designate by e' the Suslin scheme of the sets M - M,,,,,,... nk , then we have

M-A = A (r'). We here set M = E. Since all ExMf,are measurable and since w(E Ax) and all of 6.3.5 that EXA is also rp-measurable.

are finite, we obtain from the proof Thus because of 6.1.4 A = SExA.

also immeasurable. BIBLIOGRAPHY: The same as in No. 1.-To theorem 6.3.51: N. LUBIN and W. S nPIa7sla, Bull. Acad. Cracovie 1918, p. 44; N. LUBIN, Fund, math. 10 (1927), p. 25; W. SIaapi-mcI, C. R. Soc. Be. Varsovie22 (1929), p. 155; Fund. math. 21 (1983), p. 29; E. SZPILR,-IN, r una. math. 21 (1933), p. 229; S. SASS [21, p. 50; he proves this theorem quite generally, v. ithout an assuiaption as for R; cf. also the theorem 6.4.4.

4. Regular m e a s u r e functions. We always have x(A) k ov(A). If /px(A) = c(A) for all A e (1, then the measure function V is called regular. 6.4.1. If 1P is any measure function, then qpx is a regular measure/unction. According to 6.2.2 oX is a measure function. Thus we have to prove that (0) X = mx. According to (2.1) (3x)x it From the definition (2) we get (,x)X(A) = inf ,px(N), if we designate by N the sex-measurable sets. Since 0,

NQA

by 6.2.412 and 6.2.3 every So-measiurable set M is also Sex-measurable with j,X(M) = p(M), we have inf 0(N) 6 inf r(M); that is, also (wx)X(A) px(A). NQd

MR4

6.4.2. In order that the measure function p be regular, it is necessary and sufficient that for every A e (I there be a c-measurable M a A with qv(M) = ip(A). NECESSITY: This follows from 6.2.1, since qPx Strnicixwcy: This is obvious. Because of 6.1.1 and 6.1.12 we obtain from 6.4.2:

6.4.21. Every monotone increasing, totally additive set function p iv (F is a regular measure function.

The (non-additive) measure function given as an example at the beginning of No. 1 is also regular because of 6.4.2.-Example of a non-regular measure fanctiontm: Let E consist of at least three elements. Let qp(A) - 0, ,(E) = 2, and for every other B e ( let jp(E) = 1. Every such B is not so-measurable; for if a1 a B, a2 e E - B and if we set A = {at at, a:1, then c(A) - 1, p(AB) = 1, and fp(A - B) - 1; hence (1.2) is not satisfied. Therefore because of 6.4.2 the measure function V is not regular. From 6.3.1 we now get: 6.4.22. If Sp is a regular16 measure function, then in order that the measurable 16 The example given in footnote 6, p. 67, is also a non-regular measure function. 1' That this condition is essential here (for necessity) and in 6.4.8 (for sufficiency), is shown by the example of a non-regular measure function given above.

MEASURE FUNCTIONS

§6l

73

A be a measure-cover of A, it is necessary and, if p(A) is finite", also set B s icient that p(B) = p(A). 6.4.3. If Sc is a regular measure function, then in order that the set M be Q-meaaur-

able, it is necessary and at#icient that (1.2) be valid for all p-measurable A with finitep(A). Nacassri'r: This is obvious. Suvirierg scy: According to 6.1.2 we have to prove that for every B with finite p(B) we have:

p(B) - v(BM) + p(B - M).

(4)

According to 63.4 there is a measure-cover Bx of .8, and by 6.4.22 we have p(Bx) - p(B). Since Bx is p-measurable, we have by p(BXM) + p(BX - M). Here we have 9,(BXM) assumption: p(Bx) Thus let p (B) be finity.

,p(BM) and p(Bx - M) z p(B - M), and hence p(BM) + p(B - M)

p(Bx)(= p(B)). Since on the other hand certainly p(BM) + p(B -- M) k p(B), we have proved (4). 6.4.31. If p is a regular's measure function, then to every non. measurable set B there is a p-measurable set A with finite O(A), such that BA is also non-p-measurable.

Since B is not p-measurable, according to 6.4.3 there is a p-measurable set A

with finite p(A), such that p(A) < p(AB) + p(A -- B). Now AB is not tp-measurable; for otherwise A - B - A -- AB would be also p-measurable according to 6.1.25, and hence, since A - AR + (A - B), we would have:

p(A) - p(AB) + p(A - B). If So is a regular measure function, then 6.2.431 is true without any condition as to E, since now pX = p. But then the same is the case also for 6.2.441; in fact, we now have: 6.4.32. If p is a regular measure function, then every ox-measurable set is also to.measurable.

Instead of 6.4.32 we at once prove somewhat more:

6.4.321. If p is a regular" measure function and if (2.3) is valid for every p-measurable A with finite p(A), then M is p-measurable.

If in 6.2.4 we interchange M with the p measurable A and take into considera-

p, then we obtain: p(A) - px(AM) + p(A - M) >o p(AM) + tion that px px(A - M), and from this by addition:

2p(A) - px(AM) + px(A - M) +.p(AM) + p(A - M). Since by assumption (2.3) is valid and A is (p-measurable with a finite p(A), there results from it that p(A) _ cp(AM) + p(A - M). Hence the proposition follows from 6.4.3. 6.4.4. If rp is a regular ?ne-inure function, then the p-measurable sets form a Suslin system of sets.

Using the same notations as in 6.3.5 we have to prove: if the sets of the Suslin scheme C( are rp-measurable, then the nucleus A = A(e) of this scheme is also.

If A is not 4p-measurable, then according to 6.4.31 there would

17 This condition is essential according to footnote 9, p. 68. 3" That this condition is essential, is shown by the example of footnote 6, p. 67.

[CHAP. II

MEASURE

74

be a (p-measurable M with a finite cp(M), such that MA would not be p-measurable either. But this is not possible. For if we designate by 15' the Suslin scheme of

the sets M M,,,,,,...,,, , then we have MA = A(t') and, since 4e(MA) and all 5...,) are finite, again from the proof of 6.3.5 it follows that MA is 'P(M. ,p-measurable. BIBLIOGRAPHY: The same as in No. I.--By C. CARATBgODORY [1I and H. HAmr (II the

definition of the regular measure functions was given in a narrower sense, corresponding to that which is here called an ordinary regular measure function (cf. §7,1).

5. A method for construction of regular measure functions. Again Jet E be any

set and (9 be the system of all the subsets of E; furthermore let ia- be a nonempty field C (a. Now let 4, be a monotone increasing and totally additive set function defined in a'; then we can extend ¢ to a regular measure function in 11; that is, we can construct a regular measure function p in (9, such that every set Q e a is to-measurable and for all these sets: -p(Q) _ ¢(Q). Let a-, be the smallest o-system over a (§1,2); now first we define c in , in the following way: If B e , , then we set 4p(B) = sup 4,(Q) (Q a a).

(5)

Q cs

For every B e a- we have p(B) _ ¢(B), since now among the sets Q Q B the set B itself occurs and 4, is monotone increasing. 6.5.1. If ((Q,,,)) is a monotone increasing sequence of sets of a and B = LimQ,,,,

then v(B) = rim Since ((Q,,,)) is monotone increasing, ((¢(Q.))) is also monotone increasing; hence there exists rim 4'(Qm) and, since Q. Q B, it follows from (5) that

,p(B) z rim 4,(Q.). -According to (5), to every z <
that Q C; B and 4'(Q) > z. From B = Lim Q. and Q Q B it follows that Q = Lim QQm, where ((QQm)) is monotone increasing. m

Since 4' is assumed to

be totally additive in a and QQm e 6, there results from 3.2.2 that >'(Q) = Jim ¢(QQm). m

Thus rim ¢(QQm) > z and, since 4,(Q.) m

4'(QQ.), we have also

rim 4(Q.) > Z. As this is true for every z <
From the definition (5) we obtain immediately: 6.5.11. The set function (p(B) is monotone increasing in

,.

6.5.12. For every sequence ((B.)) of sets of a, we have: ip(SB.) 5 F ,p(B.). 19 Instead of that we can also say, according to 3.2.3 and 3.4.42: let i4 be monotone increasing and additive in j5, and let Q. SQm e a imply #(SQ,,,) ; o(Q,,,). w

m

,,,

75

MEASURE FUNCTIONS

$6]

We set SB., = B. Since B. a a., we have B. =

SB.,,

,

where Bm, a J.

Thus .B - S B,., . From this we get a representation B =SC; , where the C; are C

disjoint sets of tj and every Cs is a subset of a certain B,., and hence also of :_. certain B.. Now let Q e B1 a. Then Q = SC,.Q, where the C,Q are disjoint

sets of W. Thus since ¢ is totally additive in J5, we have ¢(Q) _ E 4,(C:Q).

n is a subset of a certain B. . Let, Q,ni , Qm2, Since COQ ,Q,. Ci , every set C.Q

+ Q-;) _ Q,,. , be all the sets CiQ that are QB., . Then from 'Y(Qml -{4'(Q.,,) + . + 4'(Q.,i) and from (5) it follows that 4,(Qmi) + . + J,(Q..7) p(B,.), and thus E T(Qm,) 5 ,(B..) also. Hence ¢(Q) __ E'(C:Q) S E sp(Bm) and therefore, according to (5),

IP(B) s E P(B+.). 6.5.13. The set function r(B) is additive inn. . Let B' and B" be any two disjoint sets of a.. Then in W there are monotone such that B' - Lim Q, and increasing sequences ((Q'W)) and

B" = Lim Q,. Since B' + B" = Lim (Q; + Q.), we get from 6.5.1 p(B' + B") = lim '(Q; + Q;'). Since Q;Q;' = A and ' is additive in , we have ¢(Q; + Q:') = ¢(Q;) + '(Q;). But because of 6.5.1 lim y'(Q;) _ po(B') and ,p(B" ).

Hence ,p(B' + B") = lim #(Q. + Q.') = ,,(B') +

y

p(B").

F

6.5.14. The set function ,p(B) is totally additive in &. This follows from 3.2.42 because of 6.5.11, 6.5.12, and 6.5.13. Now we extend the definition of ip to the whole system C in the following way: If A e ti, then we set

r(A) = inf o(B)

(5.1)

BPA

(B f .),

if there is no B e J5, with B Q A, then we set rp(A) _ + oo.

If in (5.1) A e a, in particular, then we have no contradiction, since now among the B Q A the set A itself occurs and hence, because of 6.5.11, we get

inf v(B) = p(A).

BPd

6.5.2. The set function sp defined by (5.1) is a measure function.

It is evident that the conditions 1.) and 2.) of a measure function are satisfied. We prove that the oondition 3.) is also satisfied. This certainly is the case if E,p(A*) = + oo. Thus we can assume that all p(Ak) are finite. Then accordk

ing to (5.1), to every e > 0 there is a Bk a a., such that Bk Q At and So(Bk) < yo(Ak) + e

.

Setting B = SBk we have B Q SAk . According to 6.5.12,

JnoAS

76

[caAP. U

p(Bb) < ,F p(A,,) + e. Since this is true for every e > 0, by (5.1)

p(B) 5

we get: p(SAb)

T p(A,,).

r

6.5.21. If p is the measure function defined by (5.1), lien every B e $, is p-measurable.

Since the p-measurable sets form a c-field according to 6.1.41, it suffices to prove that every Q e a is 9-measurable. According to 6.1.2 we have to prove

that for every A e 4 with a finite p(A) we have: 4(A) - p(AQ) + p(A - Q). To every z > 40(A) there is a B e a., such that B Q A and so(B) < z. We set B' - BQ and B" = B - Q. Because of 1.2.3 we have B' e a.. But B" a a, also; for since B = SQ., (Q» a j5), we have B" = S(Q.. - Q), where Q. -- Q e 15.

Sine B - B' + B" and B'B" = A, by 6.5.13 we have c(B) - p(B) + p(B"). From AQ C B', A - Q Q B" we get p(AQ) g p(B') and p(A - Q) p(B"), and hence p(AQ) + rp(A - Q) p(B) < z. Since this is true for every z > p(A), we have: p(AQ) + p(A - Q) S ,(A). But on the other hand, as so is a measure function, we have: ip(AQ) + p(A - Q) k V(A). 6.5.211. If p is the measure function defined by (5.1), then every set of R,+ and of as, is p-measurable. This follows from 6.5.21, since according to 6.1.41 the ss-measurable sets form a c-field and, by 1.2.4, this is also a 6-field. 6.5.22. The measure function p defined by (5.1) is regular.

We have to prove: px(A) = p(A); that is, according to (2): p(A) - inf p(M), M :)A

where M designates the p-measurable sets. But this is an immediate consequence of (5.1), since by 6.5.21 every set B e 5, is p-measurable. If is a monotone increasing and totally additive set function defined in the field , then we designate the regular measure function p defined by (5) and (5.1) in (1 as the measure function co associated with 4.. Furthermore we designate the c-field of the p-measurable sets by a. Because of 6.5.21, we have 9 Q a and, according to 6.1.511, tt is complete for p. Therefore we call a the complete cover of 9- with respect top. Thus by 6.1.42 we obtain: 6.5.23. Every monotone increasing and totally additive set function 4' defined in c (i can be Mended to a regular measure function p, defined in (, which is totally additive in the d -field a containing 6 and complete for p. the field

Nov., let a and St be two fields c (, and let J, and £ be totally additive and monotone increasing set functions defined in a and St, respectively. We designate the measure functions associated with 4, and j by p and x, respectively; and let a be the complete cover of a with respect to p. Then we have:

6.5.3. If it c ,Q Q a and if x(K) - p(K) for all K e St, then x(A) = ,(A) also for all A e (S.

At first let B e St, ; then B - SB.,(B., a St) and, if we set K, = B, , K.

B. - (B, +

L + B»_,) (m > 1), we have B - SK., , where the K. a St are dis-

161

77

MEASVBN FUNCTIONS

joint. Since according to 6.5.14 x is totally additive in R, , we have x(B) E X(Km), and hence from K. e R we get also x(B) = Y,co(K,). Since 0 is a a-field 2 R, we have K. e @ and B e a, and thus, as by 6.5.23,p is totally additive in a, we have 4p(B) = E ,(K.) also. Hence x(B) = c(B).-Now let A e W. By

(5.1) we have c(A) = inf cp(B')(B' a W,) and x(A) - inf x(B)(B a R.). Since SSA S'2A we have a. Q R. , and thus, because of the etatemsat just proved, X(B') _ cp(B') for all B' e , . Hence all values p(B') whose in5mum is p(A) are found among the values x(B) whose infimum is x(A). Therefore x(A) 5 co(A).

On the other hand, as proved, we have X(B) = y(B) for all B e R, .

Thus p(A) 5 p(B) = x(B) for all B Q A belonging to L. Hence co(A) 5 inf x(B)(B a R.); that is, v(A) 5 x(A). Therefore x(A) - q.(A).

SPA

If A is the complete cover of R with respect to x , we obtain from 6.5.3:

6.5.31. If W tr R Q § and if x(K) _ V(K) for all K e R, theca A = a.

In the remaining part of No. 5 we shall mainly use the following assumption: let E have the form E = SEE, where the EE are v-measurable with finite p(E,). (5.2) in this case we have E e l5, . For since p(Ei) is finite, there is a Bi a { with BE a Ei according to (5.1), and hence E = SEE = SBi . i

i

6.5.4. If op is the measure function (5.1), if M is .-measurable, and if c(M) is finite, then for every a> 0 there is a B e {, and a C e Ws , such that B a M Q C p(M) i' v(C) > co(M) - e. and cp(M) + e > O(B) According to (5.1) there is a B e 15,, such that B M and yo(B) < p(M) + e; then we have cp(B - M) < e. Again according to (5.1) there is a B' a a,, such

that B' Q B - M and (p(B') < e; thus cp(B - B') = p(B) - y(B'B) > p(M) - e and B - B' a M. Since B e Jy, and B' a a., we have B = SQ. and B' = SQ;, (Qm e J5 and Q; e a). Because of §1 (1.21) we have B - B' = S(Qm - B') _ SD(Q.. - Q;,) and, since to is a field, Q. - Q,, a jy. Thus B - B' = SCm , where C. e Ws, and because of 1.2.3 it can be assumed that C. C Cm+i . Hence from 6.5.211, 6.1.42, and 3.22 we obtain:.p(B - B') = lim c,(C,). Thus, since D

,P(B - B') > ,p(M) -- e, we have also ep(C.) > co(M) - e for almost all m. 6.5.41. If c is the measure function (5.1) and cx(A) is finite, then there is a measure-kernel of A which belongs to &.. According to 6.2.11 there is a rp-measurable M C A, such that p(M) = 1pc(A).

such that C,. c M and cp(Cm) > Thus because of 6.5.4 there is a C. s 1 M C A, and p(C) co(M) - m . Setting C = SCm , we have: C e a, , C ,p(M) = cpx(A). Hence, by 6.3.11, C is a measure-kernel of A.

[CHAP. II

MEASURE

78

6.5.42. If o is the measure function (5.1) and if (5.2) is satisfiedR°, then for every A e ( t ere is a measure-kernel which belongs to as, .

+ Ei_1) (i > J.); then the Mi are We set M, = E, , M; = E; - (EI + ip-measurable with finite ip(Mi) and disjoint, and E = SMi. Furthermore we i

set Ai = AMi ; then A = SAi and p>(Ai) is finite. According to 6.5.41 there i

is a measure-kernel A ix e as of A, . Because of 6.3.33 S.4 ix is a measure-kernel of A, and since Aix e t'cao , we have also SAix e as, i

6.5.421. If rp is the measure function (5.1) and if (5.2) is satisfied, then for every A e C there is a measure.-cover which belongs to a., .

According to 6.5.42 there is a measure-kernel M of E - A which belongs . Because of 6.3.31 E - M is a measure-cover of A. Since M es, , we have M = SDQ,n,(Q,a, e a), and hence by § 1 (1.21) E - M = DS(E - Qm,). to

From (5.2) we have E e a-.., say E = SQi(Qi a a). Then E - Qm, = S(Q; -- Q,,,) and, since t1 is a field, Qi - Q,,, , e j , whenceS(Qi - Q.,) e a,; that is, E - Q., e

a..

i

and thus DS(E - Qm,) = E - M Therefore S(E - Q.,) e m Now we obtain as a counterpart to (5.1) : 6.5.43. if (5.2) is satisfied, then the inner measure associated with the measure unction(5.l) is represented b y px(A) = sup c(C)(C e as). C.C A

According to 6.5.42 there is a measure-kernel Ax of A which belongs to R,;_

By 6.3.11 we have 9x(A) = p(Ax). Since Ax a as., we can set

Ax = S(',n(Cr a as , C. C Cm+,). Therefore from 6.5.211, 6.1.42, and 3.2.2 we obtain: P(Ax) = lim or(('.), whence there results the contention. M

The function m constructed above by (5) and (5.1) is totally additive and monotone increasing in the u-field g of the gyp-measurable sets, which is complete

for co, and is Q a. Now we prove that under the assumption (5.2) is the smallest u-field, containing a and complete for the function concerned, into which the function ¢ given in a can be extended to a totally additive set function, and that this extension is uniquely determined: 6.5.5. If (5.2) is satisfiedu, if e5 2 a is a c field consisting of subsets of E, 20 The assumption (5.2) is essential here as well as in 6.5.431 and 6.5.43. This is shown by the following example: Let E be an uncountable not and let 5 be the c-field of the countable subsets of E; thus - 6. - t5+ - $,+ - 1r,, . Furthermore let -*(Q) - 0 for Q e $. Then by (5.1) jp(A) - 0 orip(A) - +ao, according as the set A e (I is countable or uncountable. (5.2) is not satisfied, since E - e 8,, . An uncountable set A e It does not have a measure-kernel belonging to Iii, , nor a measure-cover belonging to $,s . Furthermore, since.p is totally additive in (1, we get i,X(A) - P(A) as a result of 6.1.12. 21 That this assumption is essential, is shown by the example of the preceding footnote; for there we can choose 6 - (f (= 1R) and X(A) - 0 for all A e %.

7(

MEASURE FUNCTIONS

§6]

if x is a totally additive and monotone increasing set function defined in C, if e is complete for x, and if for every Q e a- we have: x(Q) _ 4(Q), then c C and for all M e l we have: x(M) _ V(M).

Since 9 Q R and Z D a, and since a and C are a-fields, we have:

,a , C 2 , , and S Q

.-First let M e a.. Then there is a monotone

increasing sequence ((Qm)) belonging to a, such that M = Lim Qm . Since 0

,p and x are totally additive in and in e5, respectively, we get from 3.2.2: w(Q.) - p(M) and x(Qm) --' x(M). Since Qm a $, we have x(Qm) _ P(Q.), and hence also x(M) _ ,(M) for all M e a, .-Now let M e a-.a and let p(M) be M and P(B) finite. Then according to (5.1) there is a B e 11, , such that B is finite. Since M e }r.# , we have M - Lim M, , where M, a `u. and ((M,)) ,s monotone decreasing. If here for M, we substitute the set MB which also belongs to a. by 1.2.3, then we can assume that M, C; B and thus that (PUN is finite. Thus, since M = Lim M. and ,p is totally additive in 9, it follows from 3.2.21: v(M,) -- P(M). From M, e a, we obtain, as already proved, x(M,) = 9(M,.). Since therefore x(11,) is alsc suite and since x is totally additive in C, we get also: x(M,) --> x(:t1), and hence x(M) = sp(31) for all M e U.a with finite (p(M).-Now let M be an arbitrary set of with p(M) finite. Ac-

cording to 6.3.421 there is a measure-cover B of M belonging to aa ; then

,p(B - M) - 0. Again by 6.5.421 there is a measure-cover C of B - M belonging to a., ;.we have P(C) = 0, and hence, as already proved, x(C) - 0 also. Thus since x is monotone increasing and

is complete for x, it follows from

M Q C that B - M e C and x(B - M) = 0 also. Since B e ilf,a Q C and M = B. -- (B - M), we have also M e C. Since B e &a and ,p(B) is finite, we B

have, as already proved, x(B) = v(B). Thus from B Z M it results that x(M)

is also finite, and hence x(B - M) - 0 implies that x(M) = x(B). Therefore as to(B) = y.(M), we have x(M) = p(M) also. Thus we have proved that for every M e a with a finite p(M) we get M e ( and x(M) _ ip(M) --Finally let M be an arbitrary set of 9. We set N, = E, , Ni - Ei - (E, + + E,_,) (i > 1) ; then the Nt are disjoint sets of a with finite p(Ni) and E = SNi . Furthermore

we set MNi = N;; then M - SN; , where the N, are also disjoint sets of J with finite cp(N;).

As proved, we have Ni e C and x(N;) _ P(Ni). Thus since E is

a a-field, we have also M e Z and x(M) _ E x(I) _ E p(N;) = p(,11'). Now, under the assumption (5.2), we can improve the statements of 6.5.3 and 6.5.31 somewhat. Keeping the notations used there we obtain: 6.5.51. I f (5.2) is s a t i s f i e d, i f 9 Q .Q a 9, and i f x(Q) _ o(Q) for all Q e a, then x(A) _ p(A) also fur all A e (I and hence A _ R.

From 6.S.5 it follows for C = A that 9 Q R and x(M) = ,p(M) for all M e . Thus the contention results from 6.5.3 and 6.5.3 1.

and hence also for all K e St.

0 That this condition is essential, is shown by the last two footnotes.

80

i tsu to

[CHAP. II

We designate by a-, the system of the Borer sets over rr(41,2). Since a- is a field, a8 is the smallest v-field over r, and hence, since @ is a v-field over we obtain a, C 8. Thus -p is totally additive and monotone increasing in e ; then by the method of 44,4 we can form the smallest a-field °, over a-s which is complete for p (of. 4.4.241 and 4.4.25). Since 9 is also a v-field complete for .p and containing as, we obtain: W°a C a. 6.5.6. If (5.2) is satisfied", then = a-i As we have just seen, a-a C g. Since according to 6.5.5 a is the smallest a-field over g which is complete for e, we have also { Q ; . An important example for the method being used in §6,5 is as follows: In the n-dimensional Euclidean space R. we consider the system of all the sets which are sums of a finite number of half-open intervals f = [a, , a2 , ,a; (cf. 48,1). This system forms a field $ and each of these sets , bl , 62 , -

is also a sum of a finite number of disjoint half-open intervals L' If f = [a, , as I ... , a. ; b, , b2, ... , b,.), we set: 44(1) = (b, - al) (b2 - a,) ... (b,. + to , where the f, are disjoint half-open intervals, and if Q = I1 + f2 + we set 4'(Q) = 44(1,) + 4,(h) + P(A) ; then 4, is a totally additive and monotone increasing set function in a. Applying the preceding theory to 4,, we obtain a measure function ,.A) in R., which is called the n-dimensional outer measure of A. We will not go into details here, since we will return to this subject in §8,2. 6. Extension of a totally additive set funtion. The theory developed in No. 5 makes it possible to extend a totally additive set function 4G defined in a field 5 Q 4e into a v-field containing 5. In j we form the positive-function t4+ and the negative-function V of the given function 4, (13,4); these are totally additive and monotone increasing set functions in t5 (3.4.3 and 3.4.32). According to No. 5, they can be extended to regular measure functions ,1(A) and p2(A) defined for all A e Q (6.5.2 and 6.5.22). By 6.1.41 and 6.5.21, the pi-measurable sets form a v-field a, Q a and the rp,-measurable sets form a v-field 2 D 5. and according to 6.1.42, pi and rp, are totally additive in g, and g2, respectively. Thus TZ = , - t` 2 is a o-field in which both ce and rp, are totally additive. By 6.1.21 E e , and E e , , and hence E e 1'1 also. If and only if either 4p, (E) or V2(E) is finite, the difference V = V, - gyp, can be formed and represents a totally additive set function in 1J for which 11lrp = 46.

6.6.1. Let V = sp, - s be the totally additive set function in the v-field l l = -v2, constructed as above; then in T1 we have: P+ _ ,p, and V = (P2. We prove rp+ _ p, ; the demonstration of c = So, is analogous.-Because of 23 According to H. Hahn ['l J, theorem 38.2.8.

24 That this condition is essential, is shown again by the erairple of footnote 20, p. 78; for there $a - ltla - e7, but 5 - (Y. " Cf. R. Huhn [2;, theorein 20.3.2.

81

CONTENT FUNCTIONS

§71

(M) for M e 9)t, and hence q (M) = -I- oo implies P+(M) _ + eo, and vice versa. Thus further on we need only to consider those M for which vi(M) and p+(M) are finite.-First

3.4.61 we have cp(M) - fei(M) - , 2(M) - so+(M) -

we have So+(Q) = VI(Q) for every Q e 15. For according to §3(4) pI(Q) = J ,+(Q)

_

sup ,y(X) (X a Q1 a) and yv+(Q) = sup p(X) (X a Q1 TI). Thus since 9t Q and tp(X) _ ¢(X) for X e a-, we obtain vi(Q) So+(Q). But as j. - Sot - V for every Spz(Q), and hence p+(Q) from 3.4.62 it follows that v+(Q) Q e W: Then similarly as at the beginning of the proof of 6.5.5 (only there substituting Tt for C, V, for So, and p+ for x), one proves that V+(M) - VI(M) for M e as , and thus all the more for M e as.-Since 9)2 Q W is a c-field, we have

0 Z as,. Now let M e as. ; then M = Lim M, , where M. a as and ((M.)) is monotone increasing. Thus since rp+ and v, are totally additive in TI, from 3.2.2 we get Sp+(M,) --> p+(M) and "(M,) - SP,(M). Therefore as M, a as implies rp+(M.) = pi(M,), we obtain p+(M) = pi(M) also.-Finally let M be an arbitrary set of 0 (with 9x(M) finite). Then according to 6.5.41 there is a measure-kernel C of M belonging to &,; that is, C C; M and VI(C) = WI(M). Because of the result just obtained we have VI(C) = p+(C). Since C c M, we have p+(C) S rp+(M), and hence Spl(M) 5 v'(M). But according to 3.4.62 fv+(M) 6 SI(M). Thus m+(M) = So,(M). Sp be the above constructed totally additive set function in 6.6.2. Let So = V, the a-feld T2 - uI . & ; then 871 is complete for {p.

Let M e S1)1 and M = A; then according to 4.1.11 p+(M) = So (M) = 0, and hence, because of 6.6.1, VI(M) = Sp,(M) = 0 also. Now let X e MI 12. Since S* and Sps are monotone increasing and GUI and

2 are complete for SOI and p, ,

respectively, we obtain X e aI and X e &. Thus X 6

I

: = V also; that is,

$f1 is complete for p. BISLroOSArHY to Nos. 6 and 6: M. Fn4cawr, Fund. math. 6(1924), p. 227, (referring in part to C. DE LA VAI44E POUSsIN (21, p. 83); H. HAmr, Annali 8ouela Norm. di Piss (2) 2 (1933), p. 433; J. you NarsiANN [1), p. 95, 164; B. JsssnN, Mat. Tidoskrift B 1934, p. 78; 1938, p. 16; 0. NIaoiiu, Acad. roy. Belgique, M4moires (2) 17 (1938), feet. 7; A. P. Monse,

Bull. Am. Math. Soc. 50 (1944), p. 723; of. also Ergebnisse der Math. 5 (1937), No. 2 (E. Hors), p. 2.--(An analogous method on the bade of another notion of measure has been need also by E. Tom.izz [11, 12.)

17. Content functions. 1. Ordinary measure functions. Now let us take a metric space E (12,3) as a basis. As in §6 we designate by (rC the system of all subsets of E. We call a measure function p defined in q, an ordinary measure function if every closed set

(and hence by 6.1.23 also every open set) of the space B is measurable. Every monotone increasing, totally additive set function in Cr is an ordinary, regular measure function because of 6.421 and 6.1.12. Other not so trivial examples of ordinary, regular measure functions will be described in §8,2 and

§8,6.-The example given at the beginning of §6,1 represents a regular, but

CHAP. II

MEASURE

32

no;. ordirary measure function. The examples given in footnote 6, p. 67 and at the beginning of §6,4 are non-regular and non-ordinary measure functions. 7.1.1. In order that the measure function rp be an ordinary measure function, it i necessary and sufficient that for every two sets A and B of Cr whose distance 0 ($2, 3) we have: jp(A + B) = p(A) + p(B). NECESSITY: Let A° and B° be the closures of A and B, respectively (§2,1). By assumption A° and B° are gyp-measurable, and since At > 0, A° and B° are B°, from 6.1.3 it follows that (p(A + B) = clisjuint. Thus as A Q A° and B of A) .4- V(B). - SUFFICIENCY: Let M be closed. According to 6.1.2 we have to prove that §6 (1.2) is true for all sets A with finite p(A). Since by condition

3.) of §6,1 F(A) 5 y(AM) + p(A - M), we have only to prove: S(AM) + p(A - M) 6 e(A). (1) Since M is closed, we have aM > 0 (§2,3) for every a e A - M. Thus if we designate by At the set of all the points a e A - M satisfying the inequality aM > 1, and in general by Am(m > 1) the set of all the points a e A - M satisfying the inequality

m

_ 11 > aM

m

then A - M - SA, . 0

Since every

two of the sets A, , A, , Ac, have a positive distance, we have by assumption: .p(AI + As + .4- A2,,._1) = e(AI) +,p(As) + ... + pe(A2m-,). Furthermore, A - Bbl C A, we have ge(A,) + .o(A,) + .. + since A, + A 3 + ... + Azm--,

jo(A), and hence E p(A,;_,) < v(A). Thus as cp(A) is finite,

sp(A2m-,)

V(A2,_1) ---+ 0.

In the same manner one obtains

p(AV) --+ 0, and hence

(p(A1) - 0 also. Setting S A; = B., we get by condition 3.) of §.6,1: i>m cp(A 1), and thus so(B.,) s 0 also. Now A - M = A, + + e>m

i>m

A m -+,p(Bm).

B., and hence p(A, + . + A m) $ 9(A - M) 5 s'(A, + Thus as a result of V(B,,,) --+ 0, we obtain:

(1.1)

Since A, +

.{- A m) +

p(A, + .....{- A m) --+ p(A - M). + A. and AM have a positive distance, we have by assumption

,(A?11) + w(A, + ... + Am) _ (p(AM + A, + ... + Am) 5 c(A). Hence, because of (1.1), there results (1). If in the metric space E we designate by ta- the system of the closed sets and by (3 the system of the open sets, then the Borel sets over (zp + ®) (§l,2) are called the Borel sets in EY6, and the analytic sets over (( + (1) (§6,3) are called the analytic sets in E27, 7.1.2. If rp is an ordinary measure function, then every Borel set in E is 'P-measurable.

By 6.1.41 the system of all (p-measurable sets is a o-field which, according to " Cf. H. Hahn (2), p. 264. 91 Cf. H. Hahn [2], p. 344.

§7l

CONTENT FUNCTIONS

83

the definition of the ordinary measure function, contains all the closed sets; and" the system of the Borel sets in E is the smallest c-field containing all the closed sets. 7.1.21. If ep is an ordinary, regular measure function, then every analytic set in E is rp-measurable.

The analytic sets in E form ° the smallest Suslin system of sets (§6,3) over the

closed sets in E. These are p-measurable according to the definition of the ordinary measure function, and hence, by 6.4.4, the analytic sets in E are also. Using 6.3.51 one proves in the same manner: 7.1.211. If rp is an ordinary measure function and if E = SE; , where all 9p'<(Es) i

are finite, then every analytic set in E is rp-measurable.

Again let rp be an ordinary measure function. A gyp-measurable set Al (;-,5 A) is called always of positive measure if every set G, which is open in M and F6 A, also , A.

7.1.3. If V is an ordinary measure function and if the space E is separable (§2,2), then every ip-measurable set Ill with 1p(M) > 0 contains a measure-kernel C which is closed in M and always of positive measure. be a distinguished system of sets, open in E (§2,2), , G,,, , Let G, , G2 ,

and let H, be those G. for which 31G. =, A. Then C = M - SH, accomplishes that which we desired. Indeed since S11, is open, C is closed in M. From MII, _, A we obtain, according to 4.1.32, SMH, =, A also, and hence, because of 6.3.21, C is a measure-kernel of M. Furthermore let G be a set, open in E and with GC =, A. We represent G in the form G = SG,,;; then G.,,C =, A, and hence by 4.1.32 G.,{M = G. ,C + SG.,H,M =, A also. Thus every G,,; is an H, ,too, and hence G

SH, and GC = A.

Therefore C is always

of positive measure. We call a measure function p continuous if for every countable set A a ( we have:,p(.A) = 0. Then in the v-field of the cp-measurable sets the totally additive set function

7.1.31. A measure function p is continuous if and only if for every a e E we have:ce({aj) = 0. For if this condition is satisfied and if A = jai, a2, - , ak, is a countable set of (, then because of property 3.) of §Gj we have: ce(:l) S E o({ ak l) , {

k

and hence c(A) = 0 also. Now if tip is continuous and Al is closed, then the measure-kernel C established in 7.1.3 can be assumed to be perfect. For according to §2 (1.1) C can he decomposed into the nucleus C,r and the " According to H. Hahn [21, theorem 33.4.42. "According to H. Hahn [21, §40 (3.2) and theorem 40.2.21.

MEASURE

84

[CHAP. 11

scattered part Cs ; that is, C - Car + Cs (with C1Cs = A). There, mince C is closed, by 2.1.71 C1 is perfect; and mince C is separable (2.2.1), Cs is countable'o, and hence, as a is continuous, p(Cs) - 0. Thus by 6.3.21 Cx is a measure-kernel of M. We can also formulate the theorem just proved in the following manner: 7.1.32. If the ordinary measure function p is continuous in the sepsrsble space E, then every closed set F with p(F) > 0 contains a perfect subwt Is, iMiek is always of positive measure, with p(P) - p(F). BInuooiAPnY: The same as in *6,1; in addition: J. Araveosnevs, Portugaliae Math. 3 (1942), p. 258; A. P. Moass and J. F. RArtnoLrH, Trans. Amer. Math. Bee. 66 (1944), p. 242--

C. CAaATHSODORY, loc. cit., ealls "measure function" that which is called "ordinary measure function" here; this notation is due to H. HAHN [I], p. 430. EYamples of ordinary, but not regular measure functions may be found in C. CASAI=Aosoay (I], p. 363, and 2°d

edition (1927), p. 693.-From the result of S. SASS mentioned in the bibliography to §6,3 it follows that 7.1.211 is true also without the assumption for E.

2. Content funtions. We call an ordinary measure function a content function if for every set A e Q there is a Greet M Q A with p(M) = p(A). Since according to 7.1.2 every Ga is p-measurable, there follows from 6.4.2: 7.2.1. Every content function is an ordinary, regular measure function. The converse of this theorem is not true. Example: Let E be the space RI;

let p(A) = 0 for every countable set and p(A) = + co for every uncountable set A e (I.

Then p is totally additive in (( and hence an ordinary, regular measure

function. Now let A. be the set of the rational points of RI . Since every As has the power MV 3' we get p(Ao) = 0, p(Me) _ +-, and thus p is not a content function. Examples of content functions will be found in Gi-set Mo

§8,2 and §8,6 (cf. also footnote 33, p. 85). Now from 6.3.1 (since sx = p) there results the following improvement of the part of 6.3.4 referring to measure-cover: 7.2.2. If p is a content function in 9, then for every A e 4 with finite p(A) there is a measure-cover which is a Gs. 7.2.21. If p is a content function and M is a Go with p(M) finite, then there is a measure-kernel of M which is an F..

By assumption M = DG., where the G., are open. Every G. is an F. ,'Y 0 &qFm; , where we still can assume: F.; Q F,,, +,. say Then M = and hence by 3.2.2: p(M) - lim p(MFm;). Thus for a > 0 there is an i, such ;

that p(M) --- p(MF,,) < e. Hence, if we choose e '° According to H. Hahn [2], theorem 18.5.8, for instance. 11 Cf. H. Hahn [2], theorem 19.2.21. s Cf. H. Hahn [2), theorem 10.7.41.

, there is a closed set

85

CONTENT FUNCTIONS

17)

F. C G. , such that v(M) - p(MF.,) < -1 . Setting F = DF,. , we have F

DG., that is, F Q M, and by 11 (1.21) M - F - NS(M - MF.,), and thus M

(M - F) a rp(M) - ys(F') s

(.a(M) - ,p(MF,,.)) < 'I. So for it s 1k we

obtain a closed set F(}' a M with o(e') > 9p(M) - k . Then C

S1?0'

is a F. contained in M with p(C) = y(M), and hence by 6.3.11 C is a measurekernel of M. Now we can improve the part of 6.3.4 referring to measure-kernel: 7.2.22. If 9 is a content function, then for every A s (I with sox (A) finite there is a measure-kernel which is an F. . According to 6.3.4 there is a measure-kernel C of A, and so cp(C) = yvx(A).

By 7.2.2 there is a measure-cover C' of C which is a Go , and hence ,(Cx) = ,p(C) = rpx(A). Again by 7.2.2 there is a measure-cover Ox of Cx - C which is a Gs, and since cp(Cx) = sp(C), we have V(Cxx) = 0. According to 7.2.21 there is a measure-kernel K of Cx which is an F,, and ,p(K) = *(Cx) = rpx(A). Now we set Ax - K -- Cxx. Since Cxx is a Gs, its complement -Cxx is

an F., and hence Ax, which can be written as K (-Cxx), is also an F,.

But Ax = K -- Cxx Q Cx - CxX C Cx - (Cx - C) = C a A,

and t,(Ax) _ v(K) - i,(KCxx). Therefore since p(K) - ysx(A) and

,p(Cxx) = 0, we have: ,p(Ax) _ 4px(A).

Thus, by 6.3.11, Ax is a measure-

kernel of A. 7.2.23. If 0 is a content function and if A - SA; where the V(AS) are finite, i

then there is a measure-kernel of A which is an F. .

A{ with cp(M4) _ 9p(A;); thus by By 6.4.2 there is a p-measurable M; assumption p(M,) is finite. We set M = SM{. If we set M1 - M" also, and the M{ are M; = Mi -- (Ml + . + M,_t)(i > 1), then M = SM, { c-measurable and disjoint; tp(M;) is also finite (since C M;), and hence ,px (AM {) is finite, too.

Thus by 7.2.22 there is a measure-kernel C{ of AM; which

is an F.. Then C = SC; is also an F, and C a A. Since A = SAM, by 6.3.33 C is a measure-kernel of A. Now because of 6.3.31 we obtain from 7.2.23.-

7.2.24. If ,p is a content function in E and if " E = SEi where the q(E{) are 4

u That this condition is essential, is shown by the following example: Let v(A) - 0 and ,(A) - +oo for every non-empty A e 1,1s. Then every A is ,o-measurable (thuaqX wx - v) and v is a content function. Now let Ace 11 be a set which is neither a Gi nor an Fa , and let A c V c :Ac C B, where B is a G6 and C is an F, . Then there are two points a s Ao ,

e C and b e B, b .A,. If we set M, - jai and M, _ {b;, then F(A0M,) - -}-.o, ,(CM,) - 0, ,p(.4 M1,) 0, and v(BM,) = + ac. Thus B is not a measure-cover and C is not a measure-kernel of Ao. a

CHAP. 11

MEASURE

8(i

,finite, then for every A

E there is a measure-cover which is a Ga and a measure-

kernel which is an F. . 7.2.25. V ruler the assumptions of 7.2.24, for every 9-measurable set M C E there is an F,-set C C 31 with C. =, M and a Ga-set B L M with B =c, M. + oc 't'his follows immediately from 7.2.24, if ce(M) is finite. For y,(M) one has also to consider 6.3.3. 7.2.251. Let the assumptions of 7.2.24 be satisfied; then in order that M C F be gyp-measurable, it is necessary and sufficient that M be the sum of an F,-set and a zero-set for p.

\ECESSITY: It follows from 7.2.25. SUFFICIENCY: It follows from 6.1.24 because of 7.1.2 and 6.1.5. I n the same way we obtain : 7.2.252. Let the assumptions of 7.2.24 be satisfied, then in order that M C; E be vp-measurable, it is necessary and sufficient that M be the difference between a Ge-set and a zero-set for p.

7.2.3. If

is a content function and if A = SAi where the p(A;) are finite, then i

A with tP(F) > z. According to 7.2.23 there is a measure-kernel Ax of A which is an F.. Thus cv(Ax) = cpx(A) and Ax = SFm , where F. is closed and F. C F.,+1 . By

for every number z < cpx(A) there is a closed set F

3.2.2 cp(F,.) --> cp(Ax) (=cpx(A)), whence the contention results. From 7.2.3 we obtain immediately: 7.2.31. If cp is a content function, if A is cp-measurable, and if cp(A) is finite, then for every s > 0 there is a closed set F C A with cp(A - F) < e.

For continuous (No. 1) content functions 7.2.3 and 7.2.31 can be improved upon, because of 7.1.32, in the following way: 7.2.32. If the content function cp is continuous in a separable space E, then the closed sets F of 7.2.3 and of 7.2.31 can be assumed to be perfect.

According to 7.2.1, every content function is a regular measure function; that is, px(A) = p(A) for all A s (2. Therefore in the counterpart of 7.2.3, only cp will occur; this theorem, however, will be true only under stronger conditions. 7.2.4. If cp is a content function, if E = SEi where the E; are open sets" and the i

cp(Ei) are finite,"' and if A a E with a finite cp(A), then for every number z > (,(A) there is an open set G Q A with cp(G) < z. 3' The condition that the Ei be open sets, is essential here and cannot even be replaced by the condition that the Ei be Gi-sets. Example: Let ((a,)) be a sequence of different. paints of E with a, a, and for every .4 c E let to(A) - the number of the points a, contained in A. Then qp is totally additive in 11 and, since by 2.1.4 E - Sla,) - Be is a Ge-set

with .(Eo) - 0, v is also a content function, and we have E - Be + Sla,l. However, E is not a sum of open sets with finite cp and the contention of 7.1.4 or of 7.2.41 is not valid, =a) we have.p(G) - +oo and also rp(G - )a)) _ +°. since for every open set G 31 In particular the condition is satisfied if p(E) itself is finite.

§7)

CONTENT FUNCTIONS

87

We set z - p(A) = e(> 0). According to 7.2.24, 7.2.1, and 6.4.22 there is a G3-set B D A with gp(B) = p(A). Now we set BE; = B, . Because of 2.1.41 B; is the intersection of a monotone decreasing sequence ((G;'')) of open sets, where we can assume G; Q E; . Since p(E,) is finite, it (substituting G;"E; for follows from 6.1.42 and 3.2.21 that lim p(G;')) _ -p(B;). Thus by 3.1.11 there is

an open set G` Q B; with v(G"'' - B,) < 2 . We set SGT" = G; then G is open and, since B = SB; , we have G a B(MA) and G - B S(G`'' - B;). c

Thus w(G - B)

(p(G"' - B,) <

e, and hence by 3.1.11

,p(G) - V(B) < e. Therefore, sincep(B) _ p(A), we have also qo(G) - lo(A) < a;

that is, (p(G) < z. 7.2.41. If p is a conient function, if E = SE; where to E; are open setsd" and the p(E;) are finite,'5 and if A is cp-measurable., then for every e > 0 there is an open

set G Q A with p(G - A) < c. According to 7.2.24, 6.3.3, and 6.2.3 there is a G,-set B A with V(B - A) = 0. Now in the same way as in the proof of 7.2.4, we obtain an open set G B(MA) with p(G - B) < e. Therefore, since A is rp-measurable and hence p(G - A) = jp(G - B) + sp(B - A), we also have: p(G - A) < e. BIBLIOGRAPHY: H. HAHV 111, p. 444.

3. Non-p-measurable sets. Let (p be a measure function on E. Then it is possible that every set A C; E is gyp-measurable. An example of this is every totally additive, monotone increasing set function in Q (6.1.12). It. is also possible that, except for A and E, no set is p-measura.ble. See the example of a (non-regular) set function given at the beginning of §6,4 (if there we set v(E) = 1, then we obtain a similar example of a regular measure function). But in general there are both rp-measurable sets (different from A and E) and non-se-measurable sets. For instance, see the example given at the beginning of §6,1. In order to get more precise propositions, we first give the following definitions. A set which is a G3 (§2,1) in every metric space E Q A is called an absolute Ga or a Young set36. It can be proved36 that A is already a Young set, if there is a complete (§2,3) metric space E Q A in which A is a Ga. Therefore we could

also take as a definition: A is a Young set if A is a Gb in a complete metric space E Q A. Furthermore, a set A which does not contain any perfect subset *A is called a totally imperfect set. In a separable (§2,2) Young space E, whose nucleus (§2,1) is not empty, every set A C E is a sum of two disjoint, totally imperfect subsets, A = A' + A".37 " According to F. Hausdorff [2], p. 136; cf. also H. Hahn [2], p. 126. 87 Cf. H. Hahn [2l, theorem 19.9.9. It follows from the proof of this theorem that for

E itself the representation E - E' + E" can be chosen such that both subsets E' and B" have the power K.

CHAP. H

MEASURE

88

7.3.1. If w is a continuous content function in E, if E = SEr where the,p(E4) are finite, and if E is a separable Young space whose nucleus is not empty, then every

E is a sum of two disjoint subsets, A = A' + A", such that fox(A') = 0 and qrx(A") = 0. By taking A' and A" as totally imperfect we obtain from 72.32: ,Px(A') = 0 set A

and ,px(A") - 0.

7.3.2. Under the conditions of 7.3.1, every 4p-measurable set M with ,p(M) > 0 is

the sum of two disjoint, non-9-measurable subsets,

M = A' + All, such that

jp(A") _ v(M) ,px(A') = 0,,p(A') = p(M) and ,px(A") = 0, Let M -A'+ All be the decomposition of M according to 7.3.1. From 6.2.4 it follows that px(A') = x(A") _ ,p(M), and thus, since for a content function ,px = p, also p(A') - Se(A") = ,p(M). That A' and A" are not to-measurable

is a consequence of 6.2.32. Using 6.1.511 we obtain immediately from 7.3.2: 7.3.3. Let the conditions of 7.3.1 be satisfied; then in order that every subset of M(CE) be ,p-measurable, it is necessary and sufficient that, (M) = 0.

4. Content-fike set functions. Again let E be a metric space, let TZ be a v-field consisting of subsets of E and containing all open sets and hence also all G,-sets, and let ,p be a totally additive set function defined in D2. The absolute-function of ,p is designated by ,p (13,4). Now, if to every M e 9Y there is a Greet B D M with o(B) _ ip(M), then we call p content-like.'u If according to 16,5 we extend the definition of ,p to all A C-. E by (4)

,A(A) = m -p(M)(M a 9),

then by 6.5-22,p is a regular measure function. 7.4.1. In order that yv be content-like, it is necessary and sufficient that the regular measure function O(A) be a content function. NECESSITY: Because of (4) there is a sequence ((M,)) of SD'l, such that M, 2 A and (M,) - (A). By 1.2.4 we have DM, -A e %R and, since M, Q .A A,

we get o(A) = o(A). As w is content-like, there is a (7k-set B a A, such that p(B) = e(A) - fs(A). Su,'rxcIENCY: Let M e IV'I. Since ,p is a content function, there is a Greet B Q M, such that ,p(B) = O(M). 7.4.2. If It is content-like and M = SM; where M, e TZ and the io(Mi) are finite,

then there is a F.-set C C M, such that C -, M. By 7.4.1, ,p is a content function, and hence by 7.2.23 and 3.4.14, there is a " Every content function o is totally additive in the u-field fil of the c-measurable sets (containing also the open seta), by 61.42, and hence Ip is also content-like in 17! (since here ;P - q).-The example given at the beginning of No. 2 is a totally additive, but not content-like set function in 41.

$71

89

CuNTENT FUNCTIONS

F,-set C which is a measure-kernel of M (for O(M -- C) = 0, and thus C =, M.

gyp).

According to 6.3.3,

Because of 7.4.1 there results from 7.2.25: 7.4.21: If Tp is content-like and if E - SE4 inhere Er e TQ and the q,(E4) are I

finitei°, then for every M e St there are a Gs-set B M and an F,-set C C M, such

thatB =,MandC =,M.

Because of 7.4.1 there results from 7.2.31: 7.4.22. If t' is content-like and if M a SW and jp(M) is finite, then for every e > 0 there is a closest set F M, such that op(M - F) < e, and hence I q,(M -- F) I < a also.

Yet under the same assumptions as in 7.4.22 there need not exist an open set

G a M with $(G - M) < a; for it is possible that f(G) _ -I- oo for all open sets G. In the a-field 1Q (containing the open sets) we call the totally additive set function qp strongly content like if to every M a D1 and every a > 0 there is an open G Q M with qa(G - M) < a.f0 7.4.3. Every strongly content-like pis also content-like.

Let Me W. By assumption there is an open Gk

M with sp(GG - M) <

(k = 1, 2, ); we may suppose ((Gk)) to be monotone decreasing. We set B = DGk , and thus B -2 M. If p(M) - + co, then o(B) _ + oo also. If k XM) is finite, then the p(Ga) are also and f(G,,) -- rp(M) ; therefore by 3.2.21, -p(B) - ip(M). 7.4.31. If p is content-like and if E = SRI where the E; are open sets`' and the I

fp(E,) are finite,` then p is also strongly content-like.

Let A e W. Then because of 7.4.1 and 3.4.14, the contention follows immediately from 7.2.41 (applied to tp). 7.4.4. If c, is contenUike, if E = SE, where the ER a DI and the rp(E.) are finite, and i f the totally additive set /unction 4, in Wi is q,-continuous

then 0 is also

M with B =, M. According to 7.4.21, for every M a WI there is a Ga-set B Then B - M =, A and thus B - M =+ a also. Therefore B -#M and hence, because of 4.2.44, j(B) = (M) also. Ru3moo1u PHY : The notion of the content-like set functions (with another notation) has been used already by E, TRILLING, Monatahefte fflr Math. u. Phys. 32 (1922), p. 185; of. also: J. RADON, Sitsungeberiehte Akad. Wins. Wien 122 (1913), p.1814; C. na Ir VaLI gs POUBSIN (21, p. 86.

" That this condition is essential, is shown by footnote 33, p. 85. y In R. the pg. (defined in 1)8,2) is strongly content-like according to 8.2.53, but the uk(k < n) (defined in 1)8,6) are only con tent-like (whereby 8)1 designates the o-field of the and of the , -measurable seta, respectively). 41 The condition that the Ei be open sets, is essential here and cannot even be replaced by the condition that the E{ be O,-sets. This is shown by the example of footnote 34, p. 86. 11 In particular the condition is satisfied if c(E) itself is finite.

[CHAP. II

MEASURE

90

5. A first general method for the construction of content functions." Let E be a metric space, and in the syst-eln 05 of all sets G open in E let a monotone increasing additive set function ,p(G) be given which satisfies the condition: p(SG..) < E ,p(G,). (5) r

We extend the definition of ,p to the system of all point sets A Q E in the following way [analogous to §6(5.1)]: So(A) = inf yp(G).

(5.1)

4PA

If A itself is open, no contradiction results, since then .4 is among the G

A

and hence inf p(G) = p(A). O DA

7.5.1. The function So(A) defined by (5.1) is a content function.

First we prove that p(A) is a measure function on E. By 3.1.22 the condition 1.) of §6,1 is satisfied. That condition 2.) of 16,1 is satisfied, is evident. We prove that condition 3.) of §6,1 is also satisfied. That certainly is the case if E p(Ak) = + oo ; therefore we may assume all (p(Ak) to be finite. If e > 0, k

by (5.1) there is an open Gk Q Ak (k = 1, 2,

), such that ,p(Gk) < So(Ak) +

SAk and, because of condition 2.) (§6,1) and (5), we get Then SGk k k ,p(SAk) 5 S,(SGk) 5 E,p(G,,) < Z Se(At) + e; since this is true for every e > 0, k

k

k

k

we have: p(SAO) 5 F,.p(Ak), as contended.-Now we prove that (p(A) is an k

k

ordinary measure function, demonstrating that the condition of 7.1.1 is satisfied. Let A and B be two sets of positive distance. Since by 3.) (16,1) p(-4) + ,p(B), we have only to prove: p(A + B) > SP(A) + qv(B), ,p(.4 + B) A + B we have: and this is the case if it is established that for every open G ,p(G) > ,p(A) +,p(B). The neighborhoods UA, and (In, are open (12,3), and for p < ZAB by §2 (3.11) we have UA, URP = A.

For G a A + B we set GI = GV A,

p(G1 + G2) _ and G2 = O's, then GA = A also and by 2.) (§6,1): ,p(G) v(A) + Sp(B).-Finally we prove that ,p is a content func9,(GI) + v(G2) tion. By (5.1) there is a sequence ((Gk)) of open sets, such that Gk ,p(G,,) -+ p(A). If we set M = DGL., then M is a Ga-set with Al

A and A and

SP(M) _ (A).

A method for the construction of functions o(G) in the space R. satisfying the above-mentioned conditions will be given in §8,1. BIBLIOGRAPHY: It. IIAHN [11, p. 448.

§8. Content functions in the space R 1. Interval functions.

As space E we now choose the n-dimensional Euclidean

space R. (§2,3) or, somewhat more generally, an open set H c R . A second method will be discussed in §4,5.

91

CONTENT FUNCTIONS IN THE SPACE R.

§8l

In R, , the set of all points z satisfying the inequality a < x < b is called the b is x open interval (a, b), and the set of all a satisfying the inequality a called the closed interval (a, b]. Furthermore let halt-open intervals [a, b), (a, b] x < b, a < x b, denote the sets of all x e R, satisfying the inequalities a respectively. X A. For any n sets, A,, A2 , , A. the combinatory product A, X A, X , n). denotes the set of all n-tuples (x, , x, , , xw) for which xk s At(k +a 1, 2, Then their combina, n) be n open intervals in R, . Let (ak , bx)(k = 1, 2, X (aw , bw) is called an open interval of R. tory product (at, b,) X (a2, b,) X , a,,; bl , b2, and shall be designated by (a,, a2, , bw). Similarlythe combinatory product [a, , b,] X [02 , b2] X .. X [aw , bw] is called a closed interval of . , bw]. And analogously R. and shall be designated by [a, , a, , aw ; b, , b, , [al , bl) X la,, b:) X ... X [a,,, b,) and (a, , b1J X (a, , b2) X ... X (as , bwJ

are called half-open intervals of R. and are designated by [as, a, , , , an ; b,, bz+ respectively w b! , b: , r bw) and {a,, a2,

-

, aw :

A (finite) set of closed intervals of R. , no two of which have inner points in common, shall be called a simple (finite) system of intervals. Let e be a simple finite system of intervals consisting of the closed intervals If I, C A, IZ a A, . , I C A, then we write: Cam, C A. I,, , M

I..

If A a S I, , then we write: A C CAS. (An analogous notation is employed if the simple system e consists of denumerably many intervals I, .) Let I be a closed interval. If I = I, + I, + - . + I. and if I, , I= , , I,w form a simple finite system S of intervals, then G shall be called a finite subdivision of I. If the system Cam' of intervals results from CS by substituting a finite subdivision for each interval of 6, then Cam' shall be called a finite subdivision of e. Now let H be an open set in R,, . A finite set function #(I) defined for all closed intervals I C; H shall be called an interval function.

If (o again is a simple

finite system of intervals consisting of the closed intervals I',, Is,

, I. ,

then we define: VI(CE) _ 41(f1) + #(I,) + . + #(I,»). For the empty system V of intervals we set 4'(2) - 0. The interval function '(I) shall now satisfy the following conditions: 1.) For all I we have 1'(I) 0. 2.) For every finite subdivision G of I we have 1'((B) 4,(I). From 2.) we obtain immediately: 8.1.1. If C5' is a finite subdivision of (, then 4'(6') k -p(C ) (1)

If Cam' and Cam" are finite subdivisions of @r,, then the product C' X 06" shall denote the subdivision of C( which consists of the intersections of the intervals of e' and 6". Then Cam.' X Cs" is also a subdivision both of G' and of s", and hence we have by 8.1.1: 41 All these intervals, geometrically interpreted, are parallelopipeds whose edges are parallel to the coordinate-axes.

MEASURE

92 (1.1)

0M' X s")

*W);

(CHAP. It

1'W x e") z ks" ).

From an interval function 4(I) satisfying the conditions 1.) and 2.) we now derive a set function yo(G) defined for all open sets G a H by the following definition [analogous to §6(5)]: (1.2)

r(G) = sup d(") OgG

(where @S runs through all the simple finite systems of intervals in G). 8.1.2. The function p(G) defined by (1.2) satisfies the conditions of §7,5. That cp(G) is monotone increasing, is evident. Furthermore from (1.2)

it results immediately that q(G) is additive, if one takes the following fact into consideration: If G, and G2 are two disjoint open sets and if the interval I Q G, + G. , then either I a G, or I Q G, .-As to §7 (5) we first show that for any two open sets G, and G, :

o(G, + G,) 5 p(G,) + v(G2). For this purpose we observe that to every interval I Q G, + Gs there is a finite (1.3)

subdivision of I such that each of its intervals is contained either in G, or in G2 .

For if we set E - G, = F, and E - G2 = F2, then, since I a G, + 02, we have for every point a e I: max (aF,, aF2) > 0. Since this expression is a continuous function of a`5, there is a p > 0, such that max (aF, , aF2) z p in 14 '. If we subdivide I into subintervals with diameters smaller than p, then each of these subintervals is contained either in G, or in G2, as contended. Now in order to prove (1.3), let 1B be a simple finite system of intervals contained in G, + G,. Subdividing each interval of e5 into sufficiently small subintervals we obtain a system 6 of intervals, each of which is contained either in G, or in G2 ,

as has just been shown. By 8.1.1 4,(S) ? >G(e). Let Cam, be the system of those be the system of the other intervals of C which are contained in G, , and let intervals of 9. Then k({) = 0(60 + 4,(C2) and C5, Q G, , C2 C; G, . Thus by (1.2) '(Cam,) v(G1) and O(-52) ; .p(G2), and hence d(C) 6 o(G1) + p(G,). Since ¢(S) z '(S), we have 4,(C) ; p(G,) + y(G2) and, since this is true for

every C a G, + G, it follows from (1.2) that so(G, + G2) ; p(G1) + p(G,) also; hereby (1.3) is proved. From (1.3) there results by induction:

,(G, + G, + ... + G..) 5 p(G,) + so(G2) + ... + jo(G.). Now let e5 be a simple finite system of intervals contained in SG,. Accord ing to Borel's Covering Theorem" there is a finite number of the G, , say + G. . Then by (1.2) G, , G2, ' , G. , such that S Q G, + G, + (1.4)

v(G, + G, + . ' . + Cm), and hence by (1.4) (5) 5 'P(Gl) + iP(G2) + .. - + R(Gm) w For instance, of II. liahr. [2j, theorems 25.7.i az:d ":& .6.4I. '" For instance, cf. H. 11ahn [21, theorem 25.5.';1. 41, Cf. H. Hahn [21, theorem 16.6.1.

CONTENT FUNCTIONS IN THE BPACD R.

§81

93

also; therefore ¢(() - E .(G,). Since this is true for every C a SG, , we have by (1.2): '(SG,) 5 E ,(G,); that is, §7 (5) is also proved. BIBLIOORAP$Y: H. 11Ah N [11, p. 453; H. HAHN (21, p. 146.

2. The n-dimensional measure of a point set of R. . We obtain the most important application of theorems 7.5.1 and 8.1.2, if for II we choose the whole

, a.; then we set' (b,, -- a,). 4,(I) _ (b, - a,)(b, (2) 4('(1) (condition 2., No. 1), the equality sign now In the inequality ¢(C)

space R. and for ,G(I) the volume of I ; that is, if I is the interval [al , a2, bI , b, , .. , b,

,

always holds. The get function, originating from this interval function according

to (1.2) and §7 (5.1), is called the n-dimensional outer measure or the outer Lebesgue measure of the point set A and is designated by p.(A). From 8.1.2 and 7.5.1 there results: 8.2.1. u,.(A) is a content function. The corresponding inner µ,.-measure (§6,2) is called the n-dimensional inner measure or the inner Lebesgue measure of A and is designated by ix(A). According to 7.2.1 and §6,4 we always have: un (A) = A,.(A). The µ.-measurable sets are also called n-dimensionally measurable or measurable in the Lebesgue is called the n-dimensional sense. if M is a pa-measurable set, then measure or the Lebesgue measure of M.41 According to 6.2.3 and 6.2.31 we have:

8.2.2. In order that the set A C R. be 1A.-measurable, it is necessary and, if µ.(A) is finite, also sufficient, that µ,.(A) = u.x(A). From 8.2.1, 7.2.1, 7.1.2, and 7.1.21 there result: 8.2.21. Every Rorel set in R. is A,-measurable. 8.2.211. Every analytic set in R. is p,.-measurable. 8.2.22. For every countable set A Q R we have p.(A) = 0; that is, it. is continuous.

For every point a e R. , of course, p.(( a)) = 0; therefore by 7.1.31 A. is continuous, and hence µ.(A) - 0 for every countable set A C R. . 8.2.3. E v e r y o p e n interval J = (a,, as, , a. ; bl, b, , - - , b,J is p.-measur-

able and N.(J) _ (b, - a)(b, - a,) ... (b. - a.). Since J is an open set, it is a Borel set and hence by 8.2.21 µ,.-measurable. If V, is defined by (2), then because of (1), for every simple finite system of intervals (S Q J, we have: ¢M) (b1 ._ a,) (b, - a,) - - (b. - a.), and hence µ,.(J) (b, - at)(bt -- aa) ... (b. - a.). But considering (for sufficiently large k) the

system r5k a J consisting of the single interval 451n R,: If l - (a, b1, then we set #(I) - b - a. 49 Instead of one-dimensional, the term linear is also used.

[CHAP. 1 1

MEASURE

94

we have t'Mk) -> (b, - a,) (b, - a:) ... (b. -- a.), and hence m. (J) (b, - a,) (b, - a,) ... (b,. - a.) also.

8.2.31. Every closed interval I -- [a, , as , .. , a. ; b, , b, ,

, b.] is k.-meas-

urable and µ,(I) = (b, - a,) (b, - a,) ... (b. - a.). According to 8.2.21 1 is µ,-measurable.

Setting

b.+

(ata.-k;b,-+-

Jk= we have by 8.2.3: A.(Jk)

{b,

- a, + k) ... (b. - an + k

But since I = DJk , by 6.1.42 and 3.2.21 we obtain: µ.(Jk) -->,u.(I), and hence k

the contention follows. According to 6.1.53 from 8.2.3 and 8.2.31 there results: 8.2.32. Every point set A originating from the open interval (a, , a2, b, , b, ,

,

, a.;

b,) by adding frontier points is is.-measurable and y.(A)

(b, - a,) (02 .- a,) ... (b. - a.).

Since to every bounded set A Q R. there is an interval (at, a2, .

, a. ;

b,, b, - , b.) Z A, we obtain from 8.2.3: 8.2.33. For every bounded set A Q R. , µ.(A) is finite. From this there results immediately: 8.2.34. Every set A Q R. can be represented as A = SAk, where µ (Ak) (k = It

1, 2, . - -) is finite.

Proceeding from the closed intervals I of R. we obtained above the n-dimensional outer measure u,(A) for the sets A C. R, by means of the definitions (2), (1), (1.2), and §7 (5.1), having already indicated the analogy to the theory of

But now, surpassing a mere analogy, we can show that the notion of the n-dimensional outer measure in the R. Can be subordinated to the theory 6f §6,5, that is, that the above definition of µ.(A) really coincides with the definition suggested at the end of §6,5. For the latter one it was essential to take as a basis the half-open intervals 1 of R. instead of the closed intervals I. Thus in order to see the equivalence of the two definitions, we designate (as §6,5.

at the end of §6,5) by a the field of the sets which are the sums of a finite number , a. ; b, , b2 , of half-open intervals I = jai, a, , , b,), by 4,(Q) (Q s W) the function defined there in a, and by p(A) the measure function defined there

by means of y'(Q) in R.. Since by 8.2.32 1 is µ: measurable and µ (1) = (b, - a,) (b2 - a,) ... (b, - a.), and since by 6.1.42 p,, is totally additive in the cr-field of the p,-measurable sets, it follows that 4,(Q) is a totally additive set

CONTENT FUNCTIONS IN THE SPACE R.

§81

95

function in i (Ss already stated at the end of §6,5) and that j4(Q) = A. (Q) for Since every open set G R is a sum of a countable set of disjoint half-open intervals I'', there results from the total additivity of v in (6.5.14) and of A. in a. that ,(G) = A,(G) for all open sets G R R. Thus in particular for the open intervals J = (a, , a2 , , a ; b, , b.) we have by 8.2.3: V(./) = (b, - a,)(b2 - as) . . (b. - a.). Hence to every half-open interval and to every e > 0 there is an open interval./ D I with .p(J) 5 ,p(J) + e. Now let B e a,, say B == , where the half-open intervals Ik may be assumed to every Q

8/k

k

be disjoint. Now if J,

1k is an open interval with .p(Jk) 5 (P(Ik) + 2and k

if we set B* = SIk , then B* is an open set

B, for which by 6.5.12 and 6.5.14

It

E p(Jk) S E (p(Ik) + e = ,p(B) + e, and hence: to every B e 15. and to every e > 0 there is an open B* -2 B with .p(B*) ,.(B) + e. we have: p(B*)

A;

A;

Since, on the other hand, every open G e a,, for every A Q R. the values ce(A) inf jp(B) (B e [§6 (5.1)] and inf v(G) (G open) coincide. But the latter value B'A

OaA

equals inf tt (G) = µ (.4) [§7(5.1)], since ip(G) = µ.(G), as it has already been (7q A

Hence p(A) = ,u (A) for all A c R., as contended. The equivalence of the two definitions of µ,, as shown above now makes it possible to apply 6.5.5 to µ.. For this purpose we first remark that M. has the following properties: by 6.1.41 the µ,; measurable sets of R. form a o-field V, which by 8.2.31 contains the closed intervals; A. is a monotone increasing and, by 6.1.42, totally additive set function in 9)1; by 6.1.511 T1 is complete for µ, ; for every closed interval I = [a2 , at , , a. ; b, , bt , , b.] we have by 8.2.31: shown.

1,.(I) = (b, - a,) (6, - a2) ... (b. - a.). Now it follows that 9)1 is the smallest v-field which possesses these properties, and that µ.(M), for all M e 9)1, is uniquely determined by these properties. Besides, the closed intervals I may be replaced

throughout by the half-open intervals I - [a,, a2, For from 6.5.5 there results immediately:

, a. ; b, , b, ,

, b.).

8.2.4. If a is a a -field in R. containing all half-open intervals I [a, , a2, , a.; b, , 5s , , b.), if x is a totally additive and monotone increasing set function in 25 with x(I) = (b, - a,)(b, - a=) (b. - a,,), and if e is complete for x, then 5 contains the cr-field 9T1 of all µn-measurable sets and for all M e 91 we have

x0t) = u..(M).

8.2.41. In theorem 8.2.4 the half-open intervals I may be replaced by the dosed

intervalsI = [a,,M,...,a.;5,,b2,...,b.l.

First we show that every open interval J - (a, , a2 , , a ; b, , b, , . , b.) belongs to c. By assumption the closed Intervals Ik = [a, -{- , ... , a,, + k ;

b, - k ,

k since @5 is , b - fl (for suffxaently large k) belong to @'r; and hence, a

6O Cf. H. Habn (21, theorem 20.3.31.

[CHAP. II

MEASURE

96

v-field and J = Lim It , we have J e e also, as contended.

Furthermore, since

k

x is totally additive in 6, we have by 3.2.2: x(Ik) -' X(J). Analogously, as p (J) also, and hence, since by assumption I k e X12 and J e X71, we have x(I) also. From I e ( and x(Ik) = A.(Ik), we get x(J) = J e S it follows that I - J e C5 also and, since x(I) = x(J), we have X(I - J) = 0. Thus, since by assumption x is monotone increasing in e5, and (B is complete for x, we obtain: if A Q I - J, then A e S and X(A) = 0 also. Hence , , a ; 61, b2 , every half-open interval I = [a, , a2, belongs also to (S

and x(I) = x(J) = pa(J)

Now the conditions of 8.2.4 (concerning 1)

are satisfied.

From 8.2.34 and 7.2.24 there results: 8.2.5. To every A Q R,, there is (for a measure-cover which is a t and a measure-kernel which is an P.. Furthermore, because of 8.2.22 and 8.2.34 we obtain from 7.2.32.1' 8.2.51. If A Q R. , then to even/ number z < there is a perfect set P C A

with p (P) > z. From 7.2.4 or immediately from §7 (5.1) there results:

8.2.52. If A Q R,, , then to every z > p (A) there is an open set G Q A with p,.(G) < Z8.2. 52 1. I f A Q R. , t h e n to every z > p (A) there is a simple system 6 Q

A,

consisting of the intervals I, , with F, p,.(I,) < Z. By 8.2.52 t h e r e is an open set G A with *. ( G ) < z, andn G is the sum of a countable set of disjoint half-open intervals 1,. Because of 8.2.32 and 6.1.42, p (G) = F, p,.(1,). Furthermore, if I. is the closed interval corresponding to .

I, , then by 8.2.31 and 8.2.32 we have ;,.(I,) =

and hence

w.(I,) = u..(G) < z. The interval

I = [a,, a2, ... ,a,. ;b,,bs, ...

a,,

X[as,

X ... X [a,.,b ]

is called an (n-dimensional) closed cube or, briefly, a cube if (b, - a,) _

(b2 - a2) _ .. = (b,, 1. This number t is called the length of the edge of I. The corresponding half-open interval I= [a, , a2 , *,,,a, ; bl , bs , ... q b.) shall be called an (n-dimensional) half-open cube. 8.2.522. In theorem 8.2.521 C can be chosen so that all I, are cubes. Cover R. with a sequence of simple systems ((Ck)) of intervals, such that every f3 Theorem 7.2.32 can be applied here; for the space R is separable, since the open intervals J m (a, , as , , an ; b, , bt ) , bp), with a; and b;(i c 1, 2, , n) rational throughout, form a countable distinguished system of open sets (§2,2). '= Cf. H. Hahn [2], theorem 20.3.31.

CONTENT FUNCTIONS IN THE SPACE Rn

§81

97

ek consists Qf closed cubes with the lengthy of the edge and such that rook+1 is a 6' Let G be the open set used in the proof of 8.2.521. finite subdivision of ek

The cubes of v1 lying in G constitute a simple system 61*; for every k > I the cubes of (S 1#ying in G and containing no inner point of any cube of Chi ,

form a simple system (k. Now if I,(v = 1, 2, . ) des-

, Ck_1

Cam' ,

), then ignates the countably many cubes belonging to the 6k (k = 1, 2, G = SI,. For let a e G; then there is a pa > 0, such that S.,. G also; for a sufficiently large ko a cube of Vke containing a lies in S.,, , and hence a certainly If the I, are replaced by the belongs to one of the cubes of Gi , CRk, , C , . corresponding half-open cubes 1, , then G = S1, is the sum of a countable set of

disjoint half-open cubes, and the proof may be finished as in 8.2.521. R. be p,-measurable, it is necessary and sufficient 8.2.53. In order that A A and an open set G' G - A, such that to every e > 0 there be an open set G that A,.(G') < e a

NECEssrrT: By 8.2.34 A can be represented in the form: A = SAk, where k

According to 8.2.52 there is a Gk Q Ak, such that l+n(Gk) < fpn(As) + 2k. If we set G = SG,. , then G - A Q S(Gk -- Ak), and A.(Ak) is finite.

hence A .(G - A) < E = e. Thus by 8.2.52 there is a G' a G - A, such A and that ,1.(G') < e. SUFFICIENcT: By assumption there is a Gk (fork = 1, 2, . . ). We set DGk =Ax

a Gk Q G k - A, such that µn(Gk) <

and S(G, - Gk) = A x ; then A X and A x are µ.-measurable and A x Q A a A X. k

Furthermore by it (1.21) and §1 (1.13) we obtain:

A"- Ax

k

k

k

k

= DGk D(R. - (Gk - Gk)) = DGk((Rn - Gk) + G,EGk) = DGkGk. k

k

k

Since A .(G,',) <

,

k

it follows that µ,.(Ax -- Ax) = 0, and hence by 6.1.53 A is

µ.-measurable. 8.2.6. If the point set A' C R. is derived from the point set A a R. by a translation of the form xi = z; + c{(i - 1, 2, - , n), then µ.(A') = f+.(A) and ;cnx (A') _ A.x(A); and A' is A.-measurable if and only if A is µ.-measurable. For if the interval I' is formed from the interval I by translation, then accord61 If here 2k is substituted for -1 , then the lengths of the edges of all cubes are smaller

than a given p > 0. 51 C f. footnote 40, p. 89.

98

[CHAP. II

MEASURE

ing to (2): ¢(I') = ¢(1). Thus if the simple finite system Cam' of intervals is obtained from (a by translation, we have by (1) also: O(V) _ >y(' ). Now from

(1.2) there results: if the open set G' is derived from 0 by translation, then µ"(G') = p,(G). Finally from §7 (5.1) we obtain: if A_' is formed from A by translation, then µ"(A') = p,(A). Now from 8.2.53 it follows that A' is A.-measurable if and only if this is the case for A. Furthermore, since the translation taking A into A' transforms the p"-measurable subsets M of A into the pmeasit follows from §6 (2) that urable subsets M' of A' and since p"(M') = p"x(AO = µ"x(A) also.

8.2.61. If the point set A' C R. is derived from the point set A C R. by the similarity transformation x' = cxi(i = 1, 2, ... , n), then µ,(A') = I c J"p,(A) and (A') = I C I" p"x (A) ; and A' is µ,-measurable if and only if A is µ"measurable.

For if I' is obtained from I by the similarity transformation mentioned, then according to (2) : ',&(I') = I c 1"4,(1). Continuing as in the proof of 8.2.6 we establish the contention. Now we prove a theorem about sets of positive one-dimensional measure in R1 ,

which later we shall recognize as a particular case of a much more general theorem (18.2.32). 8.2.7. If A is a set of R1 with mix (A) > 0, then in A there are two points a, and a2 whose distance I a, - a2 I is a positive rational number.

Since by §6 (2) a set. A with µlx(A) > 0 contains p1-measurable subsets M with µ1(M) > 0, we may assume from the first that A is µI-measurable with µ1(A) > 0. Then since A certainly contains bounded MI-measurable subsets M with µ1(M) > 0, we may assume from the first that A is bounded. Designating

by AA; the set derived from A by the translation x' = x +

, we get by 8.2.6 k

p1(Ak) = µ,(A). If we set B = SAk , then B is bounded, and hence by 8.2.33 It

Thus the Ak cannot be disjoint (for otherwise we would have p1(B) = EpI(Ak) _ -}- -, since all µ1(Ak) = MI(A) > 0). Then a point comMI(B) is finite. k

neon to two different Ak furnishes a pair of points of A with a rational distance.

Let I be any (closed or open) interval of R,,. The denumerable set N of all

rational points of I has the measure p,(N) = 0 by 8.2.22.

Therefore Because of 8.2.22 there results from 7.2.3266: to every z < p"(1) there is a perfect set P c I - N with µ"(P) > z. Since N is dense in I and P is closed, it follows from 2.1.5 that P is nowhere dense in 1. Thus we obtain: 8.2.8. If I is an interval of R. and z < µ,(I), then there is a perfect set P nowhere dense in I with µ"(P) > z. 8 2.81. If I is an interval of R,, and 0 < z < p"(I), then there is a perfect set Q nowhere dense in I with µ"(Q) = z.

p"(I - N) = µ,(I) (>0).

Let Ma be the intersection of the set P of 8.2.8 with the open interval 55 Cf. footnote 51, p. 96.

99

CONTENT FUNCTIONS IN THE SPACE R

§8]

(-A, -h, MO

, -A; A, A, , A) of R. and let Ma be the closure of Ma . Then a P is nowhere dense in I and is also perfect (for Ma contains no isolated increases

point, since such a point would be also an isolated point of P).

continuously from 0 to µ (P) if A increases from 0 to + co, and hence, since z. µ (P) > z, there is a value of A for which We give an example in R, . Let I be a closed interval of R1 and 0 < 1

Choose a sequence ((S,)) of positive numbers, such that

µ1(I).

2'-'S, = 1. Now

from the middle of I we delete an open interval of length Sl ; from the middle of each of the remaining subintervals of I we delete an open interval of length S2 ; from the middle of each of the four remaining subintervals of I we delete an open subinterval of length S, ; and so on. Thus altogether a denumerable set of disjoint open intervals is deleted whose sum forms an open set G with µ,(G) = 2'-'S, = 1. Its complement I - G = Q is a nowhere dense, closed set; and since no two of the deleted intervals touch, Q contains no isolated point and hence is also perfect. Finally we have µ1(Q) = µl(I) - 1.-If we choose l = µ1(I), we get µ1(Q) = 0. An example for this case is the Cantor discontinuum, which

we obtain by choosing I = [0, 1] and S, = 3' .

Thus we have:

8.2.82. The Cantor discontinuum C is a nowhere dense, perfect set in [0, 11 with

µ1(C) = 0. 8.2.83. In R. there are perfect sets M 0 A with 0. By 8.2.82 this is correct for n = 1. For n z 2, for instance, every straight , n - 1) of R. furnishes a perfect set M with µ (M) = 0; line x; = c;(i = 1, 2, this follows from 8.2.31. 8.2.84. There is a µ1-measurable set A [0, 11, such that for every interval [a, b] c [0, 11 both A [a, b] 3,6,,, A and [a, b] - A 5,46,,,, A.

Let ((A,)) be a sequence of positive numbers smaller than 1, such that E ?., is finite. Furthermore let A, be a closed set nowhere dense in [0, 1] with µ1(A,) = A, (8.2.81); besides (adding a finite number of points), we can always obtain that 0 e A, and 1 E A, and that for every interval (pi, q,) complementary

to A, we have: qi - p; <

1

v

.

We set B, = Al ; let B2 be the set derived from

B1 by adding, in every interval (a;, b;) complementary to B1, the set obtained from A2 by the similarity transformation which maps [0, 1] on [a{, b;]; let B3 be the set derived from B2 by adding, in every interval (a; , b1) complementary to B2, the set obtained from A3 by the similarity transformation which maps [0, 11 on [a; , b;]; and so on. We set A = SB, . Now let [a, b] [0, 1]. Since ,

the length of every interval complementary to A. is smaller than

this is also

the case for the length of every interval complementary to B. , and hence we can choose m, such that an interval (p, q) complementary to B,,, is entirely contained 1, in [a, b]. But then A [a, b] contains a set similar to A m+1 and, since

100

MEASURE

[CRAP. II

we have by 8.2.61 A [a, b] p A also. As [p, q] B.,+I is similar to A,.+I , we have pl([p, q] - B,.+I) = (1 - 4+.+I) (q - p) and analogously II([p, q] - B.,42) =

(1 -(1 - )L,.+,) (q - p), and so on. But since A - Lim B, and

hence [P, q] - A = [p, q] - Um B,, we obtain from 3.2.21: pi([p, qJ - A) = II (1 - A,) . (q - p). Here II (I - A,) r' 0, since E A, is finite, and thus [p, qj -- A ,E,,, A, whence [a, bJ - A P6,, A also.

The power of the system of all the p.-measurable sets of R. is given by the following theorem (where t`t means the power of the continuum) : 8.2.9. In R. there are 2s. µ,.-measurable sets.

The perfect sets M C R. with p (M) = 0, given in the proof of 8.2.83, have every subset of such a set M is p.-measurable and the power tit. By 6.1.51 M.se there are 2s. subsets of

Since there are only tit analytic sets in R. and 2" > N,6' there results from 8.2.9 the following theorem, which also complements 8.2.211: 8.2.91. In R. there are non-analytic p.-measurable sets. BIBLIOGRAPHY: The notion of the n-dimensional (outer, inner) measure of a point set of R. is due to H. LEBESGUE, Paris These 190'2, p. 5 - Annali di mat. (3) 7 (1902), p. 235; 11}, p. 102, (121, p. 110), and therefore it is very often called (outer, inner) Lebesgue measure. Independently, the following also arrived at this notion: G. VITALI, Rendic. Instit. Lomb. (2) 37 (1904), p. 69; Rendic. Cire. mat. Palermo 18 (1904), p. 116; W. H. YOUNG, Proc. London Math. Soc. (2) 2 (1904), p. 16; (cf. also W. H. YOUNG and G. CsISnoLM YOUNG

[11, p. 76).-Older, less far-reaching theories of the content of point sets of R. have been established by: H. HANKEL, Gratul.-Programm Univ. Tilbingen 1870, p. 25 - Math. Annalen 20 (1882), p. 87; O. STOLZ, Math.'Annalen 23 (1884), p. 152; G. CANTOR, Acts. math.

4 (1884), p. 388; Math. Annalen 23 (1884), p. 473; A. HABNAcK, Math. Annalen 25 (1855), p. 241; G. PEANO ,[1 J, p. 154; C. JORDAN, Journ. de Math. (4) 8 (1892), p. 76; [11, p. 28; E.

BoREL [I], p. 46.-As for the development of the notions of content and measure, of. Encyklop. d. Math. Wise. 11 C9a (L. ZORETTI-A. ROSENTUAL), p. 962.

As to the construction of the n-dimensional measure by means of half-open intervals, of. also W. FELLER, Sitzungsberichte Akad. Berlin 1932, p. 459 and (in a much more general manner) Bull. internat. Acad. Yougoslave 28 (1934), p. 30. The theorems 8.2.4 and 8.2.41 are due to W. SIERPIfiSKI, Bull. Acad. Cracovie 1918, p. 173. Concerning theorem 8.2.7: W. SIERpn sxl, Giorn. di mat. 56 (_ (3)81 (1917), p. 272; Fund. math. 1 (1920), p. 116; H. STEINRAUS, Fund. math. 1 (1920), p. 93; M. FvsAMIYA, Science Reports Tbhoku Univ. 24 (1935), p. 332. As to theorem 8.2.8, the first example of a nowhere dense, perfect point set of R, with positive measure was given by H. J. ST. SMITE, Proc. London Math. Soc. (1) 6 (1875), p. 148.

3. Basis of the real numbers. In order to give an example for 8.2.91, we pro-

ceed from the notion of a basis of the real numbers. A set B of real numbers 't As to the propositions on 14 used here, of., for instance, H. Hahn 121, theorems 4.3.3, 5.2.32, and 18.9.1. s' Cf. H. Hahn [21, theorems 40.3.4 an(t 8.2.4.

CONTENT FUNCTIONS IN THE SPACE RA

181

101

different from 0 is called a basis of the real numbers if in B there is no finite set of numbers b; satisfying any relation of the form

rib, + ribs + ... + rkbk = 0

(3)

with rational coefficients ri Pe 0 and if every real number z sented by a finite set of numbers b, e B in the form

0 can be repre-

z= rib, +r,b2+...+rkb,,

(3.1)

0. Since there exists no relation (3), it follows with rational coefficients ri that the number z has only one representation of the form (3.1). 8.3.1. There is a basis of the real numbers.

We think of the real numbers different from 0 ordered as a well-ordered set Wk",

and we form a set B according to the following rule: the first element of W belongs to B; if z. e W and if for every element z of W which precedes Z. it is known whether z e B or z ti e B, then let zQ ' e B if za can be represented by numbers b, , bs,

,

bk of B, preceding za in W, in the form (3.1) with rational

coefficients rt ; otherwise let z e B. Then the set B defined in this way is a basis of the real numbers. As an extension of 8.3.1 we show: 8.3.2. There is a basis B of the real numbers with p1(B) - 0. Let Z' and Z" be the sets of real numbers which can be represented in the form

L' a co + M -[- Ly +

... +

+ ... and z" - 2 +

+ ... +

respectively, (where co is an arbitrary integer, and ci =20 or 1 for i?. 1). First we have to show p,(Z') = 0 and p,(Z") = 0; we prove, say, the former contention. ), then If Z,;. designates the intersection of Z' and [m, m + 1) (m - 0, f 1, f 2, it is sufficient to prove that p,(Z.,,) = 0, since Z' is the sum of the denumerable set of the Z.,. Because of 8.2.6 it suffices to prove that p,(Z;) - 0, since every Z' is derived from Zo by translation. Ze' does not contain any point of the disjoint

intervals (1,

C2 + ... +

1) ; ` 28

22

e'2 1), ...

equals I + 2.2

/r ///;

( ); r

.

.. ; ... , (

+ ... + 21, + 2Ut1 21

Since the sum of the lengths of these intervals

+ ... + T---L + ... a 1, we have p,(Z') = 0, as con1

tended.-We set Z = Z' + Z"; then since 161(Z') = 0 and p1(Z") = 0, we get

p,(Z) - 0 also. Now we think of the numbers of Z, different from 0, ordered as a well-ordered set W and, exactly as in the proof of 8.3.1, we pick out

a set B from W. Thus the numbers of B cannot satisfy any relation (3) and every number of Z can be represented in the form (3.1); and since every real number z can be written as z = z' + z" with z' a Z' and z" a Z", we obtain the result that every real number z can be represented in the form (3.1). Hence B 08 For instance, ef. H. Hahn [21, §8.1.

CHAP. II

MEASURE

102

is a basis of the real numbers and, since B C Z and µ1(Z) = 0, we have µ1(B) = 0 also.

8.3.21. For every basis B of the real numbers plx(B) = 0.

Let b e B and let B' be the set derived from B by the transformation x' =

x

b' If p,x(B) > 0, then by 8.2.61 ulx(B') > 0 also. Thus according to 8.2.7 there would he two points b, and b2' in B' for which bi - b2' = r > 0 is rational. Then for the corresponding points b, and b2 of B we have bI - b2 - rb = 0 and, since this is a relation of the form (3), this is inconsistent with the notion of the basis. As it is shown by 8.3.21, every basis B of the real numbers with µ1(B) > 0 furnishes an example of a non-pi-measurable set. One can prove that, actually, there is such a basis; but we will not give a proof for this fact here. But every basis B of the real numbers furnishes an example of a non-pl-measurable set according to the following theorem: 8.3.3. If B is a basis of the real numbers, if b e B, and if C designates the set of all real numbers representable in the form (3.1) without using b, then C is not p,-measurable.

It certainly is impossible that AI(C) = 0.

For if C, designates the set obtained

from C by the translation x' = x + br, then by 8.2.6 we would have µ1(C,) = 0 also; but this is out of the question, since SC, (for all rational numbers r) is the r

set of all real numbers. Thus AI(C) > 0.-Now suppose that C is p1-measurable. If C' designates the set formed from C by the transformation x' =

b,

then accord-

ing to 8.2.61 C' would also be MI-measurable and p,(C') > 0. Thus by 8.2.7 there would be two points p' and q' in C' for which p' -- q' = r ; 0 is rational.

If we set bp' = p and bq' = q, then p e C and q e C with p = q + rb, which is impossible, since according to the definition of C the representation of p and q in the form (3.1) must not contain b. 8.3.4. No basis B of the real numbers can be an analytic set.

Let b e B; if B were analytic, then the set B' = B - (b} would be also. By we designate the set of all real numbers z which are representable in the

Bw>

form (3.1) by k numbers b; a B'

.

Let Br be the set derived from B' by the trans-

formation x' = rx. Now if B' is analytic, then Br' is analytic alsos' and, since B(') = SB,' (for all rational r Pd 0), B'u is analytic alsoe0; but then BS=I, B(B)7 r

are also analytic,61 and hence C - SB(k) would be analytic, too.

But this is impossible according to 8.2.211, since C is the set which by 8.3.3 is not k

p1-measurable.

From 8.3.4 we see that the basis B with µ1(B) = 0, obtained in 8.3.2, is an example of s pi-measurable, non-analytic set. " Cf. H. Hahn [21, theorem 40.6.2. 00 Cf. H. Hahn (21, theorem 40.2.4. 81 Cf. H. Hahn 121, theorem 40.7.4.

§8l

CONTENT FUNCTIONS IN THE SPACE Rn

103

BIBLIOGRAPHY; Theorem 8.9.1 is due to G. HAMEL, Math. Annalen 60 (1905), p. 459; one owes the other theorems of this number to W. SiExpiksxt, Fund. math. 1 (1920), p. 105. The remark subsequent to 8.3.21 refers to C. BURSTIx, Sitzungeberichte Akad. Wiss. Wien 125 (1916), p. 209.-Furthermore we mention: W. SiERPIkSxr, Publ. math. Univ. Belgrade 1 (1935), p. 220; S. RuziEwicz, Fund, math. 26 (1936), p. 56; F. B. JoNES, Bull. Amer. Math. Soc. 48 (1912), p. 472.

4. Onedimensionally non-measurable sets. That in R. there are n-dimensionally non-measurable sets, follows from 7.3.2 because of 8.2.22 and 8.2.34. Examples of onedimensionally non-measurable sets have been met above in 8.3.3; subsequent to 8.3.21 we have remarked that every basis B of the real numbers with µ1(B) > 0 is also non-µl-measurable. Now we give other examples. We think of the real numbers mapped on the points of a circle with circumference 1, such that x and x + 1 furnish the same point. Let a be an irrational number with 0 < a < 1. To every x we form the set M. , on the circle, of all point x + ka(k = 0, f 1,:::b 2, ). Any two of these sets are either identical or disjoint. The set Ao shall have exactly one point in common with each of these sets M. (the same point, of course, with two identical sets Mz). Let Ak be the

set obtained from A0 by the transformation x' = x + ka (a rotation of the circle through an angle ka); then SAk(k = 0, f 1, f2, k

) is the whole circum-

ference of the circle. We now develop the circle onto the interval [0, 1); then Bk results from Ak and [0, 1) = SBk (k = 0, f 1, f 2, . ). The fact that k

Ak+I is formed from Ak by a rotation of the circle through the angle a, implies that the subsets of Bk+1 lying in [0, a) and [a, 1) are congruent with the subsets of BA, lying in [1 - a, 1) and [0, 1 - a), respectively. By 8.2.6 and 6.1.3 there results: µi(Bk) = 1R1(Bk+1). Hence the sets Bt are non-.&1-measurable. For if one of them were µ,-measurable, then by 6.1.41 and 8.2.6 all of them would be; but since [0, 1) = SBk , we would then obtain by 6.1.42: 1 =F, µ, (B,), which is k

impossible, as all µl(Bk) are equal. We give a second example of a non-µl-measurable set, proceeding from the following remark: 8.4.1. There is a set A of irrational numbers which contains exactly one of each pair of irrational numbers with a rational sum. We think of all irrational numbers ordered as a well-ordered set IT-12 and form a set A according to the following rule: the first element of W belongs to A ; if z, e W and if for every element z of W which precedes z, it is known whether

z e A or z r.. a A, then let z, ti e A if among the numbers preceding z, in Ti' there is one, re , such that z5 e A and z. + zz is rational; otherwise let. z, e A. Now let z' and z" be two irrational numbers with a rational sum: z' + z" = r; 62 Cf. footnote 58, p. 101.

CHAP. II

MEASURE

104

at most, one of them, of course, can belong to A. We have to show that at least e A and z" r., e A. Then in W a z* one of them belongs to A. Suppose z'

and a z** would precede z' and z", respectively, such that z* e A, z** e A, z* + z' = r', and z** + z" - r" (where r' and r" are rational). But then we would have z* + z** = r' + r" - (z' + z") = r' -}- r" - r, and hence z* + z** would be rational, contrary to z* e A and z** e A. 8.4.2. The set A of 8.4.1 is non-µl-measurable. Let I be an (open or half-open or closed) interval of R, with rational endpoints.

We designate the set of all rational points in I by N and set Al = A' and (I - N) - A = A". Then A', A", and N are disjoint and I = A' + All -)- N. Now if A is µ,-measurable, then µ1(I) = µ,(A') + u2(A") +. µ,(N) ' = µl (A') + p,(A ") . But since exactly one of any two irrational numbers with a rational sum belongs to A, it therefore follows that exactly one of two irrational

points of I symmetrical to the centre r of I belongs to A; that is, one belongs to A', the other one to A". Thus A" is derived from A' by reflection in r, that is, by the transformation x" _ --x' + 2r. Hence, by 82.6 and 8.2.61, pl(A') = µ,(A"). Therefore, since µ1(I) = µ1(A') + µ1(A"), we would have µ1(A') = µ1(A")

.=

}µ,(I); that is, for every interval I with rational endpoints

µl(AI) = 4µ,(I). Thus since every open set G( A) of R, is the sum of a countable set of disjoint intervals with rational endpoints, we would have (if µ1(G) is finite): µ1(AG) _ jµ1(G).

(4)

But this is impossible.

For let Go be an open set with 0 < µl(Gs) < + oo ; by

(4) 0 < ul(AGe) < + ao

also.

According to 8.2.52 there is an open

AGo, such that µ1(G') < 2p1(AGo). Substituting G'Ge for G' we can set G' assume G' C Go. Then AG' = AGo, and hence from µ1(G') < 2p1(AGe) we obtain µ1(G') < 2p1(AG'), contrary to (4). BISI.Iooatess: The existence of onedimensionally non-measurable sets was first proved by G. VITAM [1]. The first of the examples given in this number is due to F. HAUSDOEFF

111, p. 401; Math. Annalen 75 (1814), p. 428 (it is formed by a method rather similar to that of G. Viv.+:u); as to the second example, of. W. SiFnPIks&I, Fund. math. 10 (1927), p. 177.-

It is remarkable that all examples, known so far, of non-µ,-measurable sets of R1 (and of sets of R.) are based on the use of the Zermelo choice axiom.

5. A second general method for the construction of content functions. Let E again be a metric space and let $ be a system of point sets of E (containing the

empty set A) with the following property: to every set A C; E and to every p > 0 there are countably many sets T. e Z with diameters d(T,) < p, such that A Q ST, . To every T e Z let a number r(T) z 0 be attached; in particular let -r(A) = 0.

Now we designate by ¶(A,p) every countable system of sets T. e X with r(T(A, p)) = Sr(T1) and form

d(T,) < p and ST, Z A. We set

,

105

CONTENT FUNCTIONS IN T= SPACE R.

§8)

inf r(`, (A, p)) = r(A, p). Of course r(A, p') 9 r(A, P) if p > p'(> 0), and z(...) hence lim r(A, p) exists. Now we set p- 1-O

v(A) -slim r(A, p)

(5) and we state:

8.5.1. The set function ip(A) defined by (5) is an ordinary measure Junction-6 That the conditions 1.) and 2.) of §6,1 are satisfied, is trivial. We show that condition 3.) is also satisfied. To every set At let a system t(A5, p) be given. r(V (Ak , p)), Then SS(A5 , p) is a system 3t(SAk, p), for which r( (SAs , p)) 5 k

k

k

Thus, by letting p --* +0, we obtain: and hence: r(SA5 , p) Y r(Ak, p). k so(Ak), as contended.-Now we have to show that ,p is an ordinary (o(SAk) 5 k

k

measure function; according to 7.1.1 we have to prove: if AB > 0, then ,p(A + B) _ p(A) + oo(B). Since p(A + B) S jo(A) + p(B), as we have just shown, it has to be proved only that, ,(A + B) ? ,,(A) + ,(B). Let p < >) AB; we consider any Z(A + B, p). Those T e Z(A + B, p) which contain points of A form a yystem Z(A, p) and, since p < JAB and hence every T e %(A, p) is disjoint from B, the other T e Z(A + B, p) form a system X(B, P). Hereby

r( '(A + B, p)) - r(Z(A, p)) + r(X(B, p)), and thus r(Z(A + B, p)) ?.

r(A, p) + r(B, p). Since this is true for every system t(A + B, p), we get r(A + B, p) k r(A, p) + r(B, p) also. Thus, by letting p --, + 0, we obtain the contention oe(A + B) z p(A) + q(B). 8.5.2. If ,p is the set function defined by (5) and if the sets of the system SC are is a regular, ordinary measure function.

,p-measurable, then

According to 8.5.1 we have to show that the condition of 6.4.2 is satisfied. This is trivial if o(A) = + co ; thus we assume: p(A) < +\ao . By definition of V(A) the is a system 2 (A, 1), such that r( (A, j')) < (A) + If Tk, , TM,

, T,,

,

are the sets of (A, k ], then b\y assumption and because

of 6.1.41 Mk - STki is c-measurable, and hence M = DMk is also; and since i

Mk

k

A, we have M Q A, too. From Mk = STk,

/

M it follows that t(A, L j

k11 < ,p(A) + k . `Thus,, is also a system 2[ X! and hence r( M, A by letting k - + cc, we get p(M) `;5 (p(A), and from the fact that M we obtain oP(M)

,P(A) also; therefore oo(M) = ,p(A).

a If in the definition of p we set r(T) m d(T) in particular and if X is the system of all subsets of E, then the not function p(A) defined by (5) is also called the "Carath6odory linear measure" of A. Cf. also No. 6, for k - I (but these two definitions are different in general).

(ca.&p. 11

MEASURE

106

8.5.3. If the sets of the system Z are open, then the measure function rp defined by (5) is a content function.

According to 8.5.1 and §7,2 we have to show that to every A there is a Gs-set

M Q A with p(M) = V(A). The proof is the same as for 8.5.2; one has only to consider that now Mr = STki is open and hence M = DMk is a Gs-set. i k 6. The k-dimensional measure of a point set of R . Now let E be again the R. . Then, for instance, one can choose the system of all open spheres T of R. for Z

and the n-dimensional measure of the sphere T for r(T) (thus, if T has the

radius r: r(T) =

1

\ r" 7r"/= 64)

I'1 2 -}- 11

It is easy to prove that in this case the set function p(A) defined by (5) becomes the same as the n-dimensional outer measure of A: qp(A) = p,.(A). p"(A), we have only to show that V(A) S p,.(A). This Since certainly P(A) o o. Then by 8.2.52, is the case if p"(A) = + o o; thus we assume: to every e > 0 there is an open set G A with An(G) < p,.(A) +e. According to Vitali's Covering Theorem 17.5.5 (discussed later), to every p > 0 there are countably many disjoint closed spheres Sk Q G 'with diameters smaller than p, SSA) = 0, and hence E p"(SA) = a,,.(G). By 8.2.522 argil such that A

footnote 53, p. 97, there is a simple system ( of intervals, x

G - SSA , A

consisting of n-dimensional cubes I, with length of the edges smaller than

n-,

p"(I,) < e. If we replace every cube 1, by the closed sphere C, such that where p"(C,) < circumscribed about I, (thus d(C,) = d(I,) < p), then c" is a constant (dependent only on n). Now we replace the closed spheres SA and C, by somewhat larger, concentric, open spheres S;, and C', with diameters u"(G) + e and E ;&.(C,) < (c + 1) e. also smaller than p, such that E The countable system of all the spheres Sa and CY shall be designated by ((Ti)).

µ"(Ti) < p"(G) + (c" + 2) e < p"(A) + (c" + 3)e. Then G C STi and i i Hence it follows that (for every p > 0) r(A, p) S p"(A), and thus by (5) ,p(A) s 1 (A) also, as contended. We come, however, to new content functions if for T we choose again the system of all open spheres T of R., but by r(T), instead of the n-dimensional measure of the sphere T as above, we now mean the k-dimensional measure rk(T) of a k-dimensional sphere (k < n) of the same radius as T. Then the set function furnished by (5), which according to 8.5.3 is a content function, shall

be denoted by pk(A) and called the k-dimensional outer measure of A. The r2' ,r.

6' Le.' for n = 2v: -r(T) _ -- ; for n = 2v - 1: r(T)

2 r2.-' 1.3.5 ... (2v - 1)

107

CONTENT FUNCTIONS IN THE SPACE R.

§8)

pk-measurable sets are said to be also k-dimensionally measurable and, for these sets, pk(A) is called the k-dimensional measure of A. The inner Al.-measure

of A, designated by pkx(A), is named the k-dimensional inner measure of A. Every point set of R. has a onedimensional (or linear), a twodimensional, . , an n-dimensional outer measure. By reference to this statement we obtain: 8.6.1. If pk(A) is finite for the point set A of R,,, then pt(A) = O for k < 1 S n. For a sphere T whose diameter is smaller than p we have: rl(T) < cpt rk(T),

where c means a constant (independent of T), and hence also: rl(A, p) 5 pt(A), cpt-kTm(A, p). But for p -- 0+, we obtain by definition: r,(A, p) rk(A, p) - pk(A). Thus if pL(A) is finite, there results that AI(A) = 0. From 8.6.1 it follows immediately that AAA) S pk(A) if k < 1.

(6)

Moreover, there results from 8.6.1: If , (A) > 0 (for an 1 n), then pk(A) = + ao for every k < 1. 8.6.11. If A = SA, C R. , where pk(A,) (v = 1, 2, ) is finite, then A, (A) = 0

fork 0, there results from 8.6.11: , n - 1). 8.6.12. For every open set G of R we have pk(G) _ + ac (k = 1, 2, Hence there is no analogue to 8.2.52 for k < n.65 , n); 8.6.13. For every countable set A C; R. we have pk(A) = 0 (k = 1, 2, that is, pk is continuous. Proof as for 8.2.22. 8.6.2. Let 1 <= k < 1 5 n, let f1 be a a -field of sets of R. which are both pimeas-urable and pt-measurable, and let rp be a totally additive set function in 9.71.

If pis

pt-continuous, then rp is pk-continuous also; if p is purely pk-discontinuous, then 'p is purely pt-discontinuous also.

Because of (6) this results from 5.7.11 and 5.7.111. 8.6.3. Let be a u-field of sets of R. which are pk-7easurable for k = 1, 2, . , n, and let 'p be a totally additive set function in l2; then there is a decomposition 'p =

n + co -, + ... + vo + x, where Vk(k = 1, 2, 1, 2,

,

,

n) is pk-continuous,

(k

n) is purely pk-discontinuous, Vo is continuous, and x is purely dis-

continuous.

By 5.7.3 there is a decomposition rp = co. + x.-,, where , is p,-continuous Likewise we have X,,-2, and Xn-1 is purely and X,, i is purely p,.-1-discontinuous. Proceeding where (p.-, is in this planner we come to a decomposition to = co + -f- 'p, + xo , Where rpk(k = 1, 2, , n) is pk-continuous and xo is purely MI-discontinuous. We now show that is purely p.-discontinuous. For this purpose we have 5-4- 0, then there is an A e M l fl, such that A = q to prove: if M e 9)1 with 0. By 5.1.42 there is a subset M* of M, for A and (p.-,(A) u C4, also footnote 40, p. 81).

108

MEASURE

[CHAP. II

such that X.-,(M) = Xn_I(M*); and since (by 5.7.3) have also v,,._1(M) = x,Ml(M*). Since X,rj is purely

X**-,, we Pd 0 implies x,,..i(M*) # 0. Therefore there is an A e M*112 with A =,,,, A and

x'1(A) o 0.' Since M* and hence also A is p,,...1-regular for x._1, we have by 5.1.41 xA-1(A) - X.-I(A), and thus, since V.-i = X,.-1 , we have V*-1(A) qd 0, as contended. In the same X.-I(A) also. Hence A =,,,, A and , n - 1) is purely IAA,-discontinuous. manner one sees that sok_1(k = 2, 3, Finally by 5.3.2 there is a decomposition xo -= sPo + x, where Vo is continuous and x is purely discontinuous. Moreover we prove that cpt7 is purely p1-discontinuous. Take the countable sets of fit as the system ( of the singular sets; then by 5.1.42 for every M e )2 there is a regular subset M* of M (with respect to (), such that xu (M) = xe(M*); and since vo = xo , we have also so,(M) = xo(M*). Hence we can proceed again as above and then there results that is purely p1-discontinuous. Thus 8.6.3 has been proved. 8.6.31. For the decomposition of 8.6.3 we have:

+ {3n-1 -l- ... -i- cps + z, to

so+ = son + An-1 -l- ... + c + x+, =sv.+So*-1+...+V +z.

Indeed we had sO = ,pn -0 xn-1 x,-1 = son-1 + xn-s , , XI = Sol + XO xo = We + X. By 5.1.56 and 5.1.45 we have p = (,p)* + (,p)** = cp* + sp** _

cPn + xp_j . In the same F, + To , To - SAO + 2

manner we get x.-1 = wn-1 -i' j;-2 3C, =

8.6.32. If p(M) is finite for all M e sJ7t,6e then besides the decomposition of 8.6.3 there is no other one of the form cp = vn + cpn-1 -}+ We + x where sok (k = , n) is purely pk-discontinuous, 1, 2, . , n) is ;4-continuous, cpk'-1(k = 1, 2, soo.is continuous, and x' is purely discontinuous.

By 8.6.13 the pt-continuous functions cpk and cpp(k = 1, 2, - , n) are also + ceo and continuous; and since We and "' are continuous, by 5.2.14 V. + + VD' are also continuous. Therefore it follows from 5.3.2 that cp;, + cp. + + ce = v. + . + cpo and x' = x. By 8.6.2 the AA,-continuous func, n) are also p1-continuous, and hence by 5.2.14 tions spr and cpk(k = 1, 2, + sot and 0'.. + + V1' are pt-continuous also. Therefore, since sOo cp + and suo are purely p1-discontinuous, it follows from 5.7.3 that p'. + +- cpi = Continuing in the same manner one proves the + vi and ce = sso. con + contention. BIBLIOGRAPHY to NOS. 5 and 6: C. CARATHgODORY, Nachrichten Ces. d. Wiss. Gottingen

1914, p. 420; F. IlAUSDORFF, Math. Annalen 79 (1918), p. 157; H. HASN [11, p. 450, 480.

Concerning this and as to older definitions of a k-dimensional content in R. , due to H. M1i' owsx1, W. H. Your o, and O. JANZnN, cf. Enoyklop. d. Math. Wiss. II C9a (L. Zoam'r1A ilosENTHAL), p. 994.-A. S. BasicovITcu, Math. Annalen 98 (1927), p.'459, showed by an oxainple that p" and the"Carath6odory linear measure" (p. 105, footnote63) inR,may differ; i,esi !cs, in Acta math. 82 (1934), p. 297, on the basis of the hypothesis of the continuum,

This condition is essential. For instance, let Q mean the c-field of the zero-sets for 1A1 r.d operate otherwise as in footnote 23. p. 39 and footnote 27, p. 42.

§8]

CONTENT FUNCTIONS IN THE SPACE R,,

109

he gave a set in R2 which is p1-measurable, but not p2-measurable.-Furthermore we may especially mention the following more recent literature: S. SAas, Fund. math. 9 (1927), p. 16; A. Koi ouoROFF, Math. Annalen 107 (1932), p. 351; A. S. BEaiooviTOS, Math. Annalen 113 (1946), p. 416; A. APPERT, Paris C. R. 201 (1935), p. 186; Bull. sc. math. [711 e [

(2) 601 (1936), p. 329, 368; J. Gztzas, Fund. math. 26 (1936), p. 229; E. SLPII.LuN, Fund. math. 28 (1937), p. 81; A. P. Moasaa and J. F. RANDOLPH, Duke Math. Journ. 6 (1940), p. 408; G. N6BEUUNO, Jahresber. d. Deutsch. Math. Ver. 62 (1942), p. 132, 189; Math. Annalen 118 (1943), p. 687; A. SARI), Bull. Amer. Math. Soc. 49 (1943), p. 758; S. KASEYA, M. Mzsu, and H. YONEanRA, Proo. Imp. Acad. Tokyo 19 (1943), p. 241; D. S. MILLER, Duke Math. Journ. 11 (1944), p. 139.

CRAPTER III

MEASURABLE FUNCTIONS

§9. Measurable Functions 1. Basic properties of v -measurable functions.

Let T Z be a or-field (consisting

of subsets M of a given set E) and let co(M) be a totally additive set function in )t; the absolute-function of w is again designated by ip (§3,4). To have a simple mode of expression, we call the sets M e 831 se-measurable and we call the value of the function ,p(M) the p-measure of M (without meaning to imply that co is a measure-function in the sense of 16,1).' Replacing T1 by the extended a-field 872° over 191 (§4,3 and §4,4), if it is necessary to do so, we may assume that fit is complete for jo. Let A e 932. We say the (one-valued) point function f is V-defined (is sp$nite) on A if the set of all x e A for which f is not defined (is not finite, respectively), is a zero ;set for (§4.1). Analogously we say f is .p-bounded on A if there c zero-set N for gyp, such that f is bounded on A - N.

Let f be p-defined on A. We designates by [f(i) > y] the set of all x e it a6 which f(.r.) > y. If, in this connection, we wish to refer explicitly to the set A, we use the more detailed notation: A (f (i) > y]. The symbols A [ f(1) z yJ and so on are defined analogously. The function f, 1P-defined on A, is called cp-measurable on A if for every y : I;,

that is, for every (finite) real number y, the set [f(x) > yJ is So-measurable.' Every function constant on A is ca-measurable on A. If A =, A, then every function defined on a subset of A is also ,p-measurable. 9.1.1. If f is ce-defined on A, if Y is dense in R1, and if f f(1) > yJ is cp-measurable for every y e Y, then f is ,p-measurable an A.

For to every z e R, there is in Y a decreasing sequence ((y,)) with y, -> z. llut then [f(x) > z1 = b'(f(i) > y,J, and hence [f(t) > y,J e'02 implies r

If(1) > ZJ a 872.

9.1.2. If f is 9p-measurable on A, then each of the following sets is also rp-measarable (y a 1, , y' e fr): [f(i) > y], (f(i) < yl, 1f(i) yl, [y < f(i) < y'l,

[y 3 f(i) 5 y'], [y 49 f(i) < y'l, [y < f(i) s 0, [f(i) = y], [f(i) y). First we prove that [ f(i) z yJ a V. Since [ f (i) > - -I _ A, we may assume that y > - -. Now let ((y,)) be an increasing sequence of real numbers ' To justify this notation we remark: If r ? 0 in 911, then, according to §6,5, (o can be extended to a measure function w in E, for which then the sets M E R are f-measurable (in the sense of 48, 1). If n has negative values in 212, then the analogous statement is true at least for 0. T Cf. H. Hahn 121, p. 187 and 232. ' [f ($) > yJ is then 4o-measurable also for every y a R, (¢3,6), since If (-4) > +

lM)> -°°l-s[A(z) > -n). n

110

l - A and

MEASURABLE FUNCTIONS

*9)

with y,

111

y; then [f(l) a yl - D[f(t) > y,1.-The other parts of our proposi-

tion result from the following formulas:

A - [f(t) y], [f(t) 4 y] -,A - [f(x) > y]; [f(.f) < yl [y < f(t) < y'J = [f(x) > y'l, [y -- f(z) 6 y'J = [f(x) a y'l, [y 5 f(t) < y'l = [f(t) a y] [f(1) < y'], [y < f(t) 6 y'J = [f(s) > y] [f(t)

y'l;

[f(t) = y] is the particular case y' = y of [y 9 f(t) S y'l; [f(.) P6 yl A - [f(t) = y]. 9.1.21. The function f, defined on A, is cp-measurable on A if for all y E R, either the sets If(!) 9 y], or the seta [f(at) < y], or the sets [f(.f) yJ are tp-measurable.

Let ((y,)) be a decreasing sequence of real numbers with y, -- y; then If(i) > yj = S[ f(1') y,] is pp-measurable if the sets I f(&) y,J are. If all sets I f(&) < y] are rp-measurable, then the sets [f(I) Z y] _,, A - [ f(t) < yl are also, and hence, as we have just seen, so are the sets [f(I) > y]. If all sets [f(f) yl are measurable, then the sets [f(z) > y] =, A - If(i) y] are also. 9.1.22. The function f, p-finite on A4, is ip-measurable on A if for all y e R,

,

y' a R, either the sets [y < f(1) < y'], or the sets [y f(t) ;s y'], or the sets f(x) < y'J, or the sets [y < f(t) y'] are e-meaeurable. [y This follows from 9.1.21, since

[f(1) > yJ

Sly < f(t) < nl =, Sly
and [.f(.) ? y] =. S[y s f(.) s n) =, S[y 65 f(.f) < n).

Yet from the fact that Y(x) - yJ is "v-measurable for all y e At, it does not follow that f is r,-measurable. Example: Let B be any non-u1-measurable set of R1, which as well as its complement R, - B has the power K (the existence of such a set B follows from 7.3.2 if one takes the proof of 7.3.1 and footnote 37, p. 87, into consideration). Then there is a one-to-one mapping of B on the set of ally > 0 and of R, - Bon the set of all y 9 0. By both these mappings a non-µi-measurable function y = f(x) on R, is defined which assumes every value only once; therefore every set [f(i) = y] consists of exactly one point and hence is µi-measurable. If Q C R, and N is the set of all x e A for which f(x) e Q, then N is called the complete inverse image of Q. Now 9.1.2 is quite a special case of the following theorem: ' That this condition is essential, is shown by every function f which has the value + on a non-s,-measurable subset B of _9 and the value - ac on .1 - B.

[CHAP. III

MEASURABLE FUNCTIONS

112

9.1.3. I f f is cp-measurable on A, then the complete inverse image of every Borel set in ft1 (§7,1) is gyp-measurable.

First we show : the system C of all sets of R, whose complete inverse image is to-measurable forms a a-field. Let Q. a £1 and let M, be the complete inverse

image of Q, ; then SM, is the complete inverse image of SQ,. Since M, is , r ,p-measurable, SM, is also 1P-measurable, and hence SQr a Q. Now let Q e C ,

r

and Q' c C, and let M and M' be the complete inverse images of Q and Q', respectively. Then M - M' is the complete inverse image of Q - Q'; since M and M' are cp-measurable, M - M' is also rp-measurable, and hence Q - Q' e 0q. Thus C is a or-field. Therefore, since by 9.1.2 Cl contains all intervals of 1L1 , £1 contains also all open sets and all closed sets of R1, and hence (according to §1,2 and §7,1) also all Borel sets of RI. 9.1.31. If A = SA; where the A; are gyp-measurable with finite (p(Ai) and if f is i

p-measurable on A, then the sets of f, whose complete inverse images are tp-measurable form a Suslin system of sets (§6,3).

The immeasurable sets M Q A form a a-field 9)1' (rte 9) complete for ,p; by We designate the system of all subsets of A by A. According to the method of §6,5 (starting from 9)1') one can extend the absolute-function p to a regular measure function 0 defined in W. Let ' designate the complete cover of 59)1' with respect to gyp, that is, the a-field of the 4.1.12 9)1' is complete also for gyp.

Cp-measurable sets; then by 6.5.5 or 6.5.6 0717 = `17'.-Now let C be the system of all sets of 1?1 whose complete inverse images are gyp-measurable. Furthermore let Q,,,,, ... nk be the sets of a Sualin scheme (16,3) in Cl and let M,,,,,, ... ,, be the complete inverse image of Q,,,n, ... ,.k . Then M, = M,., . nk

... is the complete inverse image of Q, = Qn, -Qn1e, - Qn,n, ... nk ... and SM, is the complete inverse image of SQ,. Thus if Q is the nucleus of the r

Suslin scheme of the Q,,,., ... k , then the complete inverse image M of Q is the nucleus of the Suslin scheme of the nk . Since the M,.,n, .. k a 0'(= OF), by 6.3.51 or 6.4.4 (applied to gyp) we have M e 9)2' also, that is, Q e C. Thus it is shown that Cl is a Suslin system of sets. 9.1.32. Under the assumptions of 9.1.31 the complete inverse image of every analytic set in .R1 (§7,1) is to-measurable.

This follows from 9.1.31, since by 9.1.3 the complete inverse image of every open or closed set of Ri is immeasurable. Yet if f is p1-measurable, it does not result that the complete Inverse image of every p,-measurable set of ft1 is p1-measurable. For there are p1-measurable

functions f(x), such that by y = f(x) a non-p1-measurable set is mapped on a p,-measurable set.&

9.1.4. If p(A) is finite and if f is c-measurable and ,p-finite on A, then to every e > 0 there is a B C A, such that 4p(A - B) < e and f is bounded on B.

Let A,, = (f(s) > n] + tf(t) < --n]; then DA = (f(t) = +oo] + ' Cf. C. Caratheodory [11, p. 359 ; H. Hahn 111 , p. 586.

113

MEASURABLE FUNCTIONS

§9l

[f(i) = - cc]. Thus since f is re-finite, we have ,p(DAn) = 0. Therefore by 3.2.21 q (An) -+ 0, and hence for almost all n: p(AR) < e; but on A - An we have

Ift
If f is ce-defined on A, then f is So-defined also on every (p-measurable subset B of A. For if C is the set of all z e A at which f is not defined, then BC is the set of all x e B at which f is not defined, and C =, A implies BC =, A. 9.1.5. If f is rp-measurable on A, then f is also on every ce-measurable subset B of A.

For we have B[f(i) > y) =

y]; thus, if A[f(i) > y] and B are

(p-measurable, then B[f(i) > y) is also. ), then f is cp-defined also on If f is gyp-defined on every set A,(v = 1, 2, A = SAY . For if C, is the set of all x e A, at which f is not defined and if C Y

is the set of all x E A at which f is not defined, then C = SC, , and C,

A

implies C =' A.

9.1.51. If f is
also on A = S A,

), then f is cp-measurable

.

v

For we have A[f(1) > y] = SA,[ f(i) > y); thus if the A,.[f(i) > yj are cp-meas-

urable, then A [f(t) > y] is also. 9.1.6. If f is se-measurable on A, then -f is also.

This results from 9.1.21, since [f(i) > y] = [-f(-*) < -y]. 9.1.61. If f is rp-measurable on A, then I f I is also. For we have [I f(i) J > yl = [f(i) > yj + [f(i) < -y]. 9.1.62. If f is immeasurable on A, then the function S(f) derived from f by the bounding transformation (§3,6) is also So-measurable on A, and vice versa. This results immediately, since the bounding transformation (and its inverse) is monotone increasing. BIBLIOGRAPHY: The notion of measurable functions for the case p - f,,, is due to H. LEBasaun, Paris These (1902), p. 28 - Annali di mat. (3) 7 (1902), p. 258; 111, p. 110; 12j, p. 118; this notion has been extended to the case of an arbitrary totally additive set function ,p by J. RADON, Sitzungsberichte Akad. Wiss. Wien 122 (1913), p. 1325.-Further contributions were made by : C. CARATHLODORY [I], p. 374; H. HAHN 111, p. 548; 0. NIHODYK, Fund. math. 15 (1930), p. 141.

2. Relations of rp-measurable functions. Let f, and f be So-defined on A ; then max (fi(x), f2(x)) and min (f,(x), f2(x))6 are also cp-defined on A. For if C; is the

set of all x e A at which f; is not defined (i = 1, 2), then C, + C2 is the set of all x e A at which max (fj(x),f2(x)) and min (fi(x), f2(x)) are not defined, and C; =, A

(i = 1, 2) implies C, + C, =, A.-The set B = A - (Cl + C2) of all x e A at which max (f,(x), f2(x)) and min (fi(x), f2(x)) are defined is cp-measurable. Max(a, b) and min(a, b) designate the greater and smaller of the numbers a, b, respectively.

[CHAP. III

MEASURABLE FUNCTIONS

114

9.2.1. If f, and f2 are immeasurable on A, then max (fi(x), .flx)) and min (f,(x), f:(x)) are also.

Let f(x) = max (fi(x), f2(x)) and let B be the set of all x e A at which f is

defined.

Then [f(i) > yl = B ([f,(.) > yl + [f2(x) > y]) and, since B and

[ f;(t) > y](i = .1, 2) are So-measurable, [f(I) > y) is also. 9.2.2. If f, and f2 are qp-measurable on A, then the set B of all x e A at which ft + fs (or fi - f2) is defined, is vp measurable and fi + f2 (or f1 - f2) is V-meaaurable on B. .

Since

-I - A[f2(t) =

B= A - A(f1(f) = by 9.1.2 B is {o-measurable.

-1,

Furthermore, on B for every y c. R, , the inequality

fl(x) + f3(x) > y is equivalent to fi(x) > y - f2(x), and hence to fl(x) > r > y - f2(x) where r is a suitable rational number. Thus U1 U) + f2 4) > yI = S[fl (x) > r] [fl. (1) > y - r] r

where r runs through all rational numbers. Therefore fi + ft is rp-meaeurable on B. 9.2.3. If f, and f2 are cp-measurable on A, then the set R of all x E A at which f" f2 is defined, is r -measurable and f, f2 is p-measurable on B. Since

B =,, A - AIf,(i) = 01 (Alf (x) = -1- w 1 + A1f2(x) _ - oo 1)

-A[.f2(x) = 0].(A[f,(x) _ +°D1 + A[f,(x) = aol), the set B is f-measurable.-By 9.1.5 f, and f2 are r'-measurable on B. First let us consider the case where f, = 0 and f2 ? 0. Then on B for y > 0 the inequality f,(x) f,(x) > y is equivalent to fi(x) > p > y where p is a suitable positive rational number; and hence for y > 0 we have [fl(!) f2(z) > yJ = where p runs through all positive rational numbers. S[ f,(Z) > pJ [f2(t) > For y < 0 we have (f1(t) pJf2(1) > y] = B, and for y = 0 we have [f,(4 f(Z) > 01 = [f1(z) > 01 [f2(1) > Q. 0 and f2 z 0.-Now consider the Therefore f, f2 is So-measurable on B if f, case of an arbitrary f, and f2. Then, by 9.1.61, I f, J 1f2 l is p-measurable on B, as Ave have just shown. Furthermore,

(M-4)4204) > 01 - (f,(f) > 01 [f=(I-) > 01 + ]f,(x) < o] [f2(x) < 01 is co-measurable.

Thus for y > 0 the set

[f,(f) f2(x) > y] = [ (f,(x)

f2(1) I

01

is cp-measurable; and for y < 0 the set

[A(x) f:(x) > y] = [i f,(z)

I

-

I f2(x) I < l Y11 + (fi(x) 42(x) > 01

115

MEASURABLE FUNCTIONS

191

is also (p-measurable. Tb erefore 1fiW .f() > y) is tp-measurable for every y e RI , and hence ft f2 is -measurable on B. 9.2.4. If f is gyp-measurable on A, then the set B of all x e A at which

f

is c-measurable and f is gyp-measurable on B.

Since B

is defined,

A - A [ f (x) = 01, the set B is .p-measurable. For y > 0 we have :

> y] = [f(x) <

].[f(

) > 0]; for y < 0 we have:

1

Lf(x) > and furthermore

Ifffl

y] = [f(z) < Y+ If(s) > 01;

>0]=

[f(i) >

is gyp-measurable for every y s RI , and hence

1.

Thus

Lf(g)

> y]

is q)-measurable on B.

9.2.5. If f, and f2 are rp-measurable on A, then the set B of all x a A at which f1 2

is defined, is ,p-measurable and f is rp-measurable on B. 2

This follows from 9.2.4 and 9.2.3 applied to 2

From 9.2.2, 9.2.3, and 9.2.5 there results in particular: 9.2.6. If fI and f2 are rp-niea arable on A and if one of the functions ft + f2,

ft - fs , f1 f2, f=, cifi + Gzf2 (where c, and c2 are constants) is to-defined on A, then this function is also c-measurable on A.

9.2.7. If f, and f2 are immeasurable on A, then the sets [fi{t) > f2(t)], [fl(.f) z f2(z)1, [fi(x) * f2(x)1, and (fi(z) = f2()1 are (p-measurable. [ f,(t) > ff(i) I is the set of all x e A at which f, - f2 is defined and positive; hence it is io-measurable by 9.2.2. Then the other contentions follow from the for-

mulas: [f1(1) P6 f2(x)] = [fl(x) > f2(Z)] + [f2(1) > fi(t)], [fl(x) = f2(x)] A - [f&±) P1 f2(x)1, Iff(x) ? f2(1)) = [fl(l) > f2(x)1 + If1(t) - f2(1)]. Finally we remark that a t1-measurable function of a 141-measurable function does not necessarily furnish a u,-measurable function: In No. 1 (p. 112) we mentioned a µ,-measurable function f(x) by which a non-µ,-measurable set N is mapped on a p1-measurable set Q. Now we define a function g on R1 : let g = 1 on Q and let g - 0 otherwise; of course, g is µ,-measurable on R1 . But the com-

posite function h(x) = g(f(x)) is not p1-measurable, since the set [h (1) = 1] = N is not µ1-measurable. BIBLIOOBAPIIY: The same as in No. 1. In addition: CH. J. DE LA VALLEE POt;SSIN [11,

vol. I, p. 253; 121, p. 29.-That a p1-measurable function of a p1-measurable function may furnish a non-p,-measurable function, was first observed by W. SIEAPIHSKI, Paris C. R. 162 (1916), p. 716, and by C. CARATHECmORY 11 j, p. 379.

116

MEASURABLE FUNCTIONS

CHAP. III

3. Non-µi-measurable functions. We give some examples of functions in R, which are not µ,-measurable. Every non-p,-measurable set N of R, (cf. §8,4)

furnishes such an example if we define: f = 1 on N and f = 0 on Ri - N. Another example is furnished by the functional equation

f(x, + xz) = f(xl) + f(X2)We ask for the finite solutions of this functional equation. From (3) it follows immediately that f(xi + x2 f(xi) + f(x2) + ... + f(xR); there-+ (3)

+

fore f(nx) = of (x) and f (!) = n f (x), and hence for every pair of positive inte-

gersmandn: f (nx)

=

nfx-

Furthermore it results from (3) that f(x) = f(x) + f(0), and hence f(0) = 0. Thus f (x) + f (- x) = f (O) = 0, and hence f (- x) = -1(x). Together with (3.1) this furnishes for all rational r: (3.2)

f(rx) = rf(x)

If we set f(1) = c, then it follows from (3.2) for all rational x that f(z) = ex. Therefore if f(x) is to be continuous, f(x) = cx must be valid for all x, and so we have: 9.3.1. Every finite continuous solution of the functional equation (3) has the

the forn f(x) = cx. We obtain the most general (finite) solution of (3) in the following manner: Let B be a basis of the real numbers (§8,3). To every b e B attach quite an arbitrary value f(b). Then if x = rib, + r2b2 + - . - + rkbk is the representation of the real number x 0 0 by the basis B [§8 (3.1)], we set

f(x) = r,f(bu) + rtf(bs) + ... + r,.f(bk)

and f(0) = 0. It is obvious that the function f defined in this way in R, satisfies the functional equation (3). Now if b and b' are two different numbers of B and if we choose f(b') 0 f(b), b

then according to 9.3.1 f(x) is discontinuous. Hence: 9.3.11. There are discontinuous (finite) solutions of the fwutional equation (3). We shall see that every such discontinuous solution of (3) is non-p,-measurable, and we shall obtain this as a particular case of a more general theorem. The function f, finite in (a, b), will be called cotrvex in (a, b) if for every pair of points x, , x= of (a, b) we have: (3.3)

f (±x) s

(XI) + f(xa)).

§91

117

MEASURABLE FUNCTIONS

A function f finite in the whole R, is called convex if f is convex in every interval (a, b). Every function satisfying (3) is convex, since for it (3.3) is valid with

the equality sign. Thus by 9.3.11 there are discontinuous convex functions. But there are also discontinuous convex functions which do not satisfy (3). Example: Let B be a basis of the real numbers and let b e B. Then (according to §8,3) every x can be uniquely represented by finitely many numbers bi e B in the following , rk are rational + rkbk , where r, r1, r2 , form: x = rb + rib, + ribs + , rk 0 0. If we now set f(x) = r$', then hereby a discontinuous and r1 , r2, convex function is defined which does not satisfy (3).

9.3.2. If the function f(x) is convex and bounded above in (a, b), then f(x) is continuous in (a, b) .

Suppose the point xo of (a, b) to be a discontinuity of f. Then there is a sequence of points ((x,)) in (a, b) with x, -* xo and a k > 0, such that I f (x,) - f (xo) 4 ? k. We set x, = x, if f (x,) - f(xo) ? k, and x,' = 2xo - x, if f(xo) - f (x,) z k; then x,' --+ xo also, and it follows from (3.3) (if we set x1 = x' .

and x2 = x, there) that in every case f(x;) ? f(xo) + k (for all v for which x; e (a, b), and hence for almost all v). Thus there is a sequence ((x,)) with x, --+ xo and f(x,) Z f(xo) + k. If we now set x; = 24' - xo , then we have also x, --+ xo and, as a result of (3.3), f(x,) i' f(xo) + 2k (for almost all v). If we furthermore set x, = 2x, - xo, then we have a sequence ((x )) with x, --+ xo and, as a result of (3.3), f(x'") z f(xo) + 4k (for almost all P). So after n steps we come to a sequence ((x;"')) with x;" --+ xo and f(x;w') ? f(xo) + 2'"'1 k (for almost all v). Thus we see that if the function f convex in (a, b) has a discontinuity, then f is not bounded above. 9.3.21. If the function f (x) is convex in (a, b) and µ1-measurable, then f (x) is continuous in (a, b). Suppose the point xo of (a, b) to be a discontinuity of f. We choose a > 0 such that (xo - 2a, xo + 2a) (a, b). Since f is also convex in (xo - a, x0 + a),

by 9.3.2 to every n there is a point x in (xo - a, xo + a) with f(x) > n. Now x' = x, then by (3.3) if x' and x" are two points of (2 - a, x + a) with 2 we have at least one of the inequalities f(x') > n and f(x") > n. There results immediately: if R. is the set (by assumption A,-measurable) of all Z a. Thus if M is the (also x e (x - a, 2 + a) with f(x) > n, then MI-measurable) set of all xe(xo - 2a, xp + 2a) with f(x) > n, then, as a result

x

of (x - a, x + a) c (xo - 2a, xo + 2a), we have also µ1(M.) z a. M.+1 c Mn , by 3.2.21 it would follow that

Since

i= a. But this is impos-

sible; for DM. is the set of all x e (to - 2a, me + 2a) with f(x) _ + - and, since f as a convex function is finite, D.M. = A. Hence the assumption that there is a discontinuity off in (a. b) leads to a contradiction.

MEASURABLE FUNCTIONS

118

[CHAP. III

Thus by 9.3.21 a discontinuous convex function cannot be jaI-measurable. So in particular we obtain:

9.3.22. A discontinuous solution of the functional equation (3) is non-

l1-measurable. BIBLIOGRAPHY: Theorem 9.3.1 is due to A. L. CAUCHY [11, p. 103 (- Oeuvres (2) 3 (1897),

p. 98); one owes theorem 9.8.11 to R. VoLPI, Giorn. di mat. 35 (- (2) 4) (1897), p. 104, and (the most general solution of (3)) to G. HAMEL, Math. Annalen 60 (1905), p. 459. Theorem 9.3.2 is due to J. L. W. V. JENSEN, Acts, math. 30 (1906), p. 189; (the particular case, included herein, for the solutions of (3) was already given by G. DARsoUx, Math. Annalen 17 (1880),

p. 56 footnote; ef. also G. HAMEL, l.c.); see also F. BERNSTEIN and G. Dowrocn, Math. Annalen 76 (1915), p. 514. Theorem 9.3.21 is due to W. SIERPII!iaxI, Fund. math. 1 (1920), p. 125; one owes theorem 9.3.22 to H. LEBaESGUE, Atti Accad. Torino 42 (1907), p. 537; proofs for 9.3.22 have been given by M. FRACHET, Enseignement math. 15 (1918), p. 390; W. SIERPII'ISXI, Fund. math. 1 (1920), p. 116, (cf. also 5 (1924), p. 334); ST. BANACa, Fund. math. 1 (1920), p. 123; H. HAHN 111, p. 583; M. KAc, Comment. math. Helvet. 9 (1936-1937), p. 170; A. ALazzEwicz and W. ORLICZ, Fund. math. 33 (1946), p. 314.

4. "quivalent functions. If fi and f, are qp-defined on A and [.fi(x) > f2(t)]

A,

then we designate this by fi ;9, f or f2 fi. Analogously the relations f, <, f2 and f, >, fi are defined. The relation fi 6, f2 (as well as f, <, fs) is transitive; that is, we have: 9.4.1. If, f I ;5, fs and f2 S, fs , then fi ;S, fs also.

For let B be the set of all x e A at which f2 is not defined; then

[f1(t) > fa(t)] Q [fi(*) > fz(t)] + [f:(.A) > fs(t)] + B. Thus since by assumption [ft(t) > fx()] =, A, [fs() > fs(-M =, A, and B = A, we have also [fI(z) > fa(x)] =, A. If [fl(.f) 0 f,(±)] =, A, then we designate this by fi =, f2 and we say: fI is ,p-equivalent to f, on A. We have fi =, fz if and only if we have both fI 5 f, and f, ?, f2. The relation =, is reflexive and symmetric, and by 9.4.1 also transitive. This is to say: 9.4.11. The relation =, is an equivalence relation. 9.4.2. If f, is ,p-measurable on A and f, =, fi , then f, is also o-measurable on A. For every y we have [f2(x) > y] _, [f&) > y] Hence, if [ff(.f) > yj is cp-measurable, then [f2(t) > y] is also. 9.4.3. I f f I =, f2, t h e n I f I I -, I fs I also.

For [ Ifi(x) 1

P6

I ,,(.f) 11 C [ff(.f) #5 fe(d)].

9.4.4. If f, =, gl and f, =, g2, then max (fi(x), f2(x)) =r max (g,(x), g,(x)) and min (fi(x), f2(x)) =, min (gi(x), g2(x)) also. Set max (f1(x), f2(x)) = f(x) and max (gl(x), M(x)) = g(x) ; then [f(t) 0 g(t)]

[fl(i) 75 g,(t)l + 1.4(1) 0 g2(4l Thus since [fi(t) 0 gi(z)] =. A and [f2(1) 0 g2(x)] =, A, there results that [ f(x) 5-15 g(.f)] =, A.

SEQUENCES OF MEASURABLE FUNCTIONS

§101

119

9.4.41. If f, =,, g, and f2 =. g2 and if fl + f2 (A - f2 , ft f2 , f,) is c,-defined on A,

then g, + g2 (91 - 92, 91'921

Z,

respectively) is also ,p-defined on A and

respectively . f, +,12 91 + 92 (fi - f2 =, 91 - 92,fi'f2 91'92 , f2 , Let B, , B, , B, C be the sets of the points x e A at which g, , g2 , g, + g2 A + f2 , respectively, are not defined. Then g291

,

BQB,+Bs +C+[f,(i) 091(i)1 +[fO) 0g2(i)] and by asumption

[f2(i) 96 gs(i)] _ A; besides, if f, + f, is p-defined on A, then C =, A also. But then B =. A; that is, q, + g2 is also V-defined on A. Furthermore [f,(x) + f2(l) 5'-' 91(x) + g2(x)] c [f,(&') 5,6 g,(i)] + [f2(1) # g2(1)]. Thus, since [f,(x) 5-1 g,(x)1 =. A and

B, _. A, B, =, A, [f,(1)'

[f24) 96 g2(i)]

A,

and

A, there results that [f,(1) + f2(i) 96 91(i) + 92(z)]

A also;

that is, f, + f2 9.4.5. If f =, g on A and B is a immeasurable subset of A, then f -, g on B also. ), than f =. g on ,SA, also. 9.4.51. If f =, g on A,(v = 1, 2, 91 + 92 .

f

g; furthermore f =. g is equivalent to both 9.4.6. f =. g is equivalent to f 9 and f =, g simultaneously. This follows immediately from 4.1.12.

BIBLIOGRAPHY: H. LHBESGUB, Ann. Fac. so. Toulouse (3) 1 (1909), p. 38; C. CARATHtoDORY [1), p. 389.

§10. Sequences of measurable functions 1. Sequences of measurable functions. Let ((f,)) be a sequence of ,p-defined functions on A. If B, ie the set of all x e A at which f, is not defined, then

A, and hence B = SB, _,; A also. But on A - B the four functions B, sup f inf f lim f and 1im f, are defined; thus these functions are so-defined on A. 10.1.1. If ((f,)) is a sequence of so-defined functions on A and

f. =, g,(' then sup f,

1, 2, ... ),

sup g,, inf f, _v inf g, , Jim f,

lim g, , and li1n f, =, lim g,

Let sup f, = ;and sup g. = g; then [f(i) s g(i)] C S[f,(i) ; g,(i)]. Thus, ,

since by assumption 1f,(1) 0 g,(x)] _. A, we have [f(i) # g(i)] also. 10.1.2. If ((f,)) is a sequence of c-measurable functions on A, then sup f, and

(cHAP. In

MEASURABLE FUNCTIONS

120

inf f, are also immeasurable on A.

Let sup f, = f; then [f(&) > yj = S[f,(t) > y]. Thus, since by assumption [f,(.) > yj is gyp-measurable, [f(&) > yjis also. 10.1.21. If ((f,)) is a sequence of p-measurable functions on A, then lion f, and lim f, are also to-measurable on A.

If we set sup f, = fn , then lim f, = inf fA . By 10.1.21. is 0-measurable; ,

,Z n

x

thus using 10.12 once more we obtain the result that lim f, is also c-measurable.

A slight generalisation of 10.1.2 and 10.1.21, respectively, furnishes the theorems: 10.1.3. If f(x, t) for every t e (a, b) is 9-measurable on A as a function of z and

for every x e A is continuous in (a, b) as a function of t, then sup f(x, t) and to (a,b)

inf Ax, t) are gyp-measurable on A. s4(a,b)

For let r, , r2,

, r, ,

be the rational numbe rs in (a, b) and set f,(x) _

f (x, r,); then sup f (x, t) = sup f,(x), and hence by 10.1.2 this function is S -meas-

urable on A. 10.1.31. If f(x, t) for every t e (a, b) is v-measurable on A as a function of x and

for every z e A is continuous in (a, b) as a function of t, then lim f(x, t) and lim f(x, t) are cs-measurable on A. t+a+

Let t, -, a be a decreasing sequence of (a, b) and set f,(x) = sup f(x, t); :.(a.e.) then lien f(x, t) = lim f,(x). By 10.1.3 f,(x) is gyp-measurable on A, and hence t -a+

by 10.1.21

, t) is also.

The following examples show that in 10.1.3 and 10.1.31 the assumption of continuity with respect to t is essential. Example of a function f(x, t) which for every t is a Baire function of x and for every x is a Baire function of t,' but for which sup f(x, t) is not A,-measurableLet B be a non-u,-measurable set in (0, 1) Q8,4); map the points x e B one-to-one

on points t, a (0, 1), and let f(x, ts) = 1 and otherwise f = 0.-\If we map the points x e B one-to-one on points t: of every interval

then in the

same manner we obtain an example of a function f(x, t) which for every t is a

Baire function of x and for every x is a Baire function of t, but for which fu;--n f(x, t) is not )A,-measurable.

Yet if f(x, t) is a Baire function of (z, t), say in the interval J = (0, 0; 1, 1), then sup f(x, t) = g(z) is µl-measurable, and hence (cf. the proof of 10.1.31) t

As to the notion of Baire functions, cf., for instance, Encyklop. d. Math. Wiss. II C9o (M. Frechet-A. Rosenthal), p. 1168, and H. Hahn [2J, 276, 284.

SEQUENCES OF MEASURABLE FUNCTIONS

§10]

121

lim f(x, t) is also. For the set [g() > y] is the projection of the set [f(t, 1') > y] and hence' an analytic set in RI (67,1); thus by 8.2.211 this set is s1-measurable.-Here g(x), however, need not be a Baire function. Example: f(x, t) = 1 on a plane Grset a J and otherwise f = 0; the projection of the Ga-set into the x-axis can furnish an arbitrary linear analytic set.'-If we analogously 1 1\ 1, 1, J and we define f (z, t) - 1 (place a suitable Gs-set in every interval 0, n 1 n on these Greets and otherwise f - 0, then it results that 1im f (z, t) also need not be a Baire function. For a e A the sequence ((f,(a))) is called convergent if there is a b e .R , such that f,(a) --j b.10 The set of all x e A for which the sequence ((f,(x))) is convergent (is not convergent), is called the set of convergence (the set of divergence)

of ((f,)). 10.1.4. If ((f.)) is a sequence of 0-measurable functions on A, then the set of convergence and the set of divergence of ((f,)) are rp-measurable.

By 10.1.21 Iin f, and link f, are 9,-measurable. The sets of convergence and r

of divergence are given by [link f,(t) = lien f,(x)] and [link f,(t) * link f,(z)], respeotively. Hence by 9.2.7 they are o-measurable. BIBUOeaAPHT: The same as in 59,1.

2. p-convergent sequences. Let ((f.)) be a sequence of gyp-defined functions on A. If link f, is (p-defined on A, then the sequence ((f,)) is called yc-ccnvergentu

on A. Thus in order that the sequence ((f,)) be p-convergent on A, it is necessary and sufficient that link f, _, link f, . If p(A) = 0, then every arbitrary r , sequence ((f,)) on A is V-convergent.

10.2.1. If ((f,)) is gyp-convergent on A and f, aiso 9-convergent on A and lim f, =,, lim g, .

By assumption link f, lint f, ,, link g, r

.

g,(,

1, 2,

LIM f, . By 10.1.1 lim

), then ((g,)) is

link g, and

Thus by 9.4.11 link g, 1, link g, also.

, , Instead of lira f. =p f we use the simpler notation f, --, f. By 9.4.11 we obtain r

Y

immediately: Cf. R. ital.: (2], theorem 40.6.11. ' Cf. H. Hahn [2], theorem 40.6.3. u Cf. H. Hahn (I], p. 32, 231; H. Hahn [2], p, 211. This differs from the usual definition in which 6 e R, is assumed, that is, b is assumed, *to be finite. Cf. also p. 23. " Or almost everywhere convergent (if there is no doubt as toy,, in particular in R. for

[CHAP. III

MEASURABLE FUNCTIONS

122

10.2.2. f, --', f and f =, g imply f, -i,, g. Furthermore f,

and ff -'a g

imply f =, g. In the usual manner one shows:

10.2.2 1. f. --o f implies -f, -'o -f. 10.2.22. 1, --*o f implies I f, I -'o I f f 10.2.23. if f. -->, f and g, -, g, then max (f,(x), g.(x)) -+, max (f(x), g(x)) and min (f,(x), g.(x)) min (f(x), g(x)) also.

/

10.2.24. If f, -'o f and g, '-'a 9 and if f, + q, and f -(- g (f. - g, and f - g, f,g, and fg,

.f,

and 9 I are pp-defined, then f, + g. -'o f + g

g, -'o f - g,

-' f

, respectively 1 . f,g, --', fg, f' 9 9. f on A, then f, ->, f on every p-measurable subset of A also. 10.2.3. if f. ), then f, --o, f on 5A; also. 10.2.31. If f, --', f on A;(i = 1, 2, i From 10.1.21 there results immediately:

10.2.4. If ((f.)) is a w-convergent sequence of (p-measurable functions on A, then lim f, is also T-measurable on A.

As a result of 10.2.4 we obtain the following theorem which essentially contains the theorems 9.1.61, 9.2.1, and 9.2.6 as particular cases: , f are .p-measurable and c-,finite on A and if 10.2.5. If the functions f, , f2, is a Baire function on R. , then g(f,, f2, - - - , f,,) is 9-measurg(u, , u2, - , able on A. , n) are defined and finite on a set B =, A, By assumption the fk(k = 1, 2, is go-defined on A. Now first let g be continuous and hence g(f,, f2, - , in R. . Then (for every y e R,) the set Y of the points of R. for which g k y is closed; thus R. - Y is open and hence" a sum of countably many open intervals Ji = (an , a;2, - , a,,, ; b;, , b:2 , ... , bin). By 9.1.2 [a;k < fk(x) < bik] - Aik . A in and hence B - SA; is rp-measurable; therefore A; = A12. A 12. 4

, f,.(i)) ? yJ are also. Thus by 9.1.21 g(f,, f2, . . , fn) is [g(f,(x), f2(1), go-measurable on A, by which the contention is proved for the g continuous in R. .

Now if the contention is valid for every function g,(u, , u, , - - , of a convergent sequence ((g,)), then by 10.2.4 it is also valid for lim g, . Hence"` the 01

contention is valid for all Baire functions g on Rn .

That the contention of 10.2.5 is no longer valid if g is assumed to be only is shown by the example at the end of §9,2. 10.2.6. Let the functions f and f.(v = 1, 2, -) be go-measurable on A and set 14 Cf. footnote 7, p. 120.

Cf. H. Hahn [21, theorem 20.3.3, for instance. " Cf. H. Hahn [21, theorem 35.1.7.

SEQUENCES OF MEASURABLE FUNCTIONS

§101

Am(q) = S [II f,(x') - f(±) Ii

>= q].15

123

Then in order tluit f, ->, f on A, it is suf-

0

cient and, if p(A) is finite," also necessary that for every q > 0: rp(Am(q)) -+ 0.17 SUFFICIENCY: Let A' (=, A) be the set of those x e A for which f and all f, are defined. For all x e A' - A.,(q) we have I I lim f, - f (1 S q and

lIm f, -- f U 5 q, where m = 1, 2, . If we set D(q) = DAm(q), then by m §1 (1.21) this is valid also for all x e A' - D(q). But since ip(Am(q)) --+ 0, by 3.2.21 p(D(q)) = 0. We set q = k and D 1 k 1 = Dk ; then for every positive integer k we have: 11 G -a f, - f 11 5

and

U liim

f, - f

11 s k on A' - D*, with

f, = f on A' - SDk and ,p(Dk) = 0. But then by §1 (1.21) ,lim f, = lim , k did not converge to 0; then .p(SDk) = 0. NEcassrTY: Suppose that k

we would have urn o(A m(q)) > 0, and hence by 3.5.11 1p(Lim A.,(q)) > 0. But

,

m

this is impossible, since at the points of Lim Am(q) we cannot have f, M

In the same manner one obtains : 10.2.61. If to 10.2.6 f and the f, are also ce finite on A, then the set A,(q) can be replaced by the set S [I f ,(I) - f(1) I _' q]. ,i.m

10.2.62. Let ((f,)) be a sequence of (p-measurable functions on A and set B.,(q) _ S [11f,(1) - f,'(.f) 11 q]. Then in order that ((f,)) be c-convergent on A, it is sufficient and, if cp(A) is finite, also necessary that for every q > 0: sp(B,.(q)) -+ 0. SUFFICIENCY: Let A'(=, A) be the set of those x e A for which all f, are defined.

We set DB.,(q) = B(q); since lp(Bm(q)) - 0, by 3.2.21 ip(B(q)) = 0. Further0

more we set B = S13(k ; then -p(B) = 0 also. Now let x e A' - B and let

J

e > 0 be arbitrarily given; we choose k* such that k* < e. Since x

e B (j),

11 As to the definition of Hat - a2 11, of. §3 (6.1).

1 This condition is essential. Example in R1: f, - 1 in [v, a + 11 and f, - 0 otherwise; furthermore let ip = of . 17 Here, as always, 0 means the absolute-function of 9.--If one replaces the set function O, totally additive in the c-field all, by its associated measure function ;5, regular in E (according to §6,5), then in the part of 10.2.6 that refers to "sufficient" one can weaken the assumption "gyp-measurable" to "9o-defined"; for in the proof one only need replace A,(q) by a

measure-cover A;,(q) for 3 (§6,3). The analogue is valid also for the parts of 10.2.61, 10.2.62, and 10.2.63 that refer to"sufcient"- (then in 10.2.63 also B need not be p-measur-

able).-In the parts of 10.2.6-10.2.63 that. refer to "necessary", the assumption of the ,p-measurability of the f. is essential; this is shown by the example of footnote 26, p. 127.

* there isanm,such that xiaB...

we have I f, - f,

CHAP. III

MEASURABLE FUNCTIONS

124

<

k -1)

also; that is,, for v 9 m* and v' z m*

(<e). Hence ((f,)) is convergent for everyx a A' - B,

and thus, since o(B) = 0, ((f,)) is gyp-convergent on A. NEcEssrrr: This results from 10.2.6, since B..(q) C, A,.

Let the sequence ((f,)) be convergent on B (CA) and there set f = lim f, ; then ((f,)) is called uniformly convergent on B (to f) if to every 8 > 0 there is vj 18 In the usual a vj , such that II f,(x) - f(x) f I < a for all x e B and all v way one proves that ((f,)) is uniformly convergent on B if and only if to every vj 8 > 0 there is a vj , such that II f,(x) - f,, (z) I I < 6 for all z e B and all vj . and v' Now we obtain the "theorem of Egoroff": 10.2.63. In order that the sequence ((f,)) of 9p-measurable functions be 9-convergent

on A, it is sufficient and, if p(A) is finite", also necessary that to every e > 0 there

be a 9p-measurable set B a A with r(A - B) < e, suck that ((f,)) is uniformly convergent on B.

SUFFICIENCY: By 10.1.4 the set of divergence D of ((f,)) is p-measurable.

But for every 0-measurable B C A on which ((f,)) is convergent we have o(A - B) z o(D). Hence if to every e > 0 there is such a B with ip(A - B) < e, then O(D) = 0. NECESSITY: Let A' (=, A) be the set of those x e A for which all f, and lim f, = f are defined. Furthermore let ((qk)) be a sequence of positive numbers with qk ---> 0.

By 10.2.6 there is an mi., such that -p(A.,(gk)) < 2k

for m Z ink. If we set B = A' - SA.,,(gk), then A' - B a SAk(gk) ; therefore k

by 3.2.3 o(A' - B) < E

k

, and hence o(A - B) < e. But on B we have

f. - f H < qk for v ? mk ; that is, ((f,)) is uniformly convergent on B. Yet in general there is no B C A with o(A w- B) - 0, such that ((f,)) is uniformly convergent on B. Example in R, : Let A = (0, 1), p = ;&I , and f, = 1 in (0, 1) , otherwise f, = 0. Then f, -+ 0 on A ; but the convergenpe is nonuniform on every B a A which has the point 0 as point of accumulation, and thus, in particular, on every B C A with i (A - B) = 0. 10.2.64. Let p(A) be finite and let f(z, t) for every t e (a, b) be p-measurable on A

as a function of x and for every x e A be continuous in (a, b) as a function of t. If lim f (z, t) = f (x) for all x e A, then to every e > 0 there is a p-measurable set 19 Here f need not be finite; cf. footnote 10, p. 121, and p. 23 as well as H. Hahn [I], p. 251, H. Hahn ]21, p. 216. 19 That this condition is essential, is shown by the example of footnote 16, p. 123.

SEQUENCES OF MEASURABLE FUNCTIONS

1101

125

B Q A pith p(A -- B) < e, such that the convergence of f (x, t) to f (x) i uniform on B.

Let t, -- a be a decreasing sequence in (a, b) and let f,(x) - sup f(x, t) and '('.4') f,(x) = inf f(x, t); by 10.1.3 the f,(x) and f,(x) are q,-measurable and we have

It(".) lim f, = f(x) and lim f,(x) = f(x). Thus by 10.2.63 (applied to the sequence /i , fi ,1, , fz , f.. f , -- f) there is a 4-measurable B A, such that o(A - B) < e, II f',(x) - f(x) II < q and 11 f,(x) - f(x) II < q on B for v k vo

f,(x) for t e (a, t,), we have also II f(x, t) - f(z) II < q f(x, () on B for t e (a, t,) and v z vo. But since f,(x)

10.2.7. Let A = S(A,) where the ep(A,) are finite; then in every double sequence

f, of cp-measurable functions on A with lim (lim f,..w) -, f thers is a simple w

w

sequence ((f.,)) with lim flnr.nr =, f

We can assume: A,+I 2 A. . Set lim fw,w = f.. Since by assumption fw --,, f on A and hence also on A., by 10.2.4 and 10.2.63 there is a B, Q A, and an m, , such that ip(A. - B.) < and I I fw, - f 11 < , on B,. Furtheron A and hence also on B., by more since by assumption lim w

10.2.63 there is a C, C; B, and an n, , such that tp(B. - C,) < 211 and fw, II < 1 on C, . Therefore we have q(A, - C,) <

II

Now in

order to show that f.,,, w, --, f on A, it suffices by 10.2.31 to show that this is so on every single set At . Let q > 0 be arbitrarily given and let ri < q. On C. we have I I

f 11 < 1; thus if n i' n*, we have on D C.: ,II f.,.., - f 11 < .zw

v

n*(
At - C, a A, - C, for r z i. Thus as a result of .p(A, - C,) <

i we have

I by 11(1.21)and 3.2.3:p(A;-DC,)=ip(S(A,-C,))<E ,; ,, 20-1 ,tw van

- 1 for

n k i, and hence O(E,) <

=

2w-2

for n z i and n k n*. Therefore 0(9,) --i 0,

and from 10.2.6 it follows that f,,,,,

--*, f on A{ .

BIBLIOGRAPHY: The following references deal with the amp - M. ; extensions to the case of a general c were given by H. HABN (11, p. 556.-As to 10.2.6 and 10.2.61: C. AxzZLA, Mem. lot. Bologna (5) 8 (1899/1900), p. 135; E. BOREL (21, p. 37; H. LEBESaUE, C. R. Paris 137 (1903), p. 1229; 149 (2909), p. 102; (31, p. 10; D. Tn. EGOBOFF, C. It. Paris 152 (19t1),

MEASURABLE FUNCTIONS

126

[CHAP. III

p. 244. Theorem 10.2.63 is due to D. Tn. Eaoxorr, loc. cit.; other proofs were given by F. R,Esz, Acta Univ. Szeged 1 (1922), p. 18; Monatshefte f. Math. u. Phys. 35 (1928), p. 243; cf. also W. S1EEPI9sK1, C. It. Soc. Sc. Varsovie 20 (1928), p. 84. Theorem 10.2.7 is due to

M. Fa>£cnsT, C. R. Paris 140 (1905), p. 28; These, Paris 1906 - Rend. Circ. mat. Palermo 22 (1906), p. 15.

3. Asymptotic convergence. Let A again be a gyp-measurable set and first let pp(A) be finite; let the functions f, and f be c-defined on A. We say: the sequence ((f,)) is asymptotically convergent to f on A if for every q > 0 there are 9-measurB, .20 able sets B. a A with P(B,) --- 0, such that A[II f,(z) - f(z) )I -> q] If the f, and f are assumed to be V-measurable on A, then one can say in a simpler manner: ((f,)) is asymptotically convergent to f if for every q > 0

0([IIf,(!) - f(x) 11 ? q]) -- 0.

(3)

We obtain the same simple definition in general, that is, also for non-cp-measurable functions, if we extend the set function -p, totally additive in the a-field V,

.to a measure function, regular in E, according to §6,5 and we designate this measure function also by ip. If p(A) = 0, then on A every arbitrary sequence ((f,)) is asymptotically convergent to every arbitrary function f. If now ,p(A) is not supposed to be finite, let us always assume here in No. 3 that A is the sum of countably many (gyp-measurable) sets A; with finite p(A;).

Then we say: ((f,)) is asymptotically convergent to f on A if ((f,)) is asymptotically convergent to f on every (gyp-measurable) subset A' of A with finite o(A'). A necessary and sufficient condition for this (resulting from 10.3.4 and 10.3.41) is that ((f,)) is asymptotically convergent to f on every set A; 21 Hence as to

the following theorems it will suffice to prove the asymptotic convergence for the case that c(A) is finite. 10.3.1. If f is V-finite on A2, then for the asymptotic convergence of ((f,)) to f on A it is suffwient that for every q > 0 we have: ip([i f,(t) -- f(t) I > q]) --* 0. This follows immediately from II f,(x) - f(x) 11 S I f,(x) - f(x) I. Yet, even if p(A) is finite, the condition of 10.3.1 is in general (that is, for non-,p-measurable functions) not necessary for the asymptotic convergence of ((f,)) to f. Example: Let A = [0, 1) and p = A,. According to the beginning of §8,4, A = SBk where the At are disjoint non-jul-measurable sets with k

A,(Bk) _ m > 0 (for k = 1, 2,

). We set f = k on Bk , f, = k on Bk for k 9 v, and f, = v on Bk for k > v; then f, -- f on A. On Bk we have for k v:

11f- f, 1i = 0 and for k > v20 If one wants to express the dependence on ,', then one can say also more explicitly: ((f,)) is gyp-asymptotically convergent to f, or: ((f.)) is asymptotically convergent to f for V. $1 For instance, the sequence ((J,)) of footnote 16, p. 123, is asymptotically convergent

to0onR,.

--'.

This assumption is superfluous, provided we replace f.(x) - f(x) by 0 if f,(x) - f(x) or

SEQUENCES OF MEASURABLE FUNCTIONS

§10]

I-

V

v

127

1

1 - 1+-v-' 1+;,

Hence for every q > 0 and for v 1 we have kl 4 that is, ((f,)) is asymptotically convergent to f on A. But

11 ? q]) = 0;

fA1[I f,(.t) - f(x) I z 1] = µl(kSVBk) z m > 0

(for all r). For gyp-measurable functions, however, we obtain: 10.3.11. If f is ia-measurablen and cp-finite" on A, then in order that the sequence of cp-measurable ((f,)) be asymptotically convergent to f on A, it is sufficient and, if c(A) is finite,* also necessary that for every q > 0 we have:

-([I f,(x) - f(x) 1 ? qJ) - 0. SUFFICIENCY: This is already included in 10.3.1.

NucsSsiTY: Suppose there

is a q > 0 for which we do not have p([I f,(z) - f(i) I 2t qJ) -90; then there is 0 > k. By 9.1.4 a k > 0 and infinitely many v for which p([I f,(x) - f(.i) there is a B c A with p(A --- B) < 2, such that f is bounded on B. Then I

k

I ? qJ) > 2 for infinitely many v. But since f is bounded on B, there is a q' > 0, such that (for all x e B) i f, -- f I a q implies II f. - f II > q'. Therefore we have also p(B[II f,(.) - f(x) II q']) > 2 for infinitely many P.

0(1311 f,(.t) -- f(.f)

Thus o(B[II f,(x) - f(z) 11 z q'J) -- 0 is impossible, and hence

q(A[II f,(x) - f(t) II

q'])

0

is also; that is, ((f.)) cannot be asymptotically convergent to f on A. From 10.2.4 and 10.2.6 there results immediately: 10.3.2. If the f, are ip-measurableYB and f, --,q f on A, then ((f,)) is asymptotically convergent to f on A. u This condition is superfluous, as it results subsequently from 10.8.84. This condition is essential for the part referring to"necessary." Example: A - 10, 11,

p e µt , f, - r and f - +- on [0, 1 ].-As to "sufficient", cf. footnote 22. That this condition is essential, is shown by the example of footnote 16, p. 123. to This condition is essential. Example: Let A - 10, 1 ] and . - µ, . According to the beginning of ¢8,4 we have A - $Bk where the Bt are disjoint non-µ,-measureable sets with k

). We set C, - $ Bt, f,(z) - 1 for z e C,, snd f,(x) - 0 k=, for z e A - C.. Then f(x) - lim f,(x) - 0 on A, and hence µ,([IJ f,(-*) - f(t) ]] Z iJ) m > 0 (for k - 1, 2,

k m(for v - 1 , 2,

) ; that is, ((f,)) is not asymptotically convergent to f on A.

[CHAP. III

MEASURABLE FUNCTIONS

128

The converse of this is not true. Example: Let A = [0, 1] and v

2' + v'(v' = 0,1,

,

2'-') let f, = 1 in [2;,

1l

v'

for

and otherwise f, = 0.

1 we have µ,([I f,(t) I q]) _ for v = 2', Then for every positive q , 2'+' - 1. Thus the sequence ((f,)) is asymptotically convergent 2' + 1, to 0 in [0, 1], but is not convergent at any point of [0, 1). 10.3.3. If ((f,)) is asymptotically convergent to f and f, =, g, , then ((g,)) is also asymptotically convergent to f.

For p([II f,(t)

- f(j) II

q]) _ o([II 9,(1) - f(t) II

q])

10.3.31. If ((f,)) is asymptotically convergent to f and f =, 9, then ((f.)) is also asymptotically convergent to g. or

0([11 f.() - f (t) I I is q)) - o([1 I f,(.f) - g(l) I I qD. 10.3.32. If ((f,)) is asymptotically convergent to both f and g, then f =, g.

Since If- II ;9 Iif.-fII+IIf.-gII,wehave 0((I l M) - 9(t) II

q]) 65 0

11 f,(z) - f(t) I I

2D

+ ([ I I f,(t) - g(t) I I 2D Therefore .

([ II f.(±) - f(s) II

imply o([II f(x)

])_0and([IIf.(±)

- g{) I I

2 -' 0

- g(t) II k q]) - 0. Since this is true for every q > 0, we have

([ I I f(t) -- g(1) 119

= 0 for m - 1, 2,

m)

[fW

.

Hence as a result of

[II f(-) - 9(t) 11 9

we obtain off (t).* g(t)] - 0; that is, f -, g. 10.3.33. If ((f,)) and ((g,)) are asymptotically convergent to f and g, respectively,

and if f, i'r g, , then f k, g also. We have to show: [f(t) < g(:t)] ==, A. If (ri , a{) (i = 1, 2, pairs of rational numbers with rr < at, then

[f(t) < g(t)J - S[f(t) <

) are all possible

ad.

Thus it suffices to show: for every pair of rational numbers (r, a) with r < a we have [f(t) < r] [g(±) > aa] -, A. We met B = [f(6) < r] [g(t) > 81 and

B, = B [f.() <

-

a]

O) > r

2

8]. Because of the asymptotic con-

SEQUENCES OF MEASURABLE FUNCTIONS

*101

129

vergence we have:,p(B,) -- O(B); since B, c [f,(1) < g,(1)], we have by assumption #(B,) = 0, and hence O(B) = 0 also. 10.3.4. If ((f,)) is asymptotically convergent to f on A, then also on every q~measurable subset of A.

10.3.41. If ((f,)) is asymptotically convergent to f on each of the sets A;

(i - 1, 2,

), then also on A = SA i. i

It suffices to prove this for finite V(A); furthermore we can assume the Aj to be disjoint. Then p(A[II f,(1) - f(±) II ? q]) s o(A;[II f.(t) - f(t) II ql)

and the series on the right-hand side converges uniformly for all v, since its terms are not greater than the terms of the series Fr o(A;) = ,p(A). Thus' we can let v - co term by term, and hence as a result of lim -p(A i[I I f,(1) - f (l) I1

q) = 0

q1) --+ 0. we obtain p(A[11 f,(t) - f(t) II 10.3.5. If ((f,)) is asymptotically convergent to f, then ((-f,)) is asymptotically

convergent to -f. 10.3.51. If ((f,)) is asymptotically convergent to f, then ((I f, I)) is asymptotically, convergent to I f I.

For 111 f I - If 111 s II f -- f I1 implies

[II

I f,(t) I - If(l) III

q] r-

[I I f.(t) - Al) 11 ? q]. 10.3.52. If ((f,)) and ((g,)) are asymptotically convergent to f and g, respectively, then ((max(f,(x), g,(x)))) and ((min(f,(x), g,(x)))) are asymptotically convergent to max(f(x), g(x)) and min(f(x), g(x)), respectively. For 11 max(f,(x), g,(x))

- max(f(x), g(x)) II

s max(II f(x) -

g(x) 11)

implies

[II max(f,(t), g,(t)) - max(f0f)"g(l)) ll z q] a [I I f,(t) - f() I I z q] + 111 g,(:) - g(t) I I '- q]. 10.3.53. If the sequences of ,s-measurable functions ((f,)) and ((g,)) are asymptotically convergent to f and g, respectively, and if f + g and all f, + g, (f - g and

all f, - g,) are p-defined, then ((f, + g,)) is asymptotically convergent to f + g

(((f, - g,)) to f - g). Let A'(=, A) be the set of all x e A for which all f, , g, , f, and g are defined.

We set A' = A, +A,+As+A,+As, where

Ai=A'[-cc

A4 = A'[g() = +

27 Cf., e.g., H. Hahn (11, p. 291.

],

As

A'[f(t) =+cc1,

As = A'[g(.f)

ao ].

[CHAP. III

MEASURABLE FUNCTIONS

130

10.3.41 it suffices to give the proof for each of the sets A i(i = 1,2,3,4,5). To Al we can apply 10.3.11 (and footnote 23,p. 127):

According to

since :;(f,+g,) - (f+g)I s If, - fI+I g, - gI,wehave A1[I (f,(x) + g,(t)) - (J(x) + g(x)) I

4]

CA, lf,(t) - f(t) I ? 2]+ A,[I 9,(t) - g(t) i z 2] hence from gyp( A,LI f,.(.) - f(±) I ? 2

- 0 and (A1[l g,(x) - g(l) I ? 2D -' 0

it, follows that o(A1[I (f,(t) + g,(t)) - (f(t) + g(t)) I z q]) -+ 0 also.-Now

we give the proof for A2 ; (it is quite analogous for A3 , A4, and Ab). We set A2 = B + B1 + B2 + . , where B = A2[g(i) = - cc ] and Bi = A2[g(t) ? -iJ (i = 1, 2, ) ; according to 10.3.41 it again suffices to give the proof for each of the sets B and B,(i = 1, 2, ). For B the contention is trivial, since f + g is ,p-defined and f = + c* on A2 and thus P(B) = 0. On B; we have f = + 00 and f + g = + co ; therefore in order to prove the asymptotic convergence of ((f, + g,)) to f +g on B; , we have to show that, for every finite p,

lim,p(B;[f,(t) + g,(t) s p]) = 0. ,

But on B;[f,(x) + g,(r) 5 p] we have either - (i + 1) p + i + 1, or g, < - (i + 1), and hence (as a result of g i i (i 1

Ilg-g,Il>II(i+1)-ill =

1+

g, , and hence f,

-i on Bi)

±1)!1+i

1

(1+i)(2+i)'

Thus we have

Bi[f,(t) T g,.(x) < p] a Bi[f,(t)

p+i+

1J

-- Br[II g(t) - g,(.+) II

1

Jj.

(1-+ -z)(2 -I-- i)

Now because of the asymptotic convergence of ((f,)) to f, that is, to + 00 on B, ,

we have lim ry(B;[f,(t) 5 p + i + 11) = 0 and because of the asymptotic eon,

(' vergence of ((g,)) to g we have limo Bi [ 11 g(x) - g,(t) 119

1

(1 + 1)(2

i)])=

0;

from this the contention follows. 10.3.54. If the sequences of cp-measurable functions ((f,)) and ((g,)) are asymptotically convergent to f and g, respectively, and if f g and all f, g, are p-defined, then ((f, g,)) is asymptotically convergent to f g. Again, as in the proof of 10.3.53, we proceed from A' = A, + A2 + A3 + A4 + A5 and we prove the contention for each of the sets Ar . We set Al = S,4',, where A;. = A'[- n f(t) n n

order to show that the contention is valid on Al , it suffices by 10.3.41 to prove

SEQUENCES OF MEASURABLE FUNCTIONS

§10]

131

that it is valid on every set A a and for this purpose 10.3.11 (and footnote 23, on p. 127) can be used: Since I f,g, - fg 15 I g. I I f, - f I + I f I I g

A, -

g,(1) - g(i) I

11 we have:

I fg, - fgI 5 (n + 1) (If, - fI + Ig, - gI). Thus An[I f,(x)g,(t)

A;, [I f,(.t)

- f(z) I z

2(n

- f(*)g(t) I

41

+i)] + A;, [I g,(.) - g(x) I ? 2(n + 1)] + A.[I g,(.f) - g(t) I

11.

Because of the asymptotic convergence of ((f,)) and ((,q,)) the 'P-value of each of the three sets of the right side converges to 0 for v -+ oo ; hence

lim 1(A' [I f,(t)g,(t) -- f(I)g(t) I ; 4l) = 0; that is, f g, is asymptotically convergent to fg on A', .-Now we give the proof for As (it is quite analogous for As , A4, and-A5). We set: A, = C + SB;, + SB'I U

it

where

C - A=[g(1) = 0], B', = A2 [g(f) Z n], and B'* = A, [g() 5 According to 103.41 it again suffices to give the proof for each of the sets C, B' , B; B. For C the contention is trivial, since f g is c-defined and f = + co on A2 and thus q,(C) = 0. On B;, we have f = + oo and fg - + ac, and hence we have to show that, for every finite p, lim 0(B,1f,(t)g,(t) p]) = 0. But on

g, and hence f,

f,(&)g,(t) 4 p1 we have either-1 2n and hence (as a result of g

n

11g-g,Ii> iin1_

1

1

2np, or g, < 2n

on B,.) iI

2n I

ffi

1

1

n

n + 1 - 2n + 1 = (n + 1)(2n + 1)

Thus we have

B;,[f,(±)g,(t) 5 p] BK[f,(t)

2np] + B', [II g() - 9.(.f) II

(n + 1) (2n +-I) From this the contention follows because of the asymptotic convergence of ((f,)) to f + oo) and of ((g,)) to g. The proof for B is analogous. 10.3.53. If ((f,)) is asymptotically convergent to f and if 1 and all 1 are ip-defined, then (t

f, 1 is asymptotically convergent to

[CHAP. III

MEASURABLE FUNCTIONS

132

WesetA - B + SB.+C,where B=A[f(t)= 01, B,=ACIf(t) I z and C is the set of the x e A for which f is not defined. According to 10.3.41 it suffices to prove the contention for each of the sets B, B. , and C. For C it

is trivial, since O(C) = 0; it is also trivial for B, since

is gyp-defined and thus

O(B) = 0. But on B. there is to every q > 0 a p >l 0, such that on

B. - B. [I f,(t) I 6 1 1 the inequality

f-f

q implies II f, - f II iQ p;u thus II

q] Q B,[11 f,(t) - f(*) II iG p) + B. [I f.(t) I

B. I_Ilf.(z)

]

f(x) Because of the asymptotic convergence of ((f,)) to f we have I

lim O(B.[II f.(±) - f(f) II z p)) = 0 and=' limo B. [I f,() 19 2nD = 0, and hence lim (B k fi(x) - f(.t) II

I

z q]) = 0 also; that is, (G)) is asympJJJ

31

totically convergent to

on B..

From 10.3.54 and 10.3.55 there results: 10.3.56. If the sequences of p-measurable functions ((f,)) and ((g,)) are asymptotically convergent to f and g, respectively, and if 1and all - are cp-defined, then 9

9V

((i)) is asymptotically convergent to In order to obteiEn the analogue of 10.2.62 for asymptotic convergence, we first show:

10.3.6. Let ((f,)) be a sequence of (p-defined functions on A and set A,,.(q) _ III f,(1) - f.- (1) 11 k q). If to every e > 0 and q > 0 there is a vo , such that e for v vo and v' 9 vo , then in ((f,)) there is a subsequence ((f,,)) which is p-convergent on A.

Let A'(=, A) be the set of all x E A for which all f, are defined. By assumpa For if f, and f have the same sign, then flf ferent signs and if both I f. I > Zn and If I >

we can set p -

2n

8inoe on B.

121

f

f, - f II . If f, and f have dif-

then II f, - f II > 2111

!,

f

.

Hence

.

rl f>(1) 15]wehaveflf_frfl 119 n 11

11

in

(1 + n)(1 + 9n)'

133

SEQUENCES OF MEASUPAB24 FUNCTIONS

§101

tion there is a v;, such that 0 (A,,, (`;11 < 2{ for v can suppose: v;+1

We set Ak = iB A,,yt1

v;

v; and v' z v; ; here we

(i); then on A' -- Ak, for

m < n, we have:

k

it f.. -- f,,. II

E II f+,+, - f.! II .sib < E%-120 1

..S i 5 u--1

<

-i ;

iW

that is, the sequence ((f,,)) converges uniformly on A' - Ak . Now p(Ak) iE

i 1. ,p 1 A,i,i+, (i))
__

1; that is, #(Ak) -* 0. Hence from 10.2.63,

taking\\\ footnote 17, p. 123, into consideration, we obtain the qo-convergence of W-1)) -

Now for brevity we call a sequence ((f,)) asymptotically convergent on A if there is a function f to which ((f,)) is asymptotically convergent on A. Then, q], we obtain: setting again A,,, (q) = 111,,(x) - f,- (x) II 10.3.61. In order that the sequence ((f,)) be asymptoticaUy convergent on A, it is necessary, if p(A) is finitee, and it is sufficient, if the f, are gyp-measurable, that to every e > 0 and q > 0 there be a re , such that f(A,,,(q)) < e for v ro and

v,?: ro. NEczsarrr: Since saCII f.(1) - f(t) II k 2D -- 0, there is a vo, such that is

CI I f,(.1) - f(t)

112]) <

for r il. re ; then if v f(t) II

But since it f, - f,.11 O([II

f,-(1) 11 9;

s

vo, we have also

2< 2

II f, - f II + [If,, - f 11, we have 91)

(LII f.(t) -- f() II Z 2D +

f, () - f(x) II ? 1)

thus for v i' ro and v'? re we obtain p(A,,,(q)) < e. Sui'FICIENCY: It suffices to give the proof for finite rp(A). By 10.3.6 there is a subsequence ((f,,)) and an f, such that f,, -, f. Therefore by 10.2.4 and 10.2.6 there is an i, such that tp([II f,,(.) - f(i) II ? ql) < e and r; ? vo . Then by assumption for every r va we have ,p([I If.(-*) - f,,(t) 11 i= q]) < e. Now

(ql f.(1) - f(t) II

2q))

41(111 f,(1) - f,;(-A) II

ql) + 0([II f,,(1) - f(*) II

4)),

w That this condition is essential here and in 10.3.011, is shown by the example of footnote 16, p. 128.

134

[CHAP. III

DMEAS['RABLE FUNCTIONS

2ql) < 2e for v vo ; that is, ((f.)) is asymp,f,(z') - f(z) 11 and ,: ergent to f. totically 10.3.611. Let ((jr)) be a sequence of ip-measurable and gyp-finite functions on A. Then in order that ((f.)) be asymptotically convergent to a ip-finite function. on A, it is sufficient and, if ip(A) is finite, also necessary that to every e > 0 and q > 0 there be a vo , such that gypQ f.(±) - f. (x) 1 >__ ql) < e for v

vo and v' z PO .

SUFFICIENCY: Since 11 If, - f.- it 5 1 f. - f., 1, it follows from 10.3.61 that ((f.)) is asymptotically convergent on A. Let A"(=, A) be the set of all x e A for which all f, are defined and finite, and set

A. (q) _ [I f.(t) - f.-(5) I

SA+, (.).

and Ak

k

q[

Then, as in the proof 10.3.6, there results the exist-

ence of a subsequence ((f,)), such that on A" - Ak for k 5 m < n we have < 21_1 , as well as p(Ak) -+ 0. Thus ((f,;)) converges on A" - Ak to a function which is finite there; then the same as the case also on A" - DA,` .

Therefore, since by 3.2.21 ip(A ') -- 0 implies ip(DAk) = 0, by 10.3.2 ((f.)) is asymptotically convergent to a p*-finite function on A. NECESSITY: Let ((f,)) be asymptotically convergent to the pp-finite function f on A. Since

If.'-f.-I

If. -f1 + If.' -f1,

we have

-(II f.(x) - f.-(I) I

ql)

2])+ I f. (z) - f(x) I

f() I ?

2

By 10.3.11 (together with footnote 23, p. 127) there is a vo , such that

([ I f,(x) - f(x) I z 2]) < 2 and o CL I f

"

(5) - f(x) I ? 2J) < 2 for v z re

ql) < e for v z va and v' z vo vo . Hence 0 ([1 f.. (x) I 10.3.62. To every asymptotically convergent sequence ((f,)) there is a 'p-convergent

and v >_

eubsequence.

If p(A) is finite, then the contention results from 10.3.6 and 10.3.61. Thus if A = SAk with finite i,(Ak), there is on Al a cp-convergent subsequence of ((f,)), k

say ((f,,;)); there is on A2 agyp-convergent subsequence of ((f,,;)), say ((f,,,)); there is on Aa a gyp-convergent subsequence of ((f.,;)), say ((f.,;)); and so on. Each of these sequences contains almost all elements of the sequence f,,, , f,,, ,

f,,, , ... , f,,, , ... ; thus ((f,,,)) is p-convergent on every Ak(k = 1, 2, ... ) and hence also on A. 10.3.63. If the sequence ((f,)) of cp-measurable functions is aCyntptotically convergent to f on A and if ((f,;)) is a gyp-convergent subsequence of ((f.)) with f,; --w g,

then g = f.

§10]

SEQUENCES OF MEASURABLE FUNCTIONS

135

((f,)) is asymptotically convergent to g by 10.3.2 and to f as a subsequence of ((f,)). Hence by 10.3.32 g =.n f. From 10.3.62, 10.3.63, and 10.2.4 there results: 10.3.64. If the sequence ((Jr)) of p-measurable functions on A is asymptotically convergent to f, then f is also gyp-measurable on A. BIBLIOGRAPHY: The notion of asymptotic convergence for p - A. and for µ.-measurable functions is due to F. Rir:sz, C. R. Paris 148 (1909), p. 1303 (with the notation"convergence en mesure"), as well as the theorems 10.3.6-10.3.62 essentially; the notation "asymptotic convergence" originates from E. BOREL, C. R. Paris 154 (1912), p. 415; Journ. de math. (6) 8 (1912), p. 192. One owes the elimination of the assumption that the f, be µ.-measurable to Al. FR&HET, Bull. Calcutta Math. Soc. 11 (1921), p. 196 (cf. also E. J. MCSHANE [1], p. 160), and the extension to the case of a general sp to H. HAHN [1], p. 570.-Footnote 26, p. 127, is due to W. SIERPIi;ISSI, Fund. math. 9 (1927), p. 33.

4. Compact sets of rp-measurable functions. Let 9 be the set of point functions f defined on a given set A. This set 21 becomes a metric space (§2,3) whose

"points" are the functions f if the distance of two functions f, and f2 of t is defined

by f, f2 = sup II f1(x) - f2(x) II ; for it is evident that then the metric axioms 1,,, and 2,. (§2,3) are satisfied. Convergence of the sequence ((f,)) of points of RI to the point f e A is now equivalent to uniform convergence (§10,2) of the

sequence of functions ((f,)) to the function f on A.

If (I C A is a compact

set (§2,1), then, according to 2.3.11, this means that every sequence of functions ((f,)) of IS contains a uniformly convergent subsequence. Now we assume A to be gyp-measurable and we consider the c,-measurable functions on A. First we give the following characterization of the rp-measurable functions: 10.4.1. In order that the function. f, defined on A, be rp-measurable, it is necessary and sufficient that to every p > 0 there be finitely many (p-measurable sets A, and .,,

numbers c,(v = 1, 2,

, ,n,), such that A = S A, and II f(x) - c, II < p for ,_1

allxeA.. NECESSITY: We choose m,, numbers c, , c2,

,

such that - oo = c1 <

c, < ... < c.,_1 < c.,, _ + c and I I c,+, - c, I < p(y - 1, 2, ... , m, - 1) ; then we set A, = A[c, S f(i) < c,+,](v = 1, 2, , m, - 1) and Am, = A[f(&) = cm,).i1 SUFFICIENCY: We set g = c, on Al , g = c2 on A2 -- A, , g = c3 on As - (A1 + A2), ... , g = cm, on A., - (A1 + ... + A_,-4); then 1 g is ,p-measurable on A and IIf-9II

then we obtain a sequence ((g.)) of p-measurable functions with g. Hence by 10.2.4 f is ,p-measurable on A. Now we give an analogous characterization of a compact set of gyp-measurable functions: 31 We remark for later use that here the A,(v - 1, 2,

, m,) are disjoint.

[CHAP. III

MEASURABLE FUNCTIONS

136

10.4.11. In order that the set Ci of (p-measurable functions, defied on A, be compact, it is necessary and sufficient that to every p > 0 there be finitely many ro-meas, , m,) and to every f e (a£ there be m, numbers c1 , r., , 1, 2, urable sets A, (v

c.,,, such that A = S A, and II f(x) - c, II < p for all x e A, . ,-1

NECESSITY: According

gk a (Z(k = 1, 2,

,

to

2.3.12

there

finitely

are

many functions

r) forming a net in (,Y. By 10.4.1 there are finitely many 2

Vkh

(p-measurable sets Ak, and numbers ck, (r = 1, 2) .

,

mk), such that A = S Ak, ,_1

< 2 for all x e At, . We form all the intersections ; ... ; pr = Air,.A2r,..... A.., (for vi - 1) 2, ... , m1 ; v2 , 1, 2, ... and 119,,(x) --

1, 2,

ck, I I

and we designate the non-empty ones by A1, As ,

,

then A = S._1A, . Since the gi , g2 , f e CE a gk

,

, Am,,

, g. form a P-net in (9, there is to every 2

such that II f(x) - gk(x) II < 2 for all x e A. Furthermore to every

for all x e A.. Hence we

A, there is a number c, , such that II gk(x) - c, I I < 2 have

11 f(x) - c, II

it f(x) - 9k(x)11 + 119k(x) - c, 11 < p for all x e A, .

SUFFICIENCY: We have to show that every sequence of functions ((f)) of iZ contains a uniformly convergent subseq uence. We set p = n;1 then, by assump-

tion, for every positive integer n there are finitely many sets A., and numbers for all , m.), such that A = S A., and II f,(x) - c;., II < c;.,(v = 1, 2, n ,_1 x e A., (v = 1, 2, , m.). Thus for every n a system of numbers (c,.1 , C;,,2 , .. , Ci.,,..) is associated with every f; . The sequence ((f{)) contains a subsequence ((f;)), such that the associated numbers ct11 converge to a limit , the associated c (e f 1), the associated numbers ct1, converge to a limit cu , numbers c;1,., converge to a limit ci , . Then ((f;)) contains again a subsequence ((f;)), such that the associated numbers cal converge to a limit ce1 , the asso, the associated numbers ca., ciated numbers can converge to a limit CA converge to a limit cs,,,, ; and so on. In this way we obtain the sequences ; now we consider their diagonal subsequence ((f;)), ((f;)), - , ((f, )), , f,... . If the number c;., associated with fj is designated by f', , f2 , , m.. Thus there is a j,y a fn. , then lim a j., = cn, for all n and for;, = 1, 2, f

such that 11 afn, - ai'.. 11 < w

for j Z j , f z j., and r = 1, 2,

, m. .

Now since A = S,_1A,,,, for nevery x e A there is at legit one v, such that

110]

SEQUENCES OF MIASUEABI,E FUNCTIONS

11f(i) II < n and II f (x) - aa..11 < I. n

137

Hence 11 fi(x) - ft(x) II < n

for j j , f j , and all x e A ; that is, (()) is a uniformly convergent subsequence of ((f{)).

According to the above notion of the compact set of functions, 10.4.11 characterizes the sets of gyp-measurable functions in which every sequence of functions

((f,)) contains a uniformly convergent subsequence. Now in an analogous manner we shall characterize the sets of gyp-measurable functions in which every sequence of functions ((f,)) contains an asymptotically convergent subsequence. Let A be a `s-measurable set with finite -p(A). We consider the set Vj of all gyp-measurable functions on A and map 9 onto the points of a metric space 4,

according to the following rules: Two functions f and g of rr are mapped onto the same point of V' if and only if f =o g. Let the distance of two points a, fi of 4' be defined as follows: if f is one of the functions mapped onto a and g is one of the functions mapped onto #, then we define" the distance (4)

a# = inf (y + p([I!f(!) - g(x)II ?yl)). 0 vs1

This definition evidently satisfies the condition 1., of §2,3. We show that 2., is also satisfied. Let f, g, h be three functions of ar and let a, $, y be the corresponding points of C Then to every e > 0 there is a y' and a y", such that

y' +offlIAt) - g(g)II

v'])
(4.1)

y"+s([IIh(t) - W) 11 k vi)
o([II f(s) - h(s) II ig y' + if l) gio([II f(s) - g(f) 11 9 ii']) + 0([II h(f) - g(t) II

y")),

and hence we obtain from (4.1) by addition:

y' + y" + p(JIl - h(j) II ? y' + y"l) < a# + 7# + e Thus, according to (4), ay < a# + yB + e and, since this is valid for every > 0, we obtain a ' 5 4 + y#; that is, 2., is satisfied. f(_A)

10.4.2. A sequence ((f.)) of functions of $ is asymptotically convergent if and only if the associated sequence ((a,)) of points of 4" is convergent.

Let ((f,)) be asymptotically convergent to f; then for almost all r: 'P([II f.(&) - M) II it el) < e; thus by (4), if a means the point associated with f: a,a < 2e for almost all r, and hence a, -- a.-Conversely let ((a,)) be a convergent sequence of points of One can regard the point of 4, onto which f is mapped, as representing the class of all functions of $ which - v f. a One understands immediately that this expression is Independent of the choice of the functions f(u) mapped onto a($).

[CHAP. III

MEASURABLE FUNCTIONS

138

4) With a, -- a; furthermore let f, e a be a function mapped onto a, and let f e a be a function mapped onto a. Since o([II f,(I) - f(x) II ? y*])

e implies

f,(x) - f(I) i{ z y]) z e for all y S y*, by (4): a,a z min (y*, e). Hence if ((f,)) is not asymptotically convergent to f, then we cannot have a, - a; that is, from a, -> a it results that ((f,)) is asymptotically convergent to f. 10.4.2 1. Let p(A) be finite and let 21 be a set of gyp-measurable functions on A. In order that every sequence of functions of 21 contain an asymptotically convergent sub-

sequence, it is necessary and sufficient that to every p > 0 there be finitely many , m,) and to every f e 81 there be m, numbers 4p-measurable sets 4,(v = 1, 2, c, , cz ,

, c,,, , such that A = S A. and

p(A,[I I f(:)

- c, 11 ? p]) < p.

NECESSITY: In the mapping discussed above let the set A Q be associated with the set 21. If every sequence of functions of 21 contains an asymptotically

convergent subsequence, then by 10.4.2 every sequence ((a,)) of points of A contains a convergent subsequence; that is, A is compact. Thus by 2.3.12 there is a finite

net in A for every p > 0, and hence in 21 there are finitely many

2 functions gk(k = 1, 2,

,

r) with the following property": to every function

f e 41 there is a gk , such thatonsi y + -PQ f(-f) - gk(t) I

y])) < P. Therefore

I

to every f e 9 there is a gk and a y, such that y +'D([II f(t) - gk(t) II ? y]) < 2 By 10.4.1 and footnote 31, p. 135, to every gk there are finitely many disjoint

So-measurable sets Ak, and numbers ck,(v = 1, 2,

,

mk),

such that

II gk - ck, < 2 on Ak, and SAk, = A. Now we form all the (finitely many) intersections A,,, A2,, A,,, and we designate the non-empty ones by 01

A, , A2, -

,

A,,, ; then S A. = A. Now let f e 21; as we saw, there is a

,I

y(;2- 0) and a gk, such that y +'P([II f(x) -gkk(t) II k y]) < 2 ; then y < 2 and 0([I f(x) - 9k() II ay)) <2 also, and hence

I

I f(t) - gk(t) II ? 2]) <

.

Fur-

thermore to every A, there is a number c, , such that 11 gk - c, jl < 2 on A, .

Thus, since the A. are disjoint, E p(A,[II f(I) -- c, II k p]) <

(
SUF_

2 By assumption, for every positive integer it there are finitely many re-measurable sets A., and numbers FICIENCY: Let ((fe)) be a sequence of functions of K. +R

c;n,(p = 1, 2,

,

m,), such that A = S An, and ,_I

(A,,,[!I

ce,,, II z n]) <

.

" We may here assume that the gl, are defined everywhere on A (not only yo-defined).

§11]

139

MEASURABLE AND CONTINUOUS FUNCTIONS

As in the proof of 10.4.11, we now determine a subsequence ((f;)) of ((f;)), such associated with the f; and again designated by a p, , there exist that for the for all n and for v = 1, 2, . , m . We show that that is, e*, --+ limits the sequence ((f, )) is asymptotically convergent. For fixed n there is a jo , such that 116*, --- 1j,,, {{ <

1

7t

for j ?= jo

'z

, j - jo , and p = 1, 2, ... , m.. Fur-

thermore, according to the meaning of the constants a*, , there is an A;,, a A 1

1

such that at every point and an A;.,. a A with o(A1,,) < n and n, a e A - A,,, we have at least one of the inequalities:

{I ff(a) - !* {{ <

(v = 1, 2, ... , m,.),

and at every point a e A - A f.,, we have at least one of the inequalities: 1 Thus for j jo and j' k jo we have , n(v = 1, 2, 11

on A - (A11, + A j,.): ff - ff % {

o(q,.,.) < n, we have'

([I ft)

3n

{{<

-

.

Therefore, since q (A t,.) <

1

n

n < n for j ? joand j'

Now let q > 0 and e > 0 be arbitrarily given; we//choose n, such that

and jo .


and n< e. Then we have for.j ? jo and j' z jo : A(LI{ ff(-f) - f,'-(1;) {{ ? q]) < e, and hence by 10.3.61 ((ff )) is asymptotically convergent. Since by 10.3.62 every asymptotically convergent sequence of functions contains a (p-convergent subsequence, while by 10.3.2 every gyp-convergent sequence of immeasurable functions is also asymptotically convergent, we obtain: 10.4.22. The conditions of 10.4.21 are also necessary and strfcient in order that every sequence of functions of ql contain a ip-convergent subseq ence. BIBLIOGRAPHY: As to the notion of compact sets of functions (G. Ascoua, C. ARZELA, M. FaACHET) cf. Encyklop. d. Math. Wise. 11 C9c (M. FntCHET-A. ROSENTHAL), p. 1145. Theorem 10.4.1 is essentially due to L. TARDINI, Giorn. di mat. 49 (1911), p. 32; one owes

theorem 10.4.11 to P. vsnsss, Fund. math. 7 (I926), p. 244. As to the other parts of No. 4: M. Fatcusr, Bull. Calcutta Math. Soc. 11 (1921), p. 199; Fund. math. 9 (1927), p. 26.

§11. Measurable and continuous functions 1. Measurability of continuous functions.

Always maintaining the terminology

introduced in §9,1 we now in particular assume E to be a metric space (§2,3). Let the function f be defined on A and a e A. Then f is called continuous

on A at a if for every sequence ((a,)) of points of A with a, -- a we have also f (a.) -+ f (a) ' If f is continuous on A at every point a e A, then f is called continuous on A. 85 Cf. footnote 10, p. 121.

140

[CHAP. III

MEASURABLE FUNCTIONS

One proves in the usual way: The function f, defined on A, is continuous at

a e A if and only if to every a > 0 there is a neighborhood U. , such that 11 f(x) - f(a) 11 < a for allxeAU.. 11.1.1. In order that the function f, defined on A, be continuous on A, it is necessary and sufficient that for every y e J, the sets [f(t) closed in A.

y] and [f(-*)

y] be

Nncassrr : Let a e A be a point of accumulation of the set [f(f) i' y]. By 2.3.1 this set contains a sequence ((ak)) of different points with lim ak = a. k

Because of the continuity off we have lim f (at) = f (a). Thus, since f (ak)

y,

k

y, and hence by 2.1.611 the set [f(t) 2; y] is closed in A. we obtain also f(a) SUFFICIENCY: Suppose f not to be continuous on A and let a e A be a point of discontinuity of f. Then there is a sequence ((ak)) of points of A with ak -, a, such that f(ak) --- f(a) is not valid. Thus there is either a number g > f(a) or a number h < f(a), such that for infinitely many k we have f(ak) z g or f(ak) g h, respectively. If we now set y = g or y = h, respectively, then either the set [f(i) ? y] or the set U(i) S yj does not contain its point of accumulation a(eA), and hence by 2.1.611 at least one of these two sets is not closed in A. 11.1.11. If the function f is defined on A, if the set Y is dense in R, , and if for

every y t Y the sets [f(i) z y] and U(t) 5 y] are closed in A, then f is continuous on A. The proof is the same as the proof of sufficiency in 11.1.1, since we can choose g e Y and h e Y.

11.1.12. In order that the function f, defined on A, be continuous on A, it is necessary and sufficient that for every y e R, the sets [f(i) > y] and U(i) < yj be open in A.

For [f(i) z y] - A -- [f(x) < yj, and hence by 2.1.3 the set [f(t) k y] is closed in A if If(i) < y] is open in A, and vice versa. The analogous statement is true for the set U(1--) (i) 9 y]. Thus the contention follows from 11.1.1. In the same way we obtain from 11.1.11: 11.1.13. If the function f is defined on A, if the set Y is dense in R, , and if for every y e Y the sets [f (l-) > yj and f(I) < y] are open in A, then f is continuous on A.

Now let us assume the open sets of E to be cp-measurable and let A designate If B is open in A, then B is also immeasurable; for, according to §2,1, B = AG where G is an open set of the space E. 11.1.2. If the open sets are q,-measurable, then every function f which is continuagain a gyp-measurable set.

oas on the gyp-measurable set A is gyp-measurable.

For by 11.1.12 every set [f(i) > yj is open in A and hence 9,-measurable. 11.1.21. If the open sets are cs-measurable, then every Baire functionE6 on A is rp-measurable.

Let a be the set of all .p-measurable functions on A. By 11.1.2 all continuous 31 Of. footnote 7, p. 120.

fill

MEASURABLE AND CONTINUOUS FUNCTIONS

141

functions on A belong to tr; by 10.2.4 f, a and f, --> f imply f e W. From this it resultssr that every Baire function on A belongs also to j5. But the converse of 11.1.21 is not true, as we now shall demonstrate. 11.1.3. If there is a set C a A which =,, A and has the power bt, then the set of all V-measurable functions on A has a power

2".

Every function which equals 0 on A - C is p-measurable on A; but there are 2x such functionsie.

11.1.31. If A is separable Q2, 2) and if there is a set C C A which =, A and has the power bt, then there are c-measurable functions on A which are not Baire functions.

This follows from 11.1.3, since the set of all Baire functions on A has only the power 1`t", while 2" > K'e. The conditions of the theorems 11.1.3 and 11.1.31 are satisfied for A = R. and V = is. because of 8.2.83 and the fact that every (non-empty) perfect set of R, has the power W. 2. 0-contnuous functions. We call the point function f (defined on the set A) V-continuous on A if the set of all x e A at which f is not continuous on A is a zero-set for v. 11.2.1. If f is c-continuous on A, then I f I is also co-continuous on A.

For lim f(x) = b implies line I f(x) (_ I b 1. s-+.

'T_4

11.2.11. If f and g are v-continuous on A, then max(f(x), g(x)) and min(f(x), g(x))

are also tenuous on A. If f and g are continuous at a point a e A, then max(f(x), g(x)) is also continuous at a. Thus if B', B", B designate the sets of discontinuities of f, g,

max(f(x), g(x)), respectively, then B a B' + B", and hence B' =, A and B" =, A imply B =, A. In the same way one proves: 11.2.12. If f and g are c-continuous on A, then each of the functions f + g, f - g, f- g, g is also p-continuous on A, provided it is defined on A.

But a ;q-continuous function of a p1-continuous function need not furnish a pi-continuous function. Example: Let g(y) = 0 for y = 0 and g(y) = 1 otherwise; let f(x)

for x

(with p prime to q) and f (x) = 0 for irrational x.

The functions g and f are ps-continuous. But g(f(x)) (= 0 for irrational z, and 1 for rational x) is not µ,-continuous. Now let ((f,)) be a convergent sequence of functions defined on A and let f 37 For instance, of. H. Hahn (21, theorem 86.1.7. +4 For instance, of. H. Hahn 121, theorem 5.8.1. Cf. H. Hahn (2), theorem 40.3.42. For instance, cf. H. Hahn 121, theorem 8.2.4. u For instance, of. H. Hahn [2), theorem 18.3.1.

[CHAP. III

MEASURABLE FUNCTIONS

142

be the limit of this sequence. ((f.)) is called simply-uniformly convergent to f at the point a e A if to every S > 0 and every vo there is a v >_ vo and a neighborhood U. , such that 11 f,(x) - f(x) 11 < S for all x e A U. . A well-known theorem42 says: Let f, -- f and let, the f, be continuous at the point a e A. Then in order that f be also continuous at a, it is necessary and sufficient that ((f,)) be simply-uniformly convergent to f at a. 11.2.2. Let all f, be rp-continuous and f. --> f an A. In order that f be also e-contin1.wus on A, it is necessary and sufficient that the set B of all x e A at which ((f.)) is not simply-uniformly convergent to f be a zero-eel for p. NECESSITY: Let B. be the set of the discontinuities of f, and B* the set of the discontinuities of f. By assumption B, =, A; if f is (p-continuous, then we have A, and hence SB, + B* =, A. Thus since B c SB, + B*, we have also B*

also B =, A. SUFFICIENCY: B =,, A implies SB, + B =, A. Thus, since B* a SB, B, we obtain also B* = A; that is, f is p-continuous. .

Again let f, -o f on A. The sequence ((f,)) is called quasi-uniformly convergent to f on A if to every S and every. vo there is a ro, such that for every x e A at least one of the inequalities II f,(x) - f(x) II < o(vo S v < po) is satisfied. A well-known theorem" says: Let ((f,.)) be a convergent sequence of continuous functions on A and set f = rim f, . Then, in order that f be also continuous

on A, it is sufficient and, if A is self-compact (§2,1), also necessary that ((f,)) be quasi-uniformly convergent to f on A. The following theorem is an analogue for (p-continuous functions: 11.2.21. Let p be carafent-like (§7,4), let ((f,)) be a convergent sequence of ,p-continuous functions on A, and set f = lira f, . Then, in order that f be also , p-continuous on A, it is sufficient and, if A is the sum of eountably many selfcompact sets, also necessary that there be a sequence ((GM)) of sets, open in A, with A - SGM =,p A, such that ((f,)) is quasi-uniformly convergent to f on GM .

NECESSITY: Let B, and B* be the set of all discontinuities of f, and f, respectively. By assumption B, =, A; if f is gyp-continuous, then B* = A, and hence SB, + B* _ A also. Since (p is content-like, there is a C Q SB, + B* which ,

7

is a G,-set and =, A. Then by 2.1.4 A -- C is an F, in A ; that is, A - C = SC4, i

where the Ci are closed in A. By assumption we have A = S.4 i , where the A i i are self-compact; thus C; = SAC;, where AfC; is closed in Ai, and hence, by i

2.1.63, also se) f-compact.

Since A - C = SA,C; , the result is that A -- C id

11 U. Dini [11, p.107; cf. also Encyklop. d. Math. Wise. II C 9 c (M. Frcchet-A. Rosenthal), p. 1163, and If. llahn 121, p. 211.

"C. Arzelb, Rendic. lat. Bologna (1) 19 (1883/1884), p. 79; Mem. lat. Bologna (5) 8 (1899/1900), p. 138; cf. also Encyklop. d. Math. Wise., toe. cit., and H. Hahn [21, p. 213.

MEASURABLE AND CONTINUOUS FUNCTIONS

§111

143

is the sum of countably many self-compact sets; that is, we can set A - C = SF,. , where the F. are self-compact. Let a e Fm . Since at a all f, and f are 0 continuous, ((f,)) is simply-uniformly convergent to f at a; that is, to every vs and a neighborhood U. , such a > 0 and every vs there is a P. that 11 f,,(x) - f(x) II < S for all x e AU.. Now according to Borel's Covering Theorem" there are finitely many of these U., say Ual , U,, , , U., , such

U., + U,, + ... + U.,

that P.

.

We set G. = A(U., + U., + - + U,);

then G. is open in A and F. C G,.. Furthermore we set PO = max(v,l , va, , . ..

,

vak);

then for every x e G. we have one of the inequalities jJ f,(x) -- f(x) H < S v 4 P0); that is, ((f,)) is quasi-uniformly convergent to f on G. . Since (Pb

F. C G., and A - C = SF.,, we have A - SG, C;AC; hence from C =,A we obtain also A -- $G,. =, A. SUFFICIENCY: If ((f,)) is quasi-uniformly convergent to f on G. , then" ((f,)) is simply-uniformly convergent to fat all points of G,. - SB, , and hence f is continuous on Gm at all points of G. - SB, . Thus ,

since G. is openin A, f is also continuous on A at all points of SB, . Since this is true for every m, f is continuous on A at all points of $G,. - SB, . , Thus the set B* of all discontinuities of f on A is a subset of (A - SG.) + SB,, and hence, as a result of A - SG.. -, A and B, =, A(v = 1, 2,

), we obtain also

B* =, A; that is, f is ,p-continuous on A. Now in 11.2.3 and 11.2.5 let A be again a p-measurable set, as in No. 1. 11.2.3. If the open sets are ,p-measurable, then every c-continuous function f on A is p-,neaaurabte. Let C be the set of all discontinuities of f on A; by assumption C =, A, and hence f is ,p-measurable on C. But on A - C f is continuous and therefore ,p-measurable by 11.1.2. Now the contention follows from 9.1.51. 11.2.4. If there is a closed set C Q A which =, A and has the power fit, then the set of all c-continuous functions on A has a power ? 2. The proof is the same as for 11.1.3 if one takes into consideration that, since C is closed, every function which equals 0 on A --- C is rp-continuous. ` 11.2.41. If A is separable and if there is a closed set C A which =, A and has the power 'fit, then there are gyp-continuous functions on A which are not Baire functions. The proof is the same as for 11.1.31.

The conditions of the theorems 11.2.4 and 11.2.41 are satisfied for A = R. " Cf. H. Hahn (2l, theorem 15.L11. .{ If ((f,)) is quasi-uniformly convergent to f on a set M, then ((f,)) is simply-uniformly convergent to f at every point a e M at which all f, are continuous. Cf. H. Hahn (21, theorem 28.2.1.

144

MEASURABLE FUNCTIONS

CHAP. III

and ip = ,u,, because of 8.2.83 and the fact that every (non-empty) perfect set of R. has the power R46 The function f, defined on A, is called continuous above (below)" on A at the

point a e A if for every sequence ((a,)) of points of A with a, ---* a we have: f(a) (lin f(a,) z f(a)).a If f is continuous above (below) on A at lim f(a,) every point a e A, then f is called continuous above (below) on A. A function f which is either continuous above on A or continuous below on A is also said to be semi-continuous on A. Now we call the function f, defined on A, q -continuous above (below) on A if the set of all x e A at which f is not continuous above (below) on A is a zeroset for gyp.

11.2S. If the open sets are 0-measurable, then every function (p-continuous above (below) on A is ge-measurable.

Let C be the set of all x e A at which f is not continuous above on A; by assumption C =, A, and hence f is 9-measurable on C. But on A - C f is continuous above and thus a Baire function`9; therefore by 11.1.21 f is ge-measurable on A -- C. Now the contention follows from 9.1.51. BinuooiAPSY to Nos. 1 and 2: H. HAZZN (11, p. 563; L. KAxvoaovzzvn, Fund. math. 16 (19M), p. 25.

3. Continuity properties of measurable functions. Let the point function f be defined on the set A and let B a A. Then the function on B which attaches to every point b e B the value f(b) is called the partial function off, restricted to B, and we designate this partial function by.Blf. In the following we again assume the open sets (and hence also the closed sets) of the metric space E to be p.-measurable; but furthermore we now suppose ge to be content-like (17,4). a Cf. footnote 41, p. 141. 47 Instead of "continuous above (below)" often the expressions "upper (lower) semiconttnaous" are used. 48 The functions continuous above (below) at a can be also characterized in the follpwing way: In order that f be continuous above (below) on A at the point a . A, it is necessary and sufficient that to every y > f (s) (to every y < f (a)) there be a neighborhood U. of a, such

that f(x) < y(f(x) > y) for all x . U.A. NzcnssITY: If there were no such U. , then for every positive integer m there would be an X. a AS.L with f (x.) ;< y; but then x,. - a and lim f (x.) > f (a), contrary to the assumption that f is continuous above at a. SUFFICIENCY: Let ((a.)) be a sequence of poin`s of A which converges to a. Then we have for almost all v: a, a U,A, and hence f (a.) < y. Thus lim f (a,) 6 y and, since y > f (a) was arbitrary, lim f(a,) f (a) also; that is, f is continuous above at a. 49 Cf. H. Hahn [2], theorem 86.2.2.

MEASURABLE AND CONTINUOUS FUNCTIONS

§11]

145

11.3.1. If qp is content-like and (p(A) is finite and if f is c-measurable on A, then

to every e > 0 there is a closed subset B of A, such that p(A - B) < e and Blf is continuous.

be, the set of the rational numbers. We form the Let r, , r2 , - , r, , c-measurable sets A, _ [f(i) z r,J and A',' _ [f(±) S r,]. By 7.4.22 there are A',' , rp(A', - B;) < 2E.z , closed sets B', and B;' , such that B' , c A: , B;'

and o(A;' - B;') <

Ti .

We set C = S(A, - B;) + S(A,' - B;') and

r,J = Now we have M[f(i) MA; = MB; (since M(A', - B;) = A). Analogously M[f(1) 5 r,J = MB;' . Since B', and B',' are closed, MB; and MB;' are closed in M; thus for every

M = A - C; then ip(C) < 2 E 2'+2 =

2.

rational r the sets M[f(z) z r] and M[f(z) 5 r] are closed in M, and hence by 11.1.11 Ml f is continuous. By 7.4.22 there is a closed subset B of M with (p(M - B) < 2 . Thus, since M = A - C with O(C) < 2 , we obtain ,a(A - B) < e and from B C Al it follows that Blf is also continuous. But in general there is no B C A with p(A - B) = 0 on which f would be continuous. Example in R, with jp = p, : Let A = 10, 1] and let P be a perfect set, nowhere dense in A, with ui(P) > 0 (8.2.8). Set f = 1 on P and f = 0 on

A - P. Then f is p,-measurable. Now if p,(A - B) = 0, then BP * A. In every interval (b - h, b + h) there is an interval (a', a") c A - P, and thus since p,(A - B) = 0, in (b - h, b + h) there is also a point a e B(A - P). As a result of b e P and a e A - P we have f (b) = 1 and f(a) = 0; Let b e BP.

thus since b e B and a e B, Blf is not continuous. 11.3.2. If p is content-like and A = SA i where all c(A i) are finite and if f is i

f-meas zreable on A, then there is an F,-set B Q A with ip(A - B) = 0 and a sequence ((fi)) of continuous functions on A, such that fi -+ f on B.

We may assume that A; C A+1. By 11.3.1 there is a closed Ci c A i with yq(Ai - Ci) < 2' , such that Cilf is continuous. We set Bi =

+ Ci ; then Bi is also closed, Bi Q Bi+1, Ai - Bi C Ai - C; , C, + C2 + and hence p(A i - Bi) < 2; ; since the Ci are closed, Bil f is also continuous. By a well-known extension theoremS° there is a continuous function fi on A, such that Bilfi = B,l f. We set B = SB; ; then B is an F,. Furthermore i

,p(A - B) = 0. For if a e A - B, then there is a j, such that a e A, ; then since Ai+,

A; , we have a e Ai for i ? j and, since a- a B, we have also a e Ai - B

'" First proved by H. Tietze, Journ. f. Math. 145 (1914),p. 9, and C. de la Va114e Poussin

[2), p. 127; cf. also Encyklop. d. Math. Wise. lIC9c (M. Freshet-A.Roeenthal), p. 1176, and H. Hahn [21, theorem 32.4.131.

146

[CHAP. III

MEASURABLE FUNCTIONS

for i > j; that is, a e Lim(Ai - B.), and hence A -- B i

Lim(A; i

- Bj.

But

1

from,p(A 1 - B1) < 2i it results, according to 3.5.1, that 0 (Lira (Ai - B;)) = 0,

and hence rp(A - B) = 0 also, as contended. Now on B we have fi -- f. B;+1 , there is a j, such that a e B; for i For let a e B; since B; j. But since B11f1 = B;1 f, we obtain fi(a.) = f(a) for i z j, and hence fi(a) --> f(a). For further consequences of theorem 11.3.2 we have to use a few facts about semi-continuous functions and about Baire functions'' of the first and second classb2 (in the metric space E). A function f belongs to the first class if f is the limit of a convergent sequence

of continuous functions. A function f belongs to the second class if f is the limit of a convergent sequence of functions of the first class.62 The system of all functions f which are limits of monotone increasing (decreasing) sequences of continuous functions is designated by Y'.'((91) and the functions of QV + Ll are called functions of the first order. The system of all functions f

which are limits of monotone increasing (decreasing) sequences of functions of the first order is designated by (2(((:) and the functions of (E2 + (, are called functions of the second order. We denote the intersections ((' (&,, and by (Si and 4 , respectively. It is evident that the functions of (,f' and Csl belong to the first class and that the functions of V and (,J2 belong to the second class. It can be proved' that the functions of coincide with the functions

continuous below (above) ; and hence the functions of the first order and the semi-continuous functions are identical. Therefore 141 is the same as the system of the continuous functions and it can be proved" that the system C coincides with the first class.

Now using the preceding definitions we obtain immediately from 11.3.2: 11.3.21. Under the conditions of 11.3.2 there is an F, set B Q A with p(A - B) _ -0, such that f is a function of the first class on B.

11.3.211. Under the conditions of 11.3.2 there is on A a function g continuous below and a function h continuous above, such that f = g + h on B.

Let ((f;)) be the sequence of continuous functions constructed in 11.3.2; in every point of B f; = f for almost all i. Let f' = fi where -i < f; < i,

f' - i where f; Z i, and f' = -i where fi S -i. Then the f' are continuous and f' -- f on B; for in every a e B with finite f(a) we have f'(a) = f(a) for 11 Cf. footnote 7, p. 120.

62 For this purpose we do not need the functions of higher classes. 66 This is somewhat different from the original definitions of R. Baire: according to him the first class does not contain the continuous functions and the second class does. not include the first class.

" R. Baire, Bull. Soc. math. France 32 (1904), p. 125; cf. H. Hahn [1], p. 162, and H. Hahn [21, theorem 36.2.2. 66 W. H. Young, Proc. London Math. Soc. (2) 12 (1913), p. 283; H. Hahn [11, p. 346; H. Hahn [2], theorem 35.1.5.

§11]

MEASURABLE AND CONTINUOUS FUNCTIONS

147

almost all i; in every a e B with f(a) = + oo (- oo) we have f*(a) = i(-i) for almost all i. Now since the f; are finite, we have on B: f = f, + E (f,+, - f*) In every a e B with finite f(a) almost all terms of this series are zero; in every a e B with f (a) = + oo (- oo) almost all terms of this series are l (-1) . We set max(fi (x), 0) = gi(x), max(f++1(x) - f': (x), 0) = g,+,(x), min(f(x), 0) = h,(x), and min(f,+L(x) - f* (x), 0) = he+1(x); then fl* = g, + h, , f++l - f* = gj+l + h;+,(i = 1, 2, .), and on B: f = E (g: + hi). Now we set g = E g;

and h = E hi. Then g is continuous below and h is continuous above on A ; for since gi k 0 and h; 5 0, the partial sums of these series form a monotone increasing and decreasing sequence, respectively, of continuous functions on A. But since for every a e B in at least one of the two series E g; and Eh; almost

all terms are zero, on B f = E (g; + h;) can be replaced by f = E g; + E h; _

g+h. Every semi-continuous function belongs to the first class; thus g and h are functions of the first class on B and, since g + h is defined everywhere on B, g + h is also a function of the first class on B. Hence from 11.3.211 we obtain 11.3.21 again.

In spite of 11.3.21 it is not possible to conclude from the theorems above that there is a function f* of the first class on A, such that f =t, f : on Example in R, for (P = µ, : Let A be a µ,-measurable subset of [0, 1], such that for every [a, b] c [0, 1] both A [a, b] *,,, A and [a, b] - A *,,, A (8.2.84). We set f = 1 on A and f = 0 on [0, 11 - A. Now if f* f, then in every interval [a, b] c [0, 1] there are both points with f* = 0 and points with f* = 1. Thus f* is discontinuous at every point of [0, 1], and hence f* is not a function of the first class on [0, 1], sinceb7such a function has points of continuity in every subinterval of [0, 1]. A."6

But there results from 11.3.21:

11.3.22. Under the conditions of 11.3.2 there is a Baire function f* a (,f2 and a f and f** f on A. According to 11.3.21, BI f belongs to the first class and hence both to and to (o on B. But then" there are on A functions f * e q2 and f** e G2, such that Baire function f** a (F2 , such that f*

(E2

Blf = Blf* and Blf = Blf**. Since O(A - B) = 0, this is equivalent to

f* = f and f** =* f on A.

Since all functions of C2 and of C2 are Baire functions of the second class, the following theorem is contained in 11.3.22: 11.3.221. Under the conditions of 11.3.2 there is a Baire function f* of the second class, such that f * =0, f on A. 66 In particular if the function g+h of 11.3.211 is defined on A, then it is a function of the first class on A which -,,f; but in general g + h is only,'-defined on A. 67 According to R. Baire, Paris These 1899 - Annali di mat. (3) 3 (1899), p. 19; cf. H. Hahn [2], p. 302. 63 Cf. H. Hahn [2], theorem 35.3.2.

148

MEASURABLE FUNCTIONS

[CHAP. III

BIBt.ooaAPBr: All the following references deal with the case So - µ,, , while extensions to the case of a general q are due to H. HAHN [11, p. 564. As to No. 3 cf. also Encyklop. d. Math. Wiss. II CO c (M. FatcnET-A. RoSENTHAL), p. 1182.-In particular we mention the following: As to theorem 11.3.1: indications at E. BOREL, C. R. Paris 137 (1903), p. 966, and Ii. LEBSsGUE, ibid., p. 1228; explicitly formulated and first proved by G. VITALX, Rend. lat. Lomb. (2) 38 (1905), p. 601 (other proofs were given by N. LUSIN, Mat. Sbornik Moskva 28 (1911), p. 266; cf. also C. R. Paris 154 (1912), p. 1688; W. SIERP1*SKI, T6hoku Math. Journ. 10 (1916), p. 81; Fund. math. 3 (1922), p. 319; G. ScoasA-DBAGONI, Rendic. Sem. mat.

Univ. Roma (4) 1 (1936), p. 3) ; the particularly simple proof given above is due to L. W. EFund. math. 9 (1927), p. 122. One owes theorem 11.3.221 to G. VITALI, loc. cit., Comm, p. 599, and theorem 11.3.22 to C. CARATHfODORT [11, p. 406. Theorem 11.3.211 is due to W. SIEIt'1ftzi, Fund. math. 3(1922), p. 319.

CHAPTER IV

INTEGRATION §12. Integrable functions 1. The p-integral of a function.

Again let lR be a o-field, let (p(M) be a totally

additive set function in 9't, and let ip(M), cp+(M), and p -(M) be its absolutefunction, positive-function, and negative-function, respectively. We employ the terminology introduced in §9,1 and, as there, we can assume T? to be complete for gyp.

Let A e 9)t and let p(A) be finite. Then the subsets M e TZ of A also form a a-field, which we designate by W. By 3.1.21 for every M e 21, rp(M) is also finite.

Let the point function f be r *-measurable (and hence V -defined) on A. The function f is called p-integrable on A if there is a set function X(M) defined in 1K and satisfying the following two conditions: 1.) X(M) is totally additive in W. c" for all x e Mat which f is defined, then 2.) If M v. It and c' f (x)

X(M) 5 c"FP(M) if 90-(M) = 0, c"sp(M) S X(M) 5 c'(p(M) if 'v+(M) = 0; c'c(M)

herein if c = f oo and p(M) = 0, one has to set op(M) = 0.1 From condition 2.) there results immediately:

X(M) = 0 if M =,A.

(1)

Now according to 3.4.71 we form the decomposition:

A = A' + A" where A'.4" = A, 4 (A') = 0, jo+(A") = 0,

(1.1)

and we set:

A'[f(f) = - m l = AL, , A"[f(t) _ -aol = A"., A"[f() _ +-l =

A'[f(x) = + oo l = A+ ,

12.1.1. In order that f be 9-integrable on A, it is necessary that either ,p(A+,,) _ 0. ,p(A".,) = 0 or p(A".,) = If cp(A4'.,,) * 0 and hence is positive, then we obtain from condition 2.),

setting c' = c" _ + co and M = A{.. , that X(A) = + co ; analogously we get: X(A_'.,)

if

ao if 0; and co if p(A_..) * 0; h(A'+.,) 0. But, because of condition 1.), X(Ai.., + A'_'.,) = X(A j..,) +

and hence there results: if either 0 or v(A".,) * 0, then 0, then X(Ai. + A".,) _ + -; analogously if either ,p(A_,.) * 0 or co. Now the contention follows from 3.1.23. X(A + A.;..,) 12.1.2. If f is S'-integrable on A, then the value X(A) is uniquely determined by the conditions 1.) and 2.). ' We maintain this stipulation also in the following. 149

[CHAP. IV

INTEGRATION

150

0 or cp(A'_'.,) $ 0; for then, as we just saw, This is true if either oo, and hence by 3.1.2 X(A) _ + oo also. Furthermore, it is true if either p(A_.,) 4 0 or p(A';.,} 4 0; for then X(A'., + A+.,) _ - « Therefore we can assume: and hence A(A) X(A'..,, +

p(A';,) = 0. Now let X(M) and X1(M) be two set functions satisfying the conditions 1.), 2.) and let S > 0 be arbitrarily given.

Again using the

decomposition (1.1) we set: A; = A'[(i -- 1)S 5 f(I) < id) and Ai A"f (i - 1)S

f(x) < iij (i = 0, f 1, f2,

).

Then if A* is the set of all +m

x e A at which f is not defined, we have: A = S

A; + S A; + A+ +-

A_ + A+., + A + A*, and hence because of condition 1.) and (1): X(A) X(A:) + E X(A:'); X1(A) _ E X1(A.) + E X1(A,'). But because o: condition 2.) we have (i - 1)Sto(A;) <__ 1(A,) 5 i&p(A{); (i - 1)&p(A;) 6 h,(A;) 5 i&p(A;) and hence 7,1(A;) = X(A;) + B;&p(A;) with 18'1 1 S 1. Analwith B;' I 1. Thus X,(A) = 7,(A) + alogously hl(A,') _ X(A;') + +-a

BS E

J p(Ai')1) = X (A) + BP(A), where 101 ;5 1. Since herein O(A) is

finite, by assumption, and S > 0 is arbitrary, it follows that )1(A) = X(A). Now if f is So-integrable on A, the value A(A), uniquely determined according to 12.1.2, shall be called the cp-integral off on A and shall be designated' by X(A) _ (A) f f dip.

(1.2)

The function f is called the integrand, the set function p is to be named the determining function of this integral, and the set A is called the set of integration. 12.1.3. If f is p-integrable on A and if B is a ip-measurable subset of A, then f is also ip-integrable on B. Let a(M) be a set function in W satisfying the conditions 1.) and 2.). Then = B I A is also a u-field and A(M) satisfies the conditions 1.) and 2.) also in s8. Hence f is -integrable on B.

Furthermore, according to (1.2): l(B) = (B)

f f dc. Now if we replace

B by M, then because of the conditions 1.) and 2.) respectively, we have: 12.1.31. If f is cp-integrable on A, then (M)

f f dp is a totally additive set function

in the or-field of the ip-measurable subsets of A.

12.1.32. 1 f f is to-integrable on A and if c' S f (x) S c" for all x e A at which f is defined, then ' The notation f f dio is also often used.-If A S; R, and - µ, , then in the usual manner d

f f dx. b

we write also (4) f f dx; if in particular A - (a, b), then we write also

a

151

INTEGFRABI.E FUNCTIONS

1121

c'w(A)

(A)

f f dip 5 c',p(A)

c",p(A) 5 (A) f f dip

c',p(A)

if ET(A) - 0, if cp''(A) = 0.

12.1.4. If f is So-integrable on the two sets A and It, then in order that f be cp4nte-

grable on A + B, it is necessary and sufficient that (A)

f f dp and (B) f f dp be

not in finite of different signs.

Nscs;ssrrr: This follows from 3.1.23 because of 12.1.31, since A and B are

gyp-measurable subsets of A + B.

SUFFICIENCY: Lit M be a gyp-measurable subset

of A + B. We set X,(M) = (MA)

f f dip and X2(M) = (M - A) f f dp;

then XI(M) and i12(M) are totally additive in the a-field R of the 9-measurable

subsets of A + B. Since by assumption (A)

f f dsp and (B) f f dp are not

infinite of different signs, by 3.1.2 l1(M) and a2(M) do not assume infinite values of different signs either. Thus by 3.2.52 X, + X2 is totally additive in 9 and, since according to 12.1.32 A, + X satisfies also condition 2.), f is cp-integrable on

A + B. A, then every function f on A is c&-integrable and (A)

f f dsp - 0.

Since D2 is complete for p, every function f on A is immeasurable.

If we now

12.1.5. If A

define X(M) = 0 for every subset M of A, then the conditions 1.) and 2.) are satisfied, and the second part of the contention follows from 12.1.2 and (1.2). From 12.1.31 and 12.1.5 there results: 12.1.51. If f is gp-integrable on A, then the set function (M)

f f dip is V-continuous

(§5, 7) in the o-field f. 12.1.52. If f =, g and f is go-integrable on A, then g is also so-integrable on A and

(A) f f dco = (A)

f g dp.

Let A' = A[f(.i) = g(z)). On A' g is So-integrable by 12.1.3, since there f = g.

By assumption A - A' =, A; thus by 12.1.5 g is 4p-integrable on A - A'. Hence according to 12.1.4 g is also -p-integrable on A. By 12.1.31 (A) f f dip

= (A')

f f dp + (A - A') f f dtp

and

(A) f ;7 ::,; = (A') f g dco + (A - A') f g do. Herein, since f = g on A', we have (A') f f dp = (.A')

f g dp; by 12.1.5

(A - A')

[CHAP. IV

INTEGRATION

152

f f dp = (A - A') f g dp = 0; and hence (A) f f dp = (A) f

g dip.

dp = 12.1.521. If f and g are p-integrable on A, then in order that (M) P4 (M) f g dp for all M e i, it is necessary and sufficient that f =, g on A.

NmmssrrY: If A = A' + A" is a decomposition (1.1) of A, then it suffices to prove that f =, g on A' and on A"; we show, say, f =, g on A'. Let r' and r" r"]. Then by A'[f(i) 5 be rational and r' < r"; let

But by f g dp ? f g dp. This is possible only

12.1.32: (A,.,.) f f dip 5 r',p(A,,,.) and (A,.,.)

assumption we have (A,.,.) f f dcp = if

0; since A,.,.

A', this is equivalent to A,.,. =, A. Now let

g(x) > f(x); then there are rational numbers r', r", such that f(x) < r' < r" < g(x); but then x e A,.,.. Thus A'[g(x) > f(z)j c S r. ,.A,,,., where we have to sum over all pairs of rational numbers r', r" with r' < r". Since there are only denumerably many such pairs and for every such pair A,.,. =, A, by A also, and hence A'[g(x) > f(t)] =, A. In the same way 4.1.32 S one obtains ""r' A'[ f(x) > g(1)] =, A. Thus A'[f(i) * g(t)j =, A; that is, f =, g on A'. SUFFICIENCY: This follows from 12.1.52. 12.1.53. If f is p-integrable an A and if A' =, A, then f is also P-integrable on

A' and (A) f f dp = (A') f f dcp. For A' = (A - B) + C, where A Q B, AC = A, B =, A, and C =, A. By 12.1.3 f is p-integrable on A - B, thus by 12.1.5 and 12.1.4 on A' also; and, ac-

cording to 12.1.31 we have: (A') f f dp

= (A - B) f f dp + (C) f f dip =

(A - B) f f dp. But since A = (A - B) + B, we have (A) f (A - B) f f dip + (B) f f 4 = (A - B) f f

f dq

=

co also.

Because of 12.1.53, in the following we can assume f to be not only p-defined on A, but defined on A. 12.1.6. If f = c is constant on A, then f is 9 -integrable on A and

(A) f f do = ccp(A).

For A(M) = cp(M) satisfies the conditions 1.) and 2.); hence by (1.2) the contention follows. 12.1.61. If f is a p-measurable function on A which assumes only the finitely many finites values cl , c , . , C. , then f is p-integrable on A and, if we set 3 Here infinite values c{ can be admitted also, provided that infinite values of different , rn). signs are not met among the cre(Ac) (i - 1, 2, .

INTEGRABLE FUNCTIONS

§121

A[f(±) = ci] = Ai , we have: (A)

153

f f do = E cv(Ai).

By 12.1.6 f is y -integrable on A i and (A i) f f dip = cip(A i) ; thus by 12.1.4

f is also qp-integrable on A = A, + A2 +

+ A., and according to 12.1.31

(A) f f dhp = E (A j) f f dip = E c,Yo(Ai). 12.1.62. If f is (p-integrable on A and c($0) is a finite number, then cf is also cp-integrable on A and (A) f cf dp = c. (A) f f dip.

For c- (M) f f dc* is a totally additive set function in A satisfying condition 2.)

for the function cf (instead of f). 12.1.63. If f is p-integrable on A, if c(*O) is a finite number, and if ¢ = cp, then f is also 4'-integrable on A and (A) f f d4' = c (A)

f f d,p.

For c (M) f f d(p is a totally additive set function in W satisfying condition 2.) for the set function csv (instead of gyp).

12.1.7. If f and g are cp-integrable on A and f S g, then

(A) f f 4 5 (A) f g d,p, provided that 1P-(A) = 0,

(A) f f dip z (A)

f g dp, provided that rp+(A) = 0.

We furnish the proof for the case c (A) = 0. Lets > 0 be arbitrarily ), given; we set A[(i - 1)6 5 f(x) < is] = Ai(i = 0, f1, f2, and A[f(t) _ - oo ] = A-.,. A [f(t) = + oe ] = A+.,

Since A = S Ai + A+ + A_.,, , by 12.1.31

-f

(A) If f dip = E (Ai)

f f dip + (A+.) f f dip + (A-,) f f dp f

and (A) f gdsp = E (Ai) g dip + (A+.,) gdcp + (A_.,) (A j)

f

g dip.

Herein by 12.1.32:

f f d p S i&p(A i) ; since g a f, we have g k (i - 1)S on A; , and hence, again

f

by 12.1.32: (Aj) g dp

(i - 1)&p(Ai).

Furthermore by 12.1.6 (A,)

Thus (Ai) f

60(A j) _> (A;) f f d
f f drp = - - or 0, according as po(A_..,) * C.

or ,p(A_.,) = 0. In the first case certainly (A,)

f ,g d4p z

f f dip; in the

[ca. '. IV

D TEGRATIOAI

154

case by 12.1.5 (A_.,) f g dp = (A-,) ffzis m 0. Finally on

second

A+., f = g(- + OD ), and hence (A+.,) have: (A) f g dip + 8

f g dip - (A+.,) f f dp. Summing up we

(A) f f dp. Since herein 0 E i'(Ai) $ p(A)

p(A;)

and SS(A) is finite, and since S > 0 was arbitrary, it follows that

(A) f g dip z (A)

f f dp.

12.1.8. If qi (A) - 0` and if ((f,)) is a monotone increasing (decreasing) sequence of cp-integrable functions on A with" (A) f

is also p-integrable on A and (A)

f, dp > - oo (< + oo ), then f = lim f,

f f dp = lim (A) f f, dp.

First, by 10.1.21, f is cp-measurable on A. Now let M e%; by 12.1.7 the sequence (M) f f, dip is monotone increasing, and hence there exists the limit A(M) = lim (M) f f, and 2.) in 21.

(p.

We have to show that X(M) satisfies the conditions 1.)

f

We set MV, (1) < 01 = M' and p(M) = (M') f, drp; then accord-

ing to 12.1.31, 12.1.32, and 3.1.2 p(M) is totally additive and finite in It.

form the totally additive set function p,(M) = (M)

f f, dip - p(M).

We

Since by

12.1.7 (M) f f, dp ;5 (M) f f,i, dp, we have p,(M) S p,.+.1(M) also. Since (M) f fi dp = (M')

f f, dp -f- (M - M') f f, dp, we have p,(M) = (411 - ill') f f, dip

4 The condition that so be monotone is essential in 12.1.8 and 12.1.81, even if (A) f f,dp(v = 1, 2, ) is assumed to be finite. Example: Let A - [-1, +11 and f, - v;

we set [-1, Ol - Al and (0, 11 - A, . Let ' be totally additive for the µ,-measurable subsets of A and let (p - it, on A, and w = -,u, on A2. Then (A) f f dv - 0 (for v = 1, 2, ...), but f - lim f, is not ,o-integrable on A by 12.1.4, since (according to 12.1.6) (AD f f dw s -1-

and (A,) f f dv - -.o.

This condition is essential. Example in R, with ip = p, : Let A - (0, 1) and f, - -ao %

in

0, Y

and f, - 0 otherwise. Then f = lim f, - 0, and hence (A) f f dam, - 0,

but lim (A) f f, dµ,

- -.o.

INTEGRABLE FUNCTIONS

§12]

155

and, since f, z 0 on M - M', by 12.1.32 p1(M) z 0. Thus from p,(M) S p,+,(M) it follows that all p,(M) z 0. Therefore by 3.6.2 lim p,(M) is also totally additive, and hence by 3.2.521im p,(M) + P(M) = lira (M) /

f f, do = X(M) is also.-

Now we show that X(M)satisfies also the condition 2.). Let c' S f(x) s c" on M. Then since the sequence ((f,)) is monotone increasing, f,(x) 5 e" and

If c' _ - oo, then X(M) z c'c(M) is trivial; therefore let c' > - oo and p < c'. We set hence (M) f f, dip 5 c"c(M) for all v; thus X(M) 5 c"cp(M) also.

M, = M[f,(i) ? pl; then M, Q M,+, and, since f, --i f, M = SM, ; thus by3.2.2 y,(M,) --+ cs(M) and A(M,) --+ A(lt). Since f, > p on 11I, , by 12.1.32 (MI) f f, do Z pcp(M,), and hence by 12.1.7 lim (Mv) f f dp = A(,ZI,.) ? pw(M,)

Thus by letting oo we obtain: A(M) _>- pp(M) and, since herein p < c' was arbitrary, X (M) > c',p(M). Therefore X(M) satisfies also the con-

also.

dition 2.); that is, f is So-integrable on A; and by (1.2)

(A) f f do = A(A) = iim (A) f f, do. Replacing 0 by -gyp in 12.1.8 we obtain by 12.1.63:

12.1.81. If o+(A) = 0° and if ((f,)) is a monotone increasing (decreasing) sequence of ,p-integrable functions on A with 7 (A)

f f, dip < d- oc (> - ao), thrn

f - lim f, is also rp-integrable on A and (A) f f dtp = lim (A) f f, do. We shall discuss other limit theorems for So-integrals (partly without assuming c and ((f,)) to be monotone) in §15, 2; (cf. 15.2.3 in particular). 12.1.9. If o is monotone in $I8 and if f ? 0 (or :50) and o-measurable on A, then f is So-integrable on A.

Let lp`(A) = 0 and let v be a positive integer. We set A[t A,, i(i = 1, 2,

,

1

< ,f(t) <

J=

, v 2/) and A [ f (.f) s' v] = A. and we define f, in the following

manner: f, =

z

and f, ---), f.

By 12.1.61 f, is c-integrable on A and (A) f f, dip

2,

on A,,; and f, = P on A,. Then ((f,)) is monotone increasing

0; hence by

12.1.8 f is also So-integrable on A.

Again let rp be monotone in 81, but now let f be an arbitrary rp-measurable ' Cf. footnote 4, p. 154. 7 This condition is essential. Example in R, with go s -µ, : A - (0, 1), f, - - eo in 0,

, and f, a 0 otherwise.

8 By 3.4.4 and 8.4.41, this is equivalent to v -(A) - 0 or V +(A) - 0.

[CHAP. IV

INTEGRATION

156

function on A. We set A+ = A[f(t) z 0] and A-=A[f(z) < 0]; then by 12.1.9 f is 'p-integrable on A+ and A- and by 12.1.4 we obtain: 12.1.91. If (p is monotone in 21 and if f is 'p-measurable on A, then in order that f be y,-integrable on A, it is necessary and sufficient that" (A+) f f dip and (A-) f f dip

be not infinite of different signs.

Furthermore, from 12.1.3 and 12.1.4 there results immediately: 12.1.92. Let A = A' + A" be a decomposition (1.1) of A. Then in order that f be p-integrable on A, it is necessary and sufficient that f be rp-integrable on A' and

on A" and that (A')

f f do and (A") f f dip be not infinite of different signs.

By combination of 12.1.91 and 12.1.92 we obtain: 12.1.93. In. order that the 9-measurable function f be

integrable on A, it is

necessary and sufficient that among the four integrals (1.3)

(A+A') f f d', (A+A") f f dip, (A A') f f dtv, (A A") f f dw

there are no two which are infinite of different signs.

Furthermore, because of 9.1.61, there results from 12.1.9: 12.1.94. If 'p is monotone in 9'" and f is 'p-integrable on A, then

IfI

is also

'p-integrable on A. The following lemma is also related to 12.1.9: 12.1.95. If co (A) = 0" and if ((f.)) is a sequence of non-negative ,p-measurable

function on A, then

(A) f lim f, dip s lira (A) ff7dco.12

(] .4)

We set g, = inf f, (m = 1, 2, . ). The gm are non-negative and by 10.1.2 ,fin,

TLis condition can be replaced by the condition that at least one of the two integrals

(A{) f f dr and (A-) f f dip be finite. For if none of these two integrals is finite, then by 12.1.32 one of them is + oo and the other one is - a o. to This condition is essential here and in 12.1.9. Example: Let A - A' -}- A" be a decom-

position (1.1) of A with p(A') >0 and g(A") < 0; furthermore let f - +- on A' and f o on A". Then I f I is not p-integrable on A by 12.1.1.-But cf. 12.3.42.

" For 0+(A) = 0 we obtain, instead of (1.4), the inequality: (A) f Jim f,dip Z 1im

(A) f f. do.

11 The corresponding inequality (A) fnra f, do Z lira (A) f f, dw is not valid under the r

v

same conditions. Example in R, with p = Kl : A - (0, I), f, = -}-no in 0,

and ., a 0

otherwise; then lim f, - 0 and (A) r lim f, d. = 0, but lim (A) f- . dp _ + oo.-Cf. 12.3.8. 1

J

.

157

INTEGRABLE FUNCTIONS

X12]

gyp-measurable, and hence by 12.1.9 -integrable; ((gm)) is a monotone increasing

sequence converging to lim f, on A.

and (A) f lim f, dio = Jim (A) (A) f g.. dip

f

Thus by 12.1.8 Jim f, is also V-integrable

gm dip.

Since gm S f. (for v Z m), by 12.1.7

(A) f f, dp(v z m), and hence lim (A) M

f gm do 5 lim, (A) f f, dip.

Thus we obtain (1.4). Immediate consequences of 12.1.95 are: 12.1.951. Ij lp (A) = 0 and if ((f,)) is a sequence of non-negative cp-measurable functions converging to f on A, then f is o-integrable and

(A) f f dp S Jim (A) f f, drp.I" 12.1.952. If rp (A) = 0, if ((f,)) is a sequence of non-negative immeasurable functions converging to f on A, and if (A) f f, dio 5 c for almost all v, then (A) f f d(p S c

also. BiBLIOoaAPHT: The above theory of the.p-integral is due to H. HAHN, Anzeiger Akad. Wiss. Wien 1929, No. 2; Festschrift der 57. Versammlung Deutecher Philologen u. Schulmgnner in Salzburg 1929, p. 193; a hint at such an introduction of the notion of integral is already found in C. DE LA VALL4E POUSSIN [2], p. 55.-As to the development of the notion

and the different definitions of integral, of. Encyklop. d. Math. Wise. IIC9b (P. MoNTELA. RosaNTHAL), No. 27-35f. As the most important steps of this development, in so far as it leads to the theory discussed here, the following are emphasized: B. RIEmANN, Habilitationsschrift Gottingen 1854, published in Abh. Gesellsoh. Wiss. Gdttingen 13 (1867), p. 101 (- Werke, 1st ed., Leipzig 1876, p. 225; 2nd ed., Leipzig 1892, p. 239); T. J. STiELTJES, Ann. Fee. sc. Toulouse (1) 8 (1894), m6m. No.10, in particular p. 68; H. LEBESGUE, C. R. Paris 132 (1901), p. 1025; Paris These 1902, p. 18 - Annals di mat. (3) 7 (1902), p. 248; [1 ], p. 98; [21, p. 105. For H. LaBESava the Lebesgue measure pi (or p,) plays the part of the determining function p. The generalization of the Stieltjes and Lebesgue integral for arbitrary totally additive set functions 4P is due to J. RADoN, Sitzungsber. Akad. Wiss. Wien 122 (1913), p. 1322; he still takes the R. as a basis; with reference to his work, the definition of the integral was generalized to abstract spaces by M. FafcHaT, C. R. Paris 160 (1915), p. 839; Bull. Soc. math. France 43 (1915), p. 248. Other theories of the integral in abstract spaces were established by the following mathematicians: P. J. DANIALL, Annals of math. (2) 19 (1918), p. 279; (2) 21 (1919/1920), p. 203; 0. NIaonm , C. R. I. Congres math. pays slaves (Warsaw 1929), p. 304; Fund. math. 15 (1930), p. 131; A. KoLxoaosoFF, Math. Annalen 103 (1930), p. 654; S. SAKE 111, p. 247; [21, in particular p. 19 and p. 322 (note of S. BA.RAca); Duke Math. Journ. 4 (1938), p. 408; F. MAEDA, Journ. as. Hiroshima Univ. A 4 (1934), p. 60; J. voN NEUMANN [I], p. 176; J. RIDDaa, Fund. math. 24 (1935), p. 72; Acts, math. 73 (1941), p. 169; Proc. Nederl. Akad. Wetensch. 49 (1946), p.167, 175; N. DUNFORD, Trans. Amer. Math. Soc. 37 (1935), p. 441 (cf. also 88 (1935), p. 600); 44 (1938), p. 334; G. BIRKHOFF, Trans. Amer. Math. Soc. 38 (1935), p. 357; Annals of Math. (2) 38 (1937), p. 50;

B. Ji;ssEN, Mat. Tidsskrift B 1935, p. 60; 1938, p. 20; H. FREUDENTHAL, Proc. Akad. Amsterdam 39 (1936), p..741; T. OGASAWARA, Journ. so. Hiroshima Univ. A 6 (1936), p. 47;

E. ToENIER [11, p. 53; D. H. HERS, Thesis, California Institute of Technology, 1937, Chap. 11; 0. HAUPT [11, vol. III, p. 60; C. CARATH*ODORT, Sitzungsber. Bayer. Akad. WIss.1938, p. 27; Math. Zeitschr. 46 (1940), p. 181; Abh. Math. Sem. d. Hansischen Univ. 14 13 Cf. footnote 12 and 12.3.8.

[CHAP. IV

INTEGRATION

158

(1941), p. 351; B. J. PErns, Trans. Amer. Math. Soc. 44 (1938), p. 277; F. WECxsN, Math. Zeitschr. 45 (1939), p. 377; S. BocHNIIR, Annals of Math. (2) 40 (1939), p. 769; Proc. Nat. Acad. U.S.A. 26 (1940), p. 29; R. L. JEFFERY, Duke Math. Journ. 6 (1940), p. 706; G. B. PRICE, Trans. Amer. Math. Soc. 47 (1940), p. 1; R. S. Psuarrs, ibid., p. 114; S. IzvSI and M. NAKAMURA, Proc. Imp. Aead. Tokyo 16 (1940), p. 518; S. IzuMI, ibid. 17 (1941), p. 1; S. KAKUTANI, Annals of Math. 42 (1941), p. 523; H. H. GOLDSTINE, Bull. Amer. Math. Soc. 47 (1941), p. 615: J. M. H. OLMSTED, Trans. Amer. Math. Soc. 51 (1942), p. 164; M. M. DAY, ibid., p. 596; C. E. RICKART, ibid. 52 (1942), p. 498; H. NAxANO, Proc. Phys.-Math. Soc. Japan (3) 25 (1943), p. 279; K. KWNISAWA, ibid., p. 524; M. S. MACPHAIL, National Math.

Magazine 20 (1945), p. 69.-Theorem 12.1.8 is related to H. LEBESavz, loc. cit., and to B. LEVI, Rendic. Ist. Lombardo (2) 39 (1906), p. 775. As to lemma 12.1.95: P. FATOU, Acts math. 30 (1906), p. 375; C. CARATUtODORY [11, p. 442; S. SAxs [1), p. 84.

2. Other properties of yo-integrable functions. 12.2.1. If f, and f2 are cp-integrable on A and if (A) f

fi dip and (A) f f2 dcs are

not infinite of different signs, then fi + f2 is also `o-integrable on A and (2)

f

(A) f (fi + f2) dcp = (A) fi dip + (A) f f2 d
First we show that fi + f2 is p-defined on A. Thus we have to prove: if

-oo]andC=A[f1(-*) B=A[f,(1) then B =, A and C =, A. Let A = A' + A" be a decomposition (1.1) of A. Then A'B =, A; for otherwise (A'B) f fi dip = + co and (A'B) f f2 dip = - oo,

f

f

dp = - oo , contrary to the assumption. In the same manner we see that A"B =, A. Thus since B = A'B + A"B, we obtain B =, A, as contended; and analogously it results that C =, A. Therefore fi + fs is se-defined and hence by 9.2.6 also cp-measurable on A. Now we have only to show that X(M) _ (M) f fi d'p + (M) f2 dso satis-

and hence by 3.1.2 (A) f1 dip = + 00 and (A)

12

f

fies the conditions 1.) and 2.) of No. 1 for f = f1 + f2. Since (A) f fI drp and (A) f ft dip are not infinite of different signs, by 3.1.2 (M) f fi do and (M) f f2 dip (for M E 40 are not infinite of different signs either, and hence by 3.a 52 A(M) is

totally additive in 2I; that is, condition 1.) is satisfied. Now we prove the first inequality of condition 2.) (the second one can be demonstrated analogously). c" on M and j (M) = 0. We have to show: Let c' fi(x) + f2(x) (2.1)

c',(M)

(M) f fi dip

+ (M) f f2 dy, s

f

If MVV(I) - + ac) 0, A, then (M) fl dP _ + oo, and hence (M) f ft dip + (M) f f2 dp - + 00

INTEGRABLE FUNCTIONS

§121

159

also; but since fi(x) _ + oo implies fi(x) +Mx) =+-,we must have c" = +00, and (2.1) is satisfied. The cases M[f,(I) = - oc 1 0, A, MUs(i) = + oo 10, A, and Mff2(1) = - ao} , A may be handled analogously. Thus we can assume

M(f,(.f) = + -] + MV,(!) = - -} + MV2(1) = + -l + MV2(z) = - -} =,A and, since we may neglect zero-sets for gyp, we can assume that f' and f2 are finite

on M. Let a > 0 be arbitrarily given and set

M;=M[(i-1)6

f,()
Then as a result of c' S f, + fs S c" we have on Mi :

C'-ia
and hence (i - 1)6 e(Mi) S (M,) f fid p 5 i&p(M;) and (c' - ia),p(M;) 5

(M,) f f2 dp 5 (c" - (i - 1)Ap(M0. Thus

(c' - A(M.) 5 (M:)

f f, dye + (M.) f

f2 dip 5 (c" + a),p(M1).

Since M = S M; , it results by summation over i that (c' - 6),p(M) (M) f f, dip + (M) f f2 dv S (c" + 6)v(M); this is true for every 6 > 0, and hence we obtain (2.1). Replacing f2 by ---fs in 12.2.1 and using 12.1.62 (for c

-1), one obtains:

12.2.11. If f, and fs are (p-integrable on A and if (A) f f, dip and (A) f f2 dcp are not infinite of the same sign, then f, - f2 is also (p-integrable on A and

(A) f (f, - f2) dp = (A) f f, dTp _ (A) f f2 dip. Let w and rp1 be totally additive in the a-field 0 and let o,(A) and V2(A) be finite. Then by 3.1.21 and 3.2.52 91 + v2 is totally additive in ?i and q1(A) + rct(A) is finite. 12.2.2. If f is 01-integrable and V2-integrable on A and if (A)f f dip, and (A) f f dips are not infinite of different signs, then f is also (91 +

-integrable

on A and (A)

f

f d(,.1 + GPs) - (A)

f f dvi + (A)f

f dv2.

First we prove this under the assumption that gyp, and v2 are monotone increasing.

We set p = i + 02 and we have to show that A(M) = (M) f f dcp1 + (M) f f dsp, satisfies the conditions 1.) and 2.) of No. 1. By 3.1.2,and 3.2.52 X(M) is totally additive in W, and hence condition 1.) is satisfied. Since jp is also monotone

[CHAP. Iv

INTEGRATION

160

increasing, we have now only to show that for all M e 1 the first inequality of condition 2.) is satisfied. But this follows from c'(p;(M)

(M) f f dp; 5 c",p;(M),

i = 1, 2 (12.1.32) by addition.-Now in order to give the general proof, according to (1.1) we form the decomposition: A = A' + A"; in the same manner we form the decomposition A = A, + A;'(i = 1, 2) where A;A;' = A, Vpi (A,) = 0, and A + + A- where A+ = A[f(ic) k 0] and Ip; (A;') = 0. Furthermore we set A

, 16) the interA^ = A[ f(&) < 0]. Now we designate by A,(/ = 1, 2, , A A"Ai Al" ; the A. i are disjoint and , A+A'A;A;' ,

sections" A+A'A;Az 16

A = Sj1A, . Because of 12.1.4 it suffices to prove the contention for each Ai

(since by assumption and 3.1.2 no pair of the integrals is infinite of different

signs). The contention is true for A, = A+A'Al'A2 , since in A1121 V, and V2 are monotone increasing. In A21 A we have v, = go and c = - vi, and hence ,p = ,p, + = ip; - c . Thus sot = to + and, since by 12.1.9 f is .p-inte-

grable on A2 and ip and sp: are monotone increasing on A2, we have, as has

; that is,

already been proved: (A2)f f dot = (As) f f dy, + (A.) f f d

(A:) f f dal = (At) f f ds - (AZ) f f dp2. Here (A2) f f d(ps is finite; for otherwise (A,)

f f d i and (A,) f f d p, would be infinite of different signs, and hence

by 3.1.2 (A) f f d,p, and (A) f f d

would be also, contrary to the assumption.

Thus (A,) f f dw, + (A,) f f dpi _ (A2) f f d(p1 + 02), as contended.

One proves

the contention for the other Ai analogously. Replacing rp$ by -92 in 12.2.2 and using 12.1.63 (for c = -1), one obtains:

12.2.21. If f is 91 integrable and ci-integrable on A and if (A) f f dip, and (A) f f dip, are not infinite of the same sign, then f is also (rot - V2)-integrable on A and

f

(A) f f d(ci - 1Pi) _ (A) f d4oi -

f

f do,.

12.2.22. In order that f be p-integrable on A, it is necessary and sufficient that f be both ,p+-integrable and ,p -integrable on A and that (A) f

f

and (A) f f di

are not infinite of the same sign.

NECESSrry: Let A = A' + A" be a decomposition (1.1) of A. Then p+ for A'12I and p+ = 0 for A"121; hence f is cp+-integrable on A' and A" and we have: 14 Some of these intersections, for instance A+A'Ai"A" , are zero-sets for q

.

INTEGRABLE FUNCTIONS

§12]

(A') f

(A') f f

(2.2)

f dr,

161

(A")f f dco* = 0.

In the same way we obtain:

(A') f f dw = 0,

(2.21)

(A-) f f dcp.

(A-) f f duo-

Now the contention follows from 12.1.4. SuFmcisxcY: This follows from 12.2.21, since jo = w+ - 07. 12.2.23. If f is Se4ntegrable on A, then

(A) f f dsp - (A) f f dip+

f

- (A) f W.

This follows from 12.2.22 and 12.2.21, since Sp = rp+ - rp . 12.2.24. If f is c,p-integrable on A, then in order that f be also q-integrable, it is

necessary and sufficient that (A) f f dip+ and (A) f f dip- are not infinite simultaneously.

Then we have:

(2.3)

(A) f f dip = (A)

f

f dp+ -I- (A)

f f dip

.

NECEsarrY: If f is 0-integrable and A = A' + All is a decomposition (1.1) of A, then (A') f f dip = (A') f f dp+ and (A") f f 4

= (A") f f dip-. Thus by

12.1.4 (A') f f dcp+ and (A") f f dip- are not infinite of different signs, and hence

by (2.2) and (2.21) (A) f

f

and (A) f f dip- are not infinite of different signs.

But according to 12.2.22 they are not infinite of the same sign either. SvFFI-

CIENCY: This as well as (2.3) follows from 12.2.22 and 12.2.2, since (p = tp+ + q P7.

If f is v-integrable on A, then X(M) _ (M) f f dip (cf. (1.2)) is totally additive in the c-field 4i according to 12.1.31.

Now we wish to have an expression for

the positive-function, negative-function, and absolute-function of (M) f f dye, which we designate by a+(M), a-(M), and i(M), respectively. If A = A' + A" is again a decomposition (1.1) of A and A+ = A[ f(t) 0], A- = A[ f(t) < 01,

then we have: 12.2.3. The

positive function,

negative function, and

?,(M) = (M) f f dip are given by the following formulas:

absolute function

of

[CHAP. Iv

INTEGRATION

162

A+(M) = (A'-M)

f f dc+ -- (A M) f f dip = ((A+A' + A A")M) f f dw,

X --(M) _ (A+M) f f do

- (A-M) f f dp+ _ -((A +A" + A~A')M) f f dip,

(M) = (M)f IfIdo= ((A+A' + A-A") M)f f dip -

((A+A" +

According to §3(4), A+(M) = sup (X) f f dp (X e it

A A')M)

121).

f f dip.

Since

X = A+A'X + A+A"X + A--AX + A-A"X, we have

(X)f fdco

f

(A+A' X) f f dip + (A+A"X) f f dto + (A A'X) f dip + (A A"X) f f d,. Herein we have by 12.1.32:

(A+A'X) f f dp S (A+A'itf) f f dp, (A-"A'X) f

f d p s 0,

(A+A"X)

f fdv 5 0

(A-A"X) f f dip 5 (A-A"M) f f dip.

From this the second part of the formula for A+(M) follows, since for X0 _ (A +A' + A`A")M both A+A"Xo = A and A-A'Xo = A; and then, because of (2.2) and (2.21), there results also the first part of this formula. In the same way one proves the formula for AC(M). Then the formula for X(M) follows by and A = A+ + X-. addition according to 12.2.2, since 0 = op+ + From 12.2.3 there results immediately, because of §3(4.21): 122.31. If f is ,p-integrable on A, then I (A) f f dip ! 5 (A) f If I d,p.

From 12.2.31, 12.1.7, and 12.1.6 we obtain: 12.2.311. If f is cp-integrable on A and If 1 L' The inequality

I

(A) / f dy I

q, then I (A) f f do I S ggp(A)."

q I v,(A) 1, however, is not true in general for a non-

monotone.p. Example: let A - A, + A, With A,A= - A, (o(A,) - -rp(A,) * 0, f - q on A, ,

and f - -q on A:. Then (by 12.1.6) (A) f f dp - (A,) f f duo + (A,) f f dv = while PA) - 0.

INTEGRABLE FUNCTIONS

§12]

163

12.2.32. If f is
f

g dcp 1 S (A) f

I f I dw

This follows from 12.1.93, 12.2.31, and 12.1.7. 12.2.33. Let .p -(A) = 018, let f be rp-integrable on A, and let (A) f f dip

> - oo

(or < + oo)"; if g is gyp-measurable on A and g > f (or g 5 f)1% then g is also pp-integrable on A.

We set A[g(&) ? 0] = A+ and A[g(x) < 0] = A-. Since (A) f f we have by 3.1.2 also (A-) f f dip > - oo, and thus, since g z f, by 12.1.9 and 12.1.7 also (A--)

f g dgp > - oc .

Hence from (A+)

f

f g d*p z 0 it follows that

the integrals (A+) g dp and (A-) g dgp are not infinite of different signs. Then the contention results from 12.1.91. 1

Now let the set function 4, be also totally additive in the c-field s?.t (cf. the beginning of No. 1); then we have: 12.2.4. If (p and P are totally additive in Y1 and ¢ 5 'p, if f is 'p-integrable

and 4,-integrable on A and f> 0 (or f 5 0), then (A) (or (A)

f f d4,

(A) f f d p

f f d¢ z (A) f f dcc).

From 4, S lp it follows by 3.4.52 that 4+ S 4;+ and

Thus since

f

(A) f f dgp and (A) f f d4, = (A) f f dV,+ - (A) f f d# (A) f f 4 = (A) f (12.2.23), it suffices to prove the proposition for monotone increasing v and,,. Let l > 0 be arbitrarily given and set

Ai = A[(i - 1)b 5 f(x) < ib] and A.,.<, = A [ f (.f) = + oo ].

Then by 12.1.32 we have:

(A+.) f f d4,

(A;) f f dd

(i = 1, 2, ...)

ib¢(A;),

(A+.) f f dcp,

(A j) f f dgp z (i - 1)bw(Ar),

14 If ,p+(A) - 0, then in the following the inequalities g f and g 5 f have to be interchanged. 17 This condition is essential. Example: Let A = A, + A, with A,A, = A,, (A,) > 0,

and f(A7) > 0; furthermore let f - - m on A, g - +'o on A, , and g = -- co on A2. Then by 12.1.1 g is not ip-integrable. 18 Because of 12.1.5E we can here also assume g j-, , f (or g 5, f).

[CHAP. IV

INTEGRATION

164

ff d/' s- a E i¢(Aa + (A+..) fi

and hence (A)

(A') If f dip + & E sv(A.) + (A+.)

d,/5

f f dip 5 (A) f f do + &p(A).

From this the contention follows, since S > 0 was arbitrary. 12.2.41. If fp and P are totally additive in 21, if 4,+ 5 -p+ and 4'

5 rp- in 21,

and if f is cp-integrable on A, then f is also ¢-integrable on A.

If A = A' + A" is a decomposition (1.1), then ¢+ S .p+ and 0imply ¢-(A') = 0 and ¢+(A") = 0. Thus by 12.1.93 it suffices to prove that among the four integrals

(2.4) (A +A') f f do,

(A A') f f do,

(A+A") f f dv.,

(A-A") f f do

no pair is infinite of different signs. Since (A+A') f f dcp = (A +A') f f dp+

f f d,,+, it follows from 4,+ 5 p+ and 12.2.4 that 0 S (A+A') ffd* 5 (A +A) f f dip; and analogous inequalities are valid for and (A +A') f f d¢ = (A +A')

A+A", A-A', and A-A". Since f is rp-integrable on A, there are among the four integrals (1.3) no two infinite ones of different signs; then because of the above inequalities the same is true for the four integrals (2.4). Now we prove as a supplement to 12.1.4: 12.2.5. If the sets A. are disjoint and if f is p-integrable on A. (m = 1, 2, then in order that f be yo-integrable on A = SA., , it is necessary and eufcient that 0

at least one of the two sums E a+(A,) and E X 7(A.) be finite. w

w

Nzcczssrr': If f is 0-integrable on A, then h+(A) = E X+(A.,) and X -(A) _ 0

X 7(A.), and hence the contention follows from 3.4.12.

M e W. We set a*(M) = E (MA,,,)

SUFFICIENCY: Let

f f dip and show that X*(M) satisfies the

conditions 1.) and 2.) of No. 1.-First we prove that 1.) is satisfied, that is, that X* (M) is totally additive in W. Let ((Mi)) be a sequence of disjoint sets of 21

and set M = SMi . Then X*(Mi) _ we have (MA,,)

X*(M) _

f f dip =

(MiA.,) f f dip, and hence

(MA,.) f f dcp = (x+(M;A,n)

m

i

(M;Aw) f f dip; since MA,, = S M{An,

(M. A.) f f dip

- C-(M.Aw)) _ wE E X+(M.A,) - Ew X (M;A,.). ,

i

INTEGRABLE FUNCTIONS

§121

165

Since in these last series the terms are non-negative, we can replace this expression by the following:

X. (M) _

E X+(MiAm) - E Eft X (M;A,.) on

(A+(M;A.,) - )r(M;A.)) (M;AM) f f dcp = E a* (M;) ; that is, A*(M) is totally additive in f.-Now we show that 2.) is satisfied. Let

c' S f

c" on M and let j(M) = 0. By 12.1.32 c'jo(MA,.) S (MA,.) f f duo

c"O(MA,.), and hence by summation over m we obtain also c'qp(M) X*(M) 5 c"cp(M). BIBUOGRAPAY: The same as in No. 1.

3. Summable functions.

If f is -integrable on A and (A) f f dip is finite,

then we call f p-summable on A. From 12.1.3 and 3.1.21 there results: 12.3.1. If f is jp-summable on A, then f is rp-summable also on every ip-measurable subset of A.

From 12.3.1 and 12.1.6 we obtain: 12.3.11. If f is p-summable on A, then

A[f(1) = -}--j = A and A[ f(t) 12.3.12. If f is 9-summable on A, then (M)

Because of 12.1.31 and 3.1.21 (M)

f f dip is bounded for all M e W.

f f dip is finite and hence, by 3.3.21,

bounded. From 12.2.22 and 12.2.23 there results: 12.3.2. In order that f be e-summable on A, it is necessary and sufficient that f be both w+-summable and rp -summable on A.

12.3.21. In order that the so-measurable function f be V-summable on A, it is necessary and sufficient that (A) f I f I dip be finite.

NECassrry: By 12.2.3 X(A) = (A) f I f I dip. SUFFICIENCY: Because of the same formula of 12.2.3 and by 3.1,21, each of the four integrals (1.3) is finite if (A)

f I f I do is finite, and hence the contention follows from 12.1.93.

INTEGRATION

166

[CHAP. Iv

12.3.3. If f is .p-summable on A and on B, then f is gyp-summable also on A + B. This results from 12.1.4, 12.1.31, and §3(1.2). 12.3.31. If the sets A. are disjoint and if f is ip-summable on A. (m = 1, 2, then in order that f be gyp-summable on A = SA m ) it is necessary and sufficient that (Am) f I f I dip be finite.

NECESSITY: If X(A) = (A) f

f drp is finite, then X(A) is also finite.

Thus

since X(A) _ E X(Am), the contention follows from 12.2.3. SUFFICIENCY: By m

12.2.3 E (A m) f I f 14 = E a (A m) ; thus by assumption E a (A m) is finite, m w m h+(A m) and E A (A m) are also finite. Therefore by 12.2.5 f is and hence w

m

,p-integrable on A and according to 12.1.31 (A) f f d,p = E (A,,,) f f d4p; thus,

f f d,p 15 a(Am), (A) f f d,p is finite. 12.3.4. If f is (p-summable on A, then cf (for every finite number c) is also ,p-summable on A and (A) of dp = c (A) f d,p. This follows for c 0 0 from 12.1.62. If c = 0, then 12.3.11 implies that cf since by 12.2.31 f (A

f

f

is ,p-defined, and the contention results from 12.1.6 and 12.1.53. 12.3.41. If f, and fa are
max (f,(x), f,(x))

and

min (f,(x), fa(x))

are also c-summable on A.

Set A' = A[ f1(x) is ft(-+)] and A" = A{f1(f) < fa(x)]. Then A = A' + A" and max (f1(x), fa(x)) = fj(x) for x e A', max (f,(x), f,(x)) = f2(x) for x e A". Thus by 9.2.7 and 12.3.1 max (f,(x), fa(x)) is ip-summable both on A' and on A", and by 12.3.3 also on A. 12.3.42. If f is ,p-summable on A, then I f I is also c-summable on A. on A, and hence by 12.3.41 For by 12.3.4 -f is I f(x) I

= max (f(x), -Ax))

is also rp-summable on A. From 12.2.1 and 12.2.11 there results:

12.3.43. If f, and fa are gyp-summable on A, then f, + fs and f -- fa are also So-summable on A.

12.3.5. If g and h are p-summable on A, if f is gyp-measurable on A, and if g 5 f 5 h, then f is also summable on A." Let A = A' + A" be a decomposition (1.1) of A. By 12.2.33 f is cp-integrable ,9 Because of 12.1.58, g

f

h can here be replaced by g ;S 0 f S, h.

INTEGRABLE FUNCTIONS

§12]

on A' and A". Moreover, by 12.1.7, (A') Since g and h are Sp-summable, (A')

f

g duo

167

(A') f f drp S (A') f h d'p.

f g dip and (A') f h dp are finite, and hence

(A')f f dip is also finite. In the same way one sees that (A")S f d4p is finite. Therefore by 12.3.3 f is rp-summable on A = A' + A". Setting g = -h in 12.3.5 we obtain: 12.3.51. If h is p-summable an A, if f is (p-measurable on A, and -if 1 f 1 S 1 h t, then. f is also

summable on A.YO

Since by 12.1.6 every finite constant is gyp-summable on A, 12.3.51 implies a;: a particular case: 12.3.52. Every bounded21 and rp-measurable function on A is also p-summable on A. 0)!2, and if (A) f f dco = 0, 12.3.6. If 9 is monotone in 21, if f >_ 0 (or f

thenf =eOanA. Let,p be monotone increasing. For .11 O q we have

(A) f f dip = (M)

f f d,o -}- (A - M) f f duo

= 0.

Since herein by 12.1.32 as a result of f > 0 both the integrals of the right side are non-negative, we obtain: (D1)

f f dip = 0 for all M e A.

Hence the proposi-

tion follows from 12.1.6 and 12.1.521. 12.3.61. I f p is monotone in 2(, if f and g are p-summable ors A, if f ?, g (or f "g),

and if (A) f f dip = (A) For by 12.2.11 (A) f

f g dp, then f =, g on A.

(f - g) dtp = 0 and f - g z v 0; hence the contention

follows from 12.3.6 and footnote 22. 12.3.7. If f is (p-:summable on .4, then to every e > 0 there is an n > 0, such that, for M e 2(, O(M) < 17 implies: (211) f 1 f ; drp < e and hence also 9 (li)

By 12.3.1 and 12.3.21 (M) f

f f dip ` < e.

If I do is finite, by 12.1.31 it is totally additive,

and by 12.1.51 gyp-continuous in 2(. Thus the proposition results from 5.7.22. Finally we generalize 12.1.951: 12.3.8. If w_(A) = 023, if ((f,)) is a sequence of (p-measurable functions pp-conI h I can here be replaced by I f 101 h 1 "' According to footnote 20, "bounded" can here be replaced by "v-bounded." 94 Because of 12.1.63, f k 0 (or f 0) can here be replaced by f iGr 0 (or f ,. 0). .a An analogous theorem holds fore+(A) - 0; then only f, g and f, g have to be interchanged. S° Because of 12.1.53, 1 f I

INTEGRATION

168

tCHAP. IV

vergent to f on A, and if there is a v-summable function g'", such that for all v f, Z g (or f, S g)n, then f and all f, are rp-integrable and

(A) f f dip < line (A) f f, d v (or (A) f f dip ? lim (A) f f, d`p) 26 Since by 12.3.11 g is rp-finite, f, - g is 9-defined and hence by 9.2.2 vp-measur-

able on A ; moreover, by assumption f, - g >_ 0, and thus according to 12.1.9 f, - g is also vp-integrable. Therefore by 12.2.1 f, _ (f, - g) + g is also V-integrable. Since f, - g -, f - g, by 12.1.53 and 12.1.9 51 f - g is vp-integra-

ble and (A) f

(f - g) dip 5

lim (A) f (f, - g) dvp. Now adding on both sides

the finite value (A)f g dvp we obtain the following, according to 12.2.1: f = (f - g) + g is yrintegrable and (A) f

f dip 5 lim (A) f f, dip.

The corresponding

second part of the proposition results if we replace f, , f, and g by and -g, respectively. For later use we mention: 12.3.81. In theorem 12.3.8 "v-convergent" can be replaced by "asymptotically eoiwergent".

First, as in the proof of 12.3.8, one sees that the f, are 4-integrable. By 10.3.64

f is .p-measurable and by 10.3.33 f z, g; thus according to 12.2.33 f is also rp-integrable. Now suppose (A) f dip > lim (A) f f, d1P. Then there would

f

a (A) f be a subsequence ((f,.)), such that lim

f., dp = lim (A) f f, d'; moreover,

since ((f,.)) is asymptotically convergent to f, by 10.3.62 ((f,,,)) contains a subsequence ((f,.,)) vp-convergent to f. Then we would have (A) f f dip > lim (A) f f,, &p, contrary to 12.3.8 which gives (A) f f dip S lim (A) f f, dip. Thus the above supposition is impossible. BIBLIOGRAPHY: The same as in No.1.-The expression " summable" is due to H. LEBasGuz (1), p. 115.

4. Characterization of the set functions that are vp-integrals. In defning (in No. 1 above) the 9-integral we started from a given point function f(x) and, "' It would suffice to assume that g is c-finite and y -integrable with (A) f g do > - co

(or < -i- "). '" Because of 12.1.53 one can here also assume f, k, g (or f, ;9, g). " In these integrals A can be replaced by any ip-measurable subset M of A, since for M all conditions of the theorem are satisfied.

169

INTEGRABLE FUNCTIONS

§121

using the determining function P, we looked for a suitable set function

/ = (M) f f 4) satisfying the conditions 1.) and 2.).

a(M) t

Now we ask con-

versely` what set functions X(M) can be represented as integrals. p and 1l2 (or EC) the assumptions of No. 1 are to be satisfied.

Again for

12.4.1. In order that a set function X(M) be a p-integral, it is necessary and sufficient that h(M) be totally additive and rp-continuous.

Or stated more explicitly: In order that to a set function X (M) defined in the a -field 81 there be a rp-iniegrable function f (z) on A, such that X(M) = (M) f f d p

for all M e W, it is necessary and sufficient that X(M) be totally additive and So-continuous in 1. NBCESSITT: This follows immediately from condition 1.) of No. 1 (or 12.1.31) and from -12.1.51. SUFFICIENCY: Since X(M) already satisfies the condition 1.) of No. 1, we have only to show: there is a 9-measurable function f(x) on A, such

that for f(x), together with X(M), the condition 2.) of No. 1 is also satisfied. According to (1.1) we form the decomposition: A = A' + All where A'A" = A, ,p (A') = 0, and p'(A") = 0. Because of the structure of condition 2.) the definition of f (z) and the whole proof can be given separately for A' and A"." We define f, say, on A'; on A" all is analogous. Thus let M s A'11 and hence 0. We have to show: if c' f(x) S c" for all x s M, then co(M)

c'c(M) s X(M)

(4)

c"v(M)

If .p(M) = 0, then since h(M) is m-continuous, we have also X(M} = 0; thus (4) is then satisfied for all values c' S c", and hence it does not matter at all how f is defined. Therefore it suffices to consider only such M for which 'P* (M) > 0. Moreover, (4) is certainly satisfied for c' = - w or c" = + co. Now let c be a

finite constant; we set h - cp = he ; since p is finite, by 3.2.52 X. is totally additive in W. If X., (M)

0 and c1

cs , then as a result of o (M) is 0 we have

also ),(M) is 0; if Xi(M) 5 0 and cs .1 c, , then X., (M) 5 0 also.-Let (where all terms are different) be the set of all rational , r, , rl , r, , numbers r. For every r, according to 3.4.71, we form the decomposition: where A', Ar' = A, ), (A;) = 0, and h+(A,,') = 0. We write A' = Ar -}AA .r

A;, - Br and 41 = B;t ; furthermore we set for every r: A' = Br' + B'r' with B,B' = A and define Br' in the following way by induction. Let B,. , r.. If r,,.+, > r;(i = 1, , m) and be already defined for r = ri, r,), then we set B;,,+, = A'-+1Br.. If rm+, < mss (r, , rx Brt , rm), then we set B".+1 = r{(i a 1, .. , m) and ri = min (r, , -

r For M e 5 let v(M) - 0, for instance. Since M - MA' + MA" and 4v (AtA') - 0, we have also V ;-(MA') - 0; thus p+(MA") = 0 implies O(MA") - 0 and, since x is 0. Therefore, as to condition 2.), one can restrict oneself to continuous, also ).(MA") sider only the sets M f A'1 4f for (B!) - 0 and analogously only the sets M e A"l ll for 0 +Of) - 0.

[CHAP. IV

INTEGRATION

170

, m), then let rk be the greatest and If rm,., lies between two of the r;(i = 1, , m) for which rk < r..+i < rz let r, be the smallest of the numbers r{(i = 1,

in this case we set Br'.+, = (Ar.+IB*k) + B,, According to §1(,.2) we + or B,_+i _ B,k or then have B;;,+l = A+, Br..+1 = (A;;,+, + B*;) Br, , respectively.

As one sees immediately by induction,

B. and B,1, Q B;% for r" > r'.

(4.1)

S .lee £ a It, we have also all Br`* Wand all B;' a W.

Moreover, we show by induc-

tion: If M e B'r 1st, then X,(M) 9 0. This is true for r - r, . Let the contention be already satisfied for r;(i = 1, - , m) and let M e Br ,+1 12C; we ,and hence X,.+,( MI) iG 0. set M = M, }die with 1t1,M2 = A and M, Then either M2 = A or (using the above notation) M2 Q B;, ; in the latter case,

since t 5 m, we have already X,,(M2) g 0, and thus, since r.+, < r, , also 0" Therefore, indeed, X,,,+,(M,) + k, (MI) k 0. X,M+,(1H2) and reIn the same way one shows (using the above representation of 0.-Now we define f(x) for placing B*, by Brt) : If M e BP' 1 Ri, then X,(M) every x e A' in the following way: If there is a rational number r, such that x e B'r , then we set f(x) = supremum of all rational r for which x e B' ; (hence if for all r x e Br' , then we set f(x) = + oo) ; yet if there is no r, such that s e B'r , then we set f(x) = - co. First we show that this function f is o-measurable on A' ; that is, we have to prove that the sets A'(f (x) > y) = M are se-measurable for all y e R, . For every x e Br' M.with r > y we have f(x) > y, thus B' Q; Me for also. On the other hand, if x e M, , that is, all r > y, and hence S Br C r>Y

f (x) > y, then there is an r > y, such that x e Br' ; thus M, Q S B* also.

M = S B; . ">5

>V

Hence

Therefore since every Br a 21, we have M. a Yf also and f is

(-measurable on A'.-Finally we show that by means of our f the condition 2.) of No. 1 is satisfied for all M e A'1% with pp(M) > 0: a) Let c' be finite and let c' ;5 f on M; then according to the definition of f and because of (4.1), for every r < c' we have M Q B; ; thus X,(11) z 0, and hence X,.(M) Z 0 also; that is, X(M). 0) Let c" be finite and let f _5 c' on M; then for every r > c' c qo(M) Br' ; thus X,(M) S 0, and hence we have M 0 also, that is, X(M)

we have M !

y) Let c' = + co, that is, f = + co on M; then for every r B; ; thus X,(M)

0, that is, X(M) Z rgo(M) for every rational r, S) From c" co,

and, since 4'(M) > 0, it follows that: X(M) = +

that is, f = - cc on M, it follows in the same way (by means of M C B;') that

X(M) _ -

. Therefore condition 2.) of No. 1 is satisfied for every M e A' 1 ff. From 12.4.1 there results, because of 12.3.12, immediately: 12.4.11. In order that to a set function defined in the u-field W there be a

se In passing we note that, since M, Q A rm+ 0. always

we have also \,,,,+, (M:)

0, and hence

INTEGRABLE FUNCTIONS

§121

p-summable function f(x) on A, such that X(M) = (M)

171

f f dp for all M s I(, it is

necessary and sufficient that h(M) be totally additive, (p-conanuous, and bounded in ?I.'0

21

BIBLIOGRAPHY: For rp - µ, and Mo : H. LEBESGUE [1], p. 129 footnote; Rendic. Aocad. Lincei (5) 16, (1907), p. 286; Apn. Ec. Norm. (3) 27 (1910), p. 399; G. VITAI4, Atti Accad. Torino 40 (1904/1906), p. 1021; 43 (1907/1908), p. 237. For general'p, but with A Q R,. (and if V( contains the Borel sets in A) : J. RAnoN, Sitaungsberichte Akad. d. Wise. Wien 122 (1913), p. 1349; (also P. J. DANIELL, Bull. Amer. Math. Soc. 26 (1920), p. 444; Proc. London Math. 0) in abstract spaces O. NIxoDYM, Fund. math. Soc. (2) 26 (1(27), p. 95). For general w

15 (1930), p. 168, first succeeded in proving theorem 13.4.1171; subsequent to him also: S. SAxs 111, p. 255; 121, p. 30; J. voN NEVMAxN 11], p. 186; Annals of Math. 41 (1940), p. 127; B. JESSRN, Mat. Tidsskrift B 1938, p. 21; C. CARATHI`E.ODORY, Math. Zeitschr. 46 (1940), p. 181; K. YosrnA, Proc. Imp. Acad. Tokyo 17 (1941), p. 228; A. D. ALExANnaoFr, Rec. Math. [Mat. Sbornik], N.S. 9 (1941), p.576; J. M. H. OLMSTED, Trans. Amer. Math. Soc. 51 (1942), p. 164; R. S. PHILIPPS, Amer. Journ. of Math. 65 (1943), p. 130; C. E. RICKART, Trans. Amer. Math. Soc. 56 (1944), p. 50; of. also A. KOLMOGOROFF, Math. Annalen 103 (1930), p. 694.

5. Upper and lower integral Now let f be a quite arbitrary p-defined (but not Let A = A' + A" be again a decomposition (1.1) of A. Then there are e-integrable functions which are >_, f on A' necessarily Se-measurable) function on A.

and ;S. f on A" (§9, 4) ; we designate them by g (g = + - on A' and = - on A" is such a function). Analogously there are -integrable functions which are 6, f on A' and z, f on A"; we designate them by h. From 4.2.45 it follows

that if one replaces A = A' + A" by another decomposition A = A* + A** having the properties of the decomposition (1.1), this does not influence the definition of the functions g and it.

Now we set

inf (A) f p duo = (A)1 f dhp,

sup (A) f it dcc = (A) f f dcp

and we call these two numbers the upper and lower p-integral of f on A, respectively.' From 12.1.7 and 12.1.53 it follows that 11 Here "bou,uled" can be replaced by "finite" (cf. also 3.3.21). ao Cf. also p. 155 where, under more restrictive conditions, another proof for this theorem is given. 41 It is often called the Radon-Nikodym theorem. B Without changing the values of the integrals we can here always assume g ;a f on A' (analogously g f on A", h 5 f on A', and h i' f on A'). For sinceg , f on A',

we have A'1g(.t) < f(.f)) =, A; if now on A'Ig(f) r< f(;E!1 we replace the values of g by +m, then according to 12.1.52 the value of (A) J g dip is not changed.

[CHAP. IV

IIVTEGRATION

172

(A)f f dip

(5)

(A) f f dim.

If f is tp-integrable on A, then

f

(A) f f dp = (A) f f do = (A) f dip;

(5.1)

for then f is both a function g and a function h. 12.5.1. If f is q,-measurable, but not 9-integrable on A, then (A) f f do = -00

and(A) ffdip = +00. Then among the four integrals of 12.1.93 at least one is + - and one is - oo

f f dsp = + 00, then, by 12.1.7 and 12.1.53, for each of our functions g we have (A +A') f g d4p = + co, thus by 3.1.2 also (A) f g dip = + , If for instance (A +A')

and hence (A) f f dip = + 00 also. In the same way one sees that, if one of the

f

four integrals of 12.1.93 has the value - co, then (A) f d 4p = - 00 also. 12.5.2. Among the functions g there is a function g* and among the functions h there is a function h*, such that (A)

f f dp = (A) f g* dco

and

(A) f f dyo = (A) f h*dv.

If (A) f f dip = + oo, then it suffices to set g* _ + co on A' and g* _

-w

on A'. Thus we can assume: (A) f f d4' < + co . Then certainly there is a

f

sequence ((g,)) of functions g, such that (A) g, dp -* (A) f f drp. If we set

g*(x) = inf g,(x) for x e A' and g*(x) = sup g,(x) for x e All, then by 10.1.2 r

g* is gyp-measurable on A, and hence by 12.2.33 and 12.1.4 g* is also a function g;

moreover we have (A) f f dip

(A) f g* d-p

(A) f g, dep.

From this the con-

tention follows for g*. Analogously for h*.

f

12.5.21. If (A) f f dip (or (A) f dip) is finite, then among the functions g there

is a function g* (or among the functions h there is a function h*), such that for aU Me $I (M) f .f &p = (M)

1

g* dip

((M) f f dip = (M) f h* do, respectively').

173

INTEGRABLE FUNCTIONS

§12]

f

Let g* be .,he function of 12.5.2 and let (A)f f dce = (A) g* dgo be finite;

then for every M e 2t by 3.1.21 (M) f g* dco is also finite, and (M) f f dip 5 (M) f g* thp.

If now (M) f f dp < (M) f g* dco, then by 12.5.2 q* could be

replaced by a function g** on M, such that (M)

f f dcp = (M)fg** dp.

Let us

set g*** = g** on M and g*** = g* on A - M; then by 12.1.4 g*** would be also a function g, and we would have (M) f g*** dip < (1!1) g* dip and

f

(A -- Al') f g*** dip = (A - M) f g* dp; thus (A) f g*** d(p < (A) f g* dip, and hence (A) f g*** duo < (A) f f dcp, which is impossible.

We can prove the contention of 12.5.21 under more general conditions:

f

12.5.211. If either (A)] f dip > - o0 or (A) f d,p < + x, then among the functions g there is a function g* and among the functions h there is a function h*, such that for all M

(M) f f drp = (M) f g* dip and (M) f f d, = (AI) f h* dcp.

We give the proof for g*; it is analogous for h*.-First let (A) f f dp

> - -o.

We define the desired function g* first on A'. For every rational r we set A, = A'[f(x) > r) and designate by Ax a measure-cover of A, (§6, 3). Moreover, we set B, = S A, (for rational s); then since A, = S A, , by 6.3.32 B, is +>r s>r also a measure-cover of A,, and B, = S B. . Thus's there is a function g* for t>r

which A tg*(t) > r) = B, ; by 9.1.1 g* is 9-measurable on A'.

Since

(A) f f dcp > - oo, there is a function h for which (A) f h dso > - eo ; then by 3.1.2 (A') fh d e

> - - also. Thus since h 5 f 5 g*", by 12.2.33 g* is (p-inte-

grable on A' and by 12.1.7 (A') f g* dp > - oo. Analogously (starting from the

sets A"[f(rt) < r]) we define the function g* on A"; then (A) f q* d

- ao also, and by 12.1.92 g* is rp-integrable on A. Since g* ? f on A' and g* S f on A", g* is one of our functions q. Now let M e ' t and let g be any So-integrable 33 Ac.ording to H. Hahn [2), theorem 30.1.41. H As to 1i f, ef. footnote 32, 0.171.

[CHAP. IV

INTEGRATION

174

function on M which is ;->f on MA' and 5f on MA"." We set C, = M.4'[g(,x) > rJ; then C,. is a rp-measurable set containing MAr and, since MB, is a measure-cover

of MA, , we have MBr - Cr =* A. Since MA'[y*(i) > g(i)] C S(MBr - Cr), r

A also; and in the same way one sees that we obtain MA'[g*(i) > MA"[g*(i) < g(i)] = , A. It follows by 12.1.53 and 12.1.7 immediately that (MA') f g* dip 6 (MA') g d(p and (MA") f g* dip 5 (MA") g dip, and hence

f

f

(M)fg* dp S (M)fg dip also. But this means: (M) f f dhp = (M) f g* dhp, as contended.-Now let (A)f f dip < +oo. By 12.5.2 there is a function g' with

(A) f g' dip = (A) f f dp(< + co). If g* means the same function as above, we set g**(x) = min (g'(x), g*(x)) for x e A' and g**(x) = max (g'(x), g*(x)) for x e A".

Since g** S g' on A' and g** z g on A", by 12.2.33 g** is'p-int.egrable

both on A' and on A" and, since (A') f g** dip < (A') f g' dcp < + ao and the

f same is true for A", by 12.1.92 g** is rp-integrable also on A. Thus since g** on A' and y** 5 f on .4", g** is one of our functions g. Now we continue in the g(i) J same manner as above for g*. Then since A'[g**(i) > g(i) ] c and thus MA'[g**(i) > g(i)] =* A and analogously MA"[g**(i) < g(i)] _,, A,

we obtain: (M) f f dip = (M) f g** dip. 12.5.3. In order that f be p-integrable on A, it is necessary and sufficient that

for allMe21 (5.2)

(M) f f d4o = (M) f .f gyp.

NECESSITY: This follows from 12.1.3 and (5.1).

SUFFICIENCY: According

to 12.5.211, it follows from (5.2) that (M) f g* dip = (M) f h* dip for all M e %.

Therefore by 12.1.521 g* = h* on A and, since h* S f 9 g* on A' and g* 5 f S h* on A", we have also f =m g* on A. Thus since g* is p-integrable on A, by 12.1.52 f is also
(A) f f dsp and (A)f f d{p have the same finite value. NECESSITY: This follows from (5.1).

SUFFICIENCY: By 12.5.2 (A) f f do =

(A) f g* dip and (A) f f dip = (A) f h* dp, and hence by assumption (A) f g* dip = 35 Cf.foot.note 32, p. 171.

175

INTEGRABLL MNCPIONS

§12]

(A) f h* dip; that is, (5.3)

W) f g* d y, + (A") f g* dsp = (A') f h* dip + (A") f h* dp.

Since h* ;5, g* on A' and g* 5, h* on A", we have: (A') f h* (IV 5 (A')

f

g* dip

and (A") f h* dp :5 (A") f g* dp; and, since by assumption both sides of (5.3) r are finite, we obtain from (5.3): (A') f h* 4 = (A')1 g* dcp and (A")] h* (14 _

But by 12.3.61 it results that h* _, g* both on A' and on A", and hence on A also. As in the proof of 12.5.3, it follows that f =, g* on A. (A ") f g* dip.

Therefore by 12.1.52 f is p-integrable on A and (A)f f dip = (A) f 9* (Ip = (4) f f dip, which by assumption is finite.

BIBLio6RAPHT: The above definition of the upper and lower integral evolves from H. LEBrfol*r:, Paris These 1902, p. 25, 45 - Annali di mat. (3) 7 (1902), p. 255, 275.-As to other definitions of the upper and lower integral, most of them suitable for the Riemann integral, cf. Encyklop. d. Math. Wiss. IIC9b (P. MoNTaL-A. RosENTHAL), Nos. 29 and 35a, as well as the bibliography to §13, 4.

6. Content-like determining function. We restrict ourselves again to the consideration of *-integrable functions f and we will now assume in particular that ,p is content-like (§7, 4).

12.6.1. If w is content-like and f is gi-integrable on A, then (M) f

f dip is also

content-like in It. This follows from 12.1.51 and 7.4.4. Strengthening 12.1.521 we can show: 12.6.2. If jp is content-like, if f and g are cp-integrable on A, and if for all closed

f

sets F s W: (F) f dyo = (F)

f g dip, then f -, g on A.

Since tp is content-like, by 7.4.2 to every M s 1 there is an F,-set C C M, such

that C -, M. Since for all closed sets F e 9 we have (F) f f dy - W) f g d v, by 12.1.31 and 3.2.2 we have also (C) f

f duo = (C) f g dto.

But from C -, M

it follows by 12.1.53 that (M) f f dip = (C) f f dip and (M) f g djc = (C) f g dw.

[CHAP. IV

INTEGRATION

176

Thus for all M e Al we have (M) f f dip = (M)

f g dip, and the proposition results

from 12.1.521. 12.6.21. If w is content-like, if f and g are So-summable" on A, and if for all sets G open in A: (G) f f dip = (G)

f g 4i then f =, g on A.

Since is content-like, to every M e `?i there is a Gs-set B Q_ M, such that .p(B) = g,(M) ; thus (since o(M) is finite) B =, M, and hence BA =, M also. The proof now proceeds as in 12.6.2, except that 3.2.21 has to be employed

instead of 3.2.2. From 11.3.221 and 12.1.52 we obtain: 12.6.3. If ip is content-like and f is rp-integrable on A, then there is a Baire function f* of the second class, such that for all ,I11 a Q.l: (1L1) f f dcp = (M)f f* dcp.

_

Because of 12.6.3 we can now strengthen 12.5.211: 12.6.31. If ,p is content-like and either (A)

f f dip > - oo or (A) f f dip < + cc

,

then there are Baire funeiions g* and h* of the second class, such that for all M e K

(M) f f d.p = (M) f g* dip and

(41) f f do = (M) f h* dip.

7. Improper integrals. Now let ip be again an arbitrary totally additive set function in P. Hitherto we have assumed that for the set A of integration ,p(A) is finite.

Now we will define also integrals for whose set of integration tp(A)

is infinite; we call them, as contrasted with the integrals with finite ip(A) discussed hitherto, improper integrals. The integrals with finite p(A) can then be called also proper integrals; where the contrary is not expressly stated, the word "integral" is always to mean proper integral.

Thus let .p(A) = f -. We restrict ourselves to the only interesting case in which A = SA m where all

are finite.

Replacing A. by

A. - (Ai + ... -}.. A,,,_1) one can assume, moreover, the A, to be disjoint. Again we define: the function f 0-measurable on A is called rp-integrable on A

if there is a set function A(M) defined in 21 and satisfying the conditions 1.) and 2.) of No. 1." We have also here: If f is ro-integrable on A, then the value " Here "9o-summable" cannot be replaced by "9o-integrable". Example: Let A - (0, 11 p, . By 8.2.84 there is a p,-measurable set B Q (0, 11, such that for every interval

and,,

(a, b) C [0, 11 both B (a, L] * 1 A and (a, b] - B $,,1 A. Now we set f g - +go on B; furthermore f - 1 and g - 0 on 10, 11 - B; then f and g are p,-integrable. For every set G open in A now (0) f f do,

- (0)

f g dp, - +

, while f

g on A.

11 If in 2.) c - 0 and 9,(M) - ± o, then we have to set c.p(M) - 0.

177

INTEGRABLE FUNCTIONS

§121

X(A) is uniquely determined by the conditions 1.) and 2.). For if A = SA m

where the A. are disjoint and the cp(A.) are finite, then by condition 1.) X(A) _ E ).(Am); herein by 12.1.2 and (1.2) a(Am) - (Am)f f dip; hence A(A)

is determined by X (A) _ E (A m) f f 4. If f is p-integrable on A, then we designate the uniquely determined value of X (A) also here by (A)

f f dip and call it the (improper) tp-integral of f on A. Thus

we have: 12.7.1. If ,p(A)

if A = SAm where the A. are disjoint and the cp(Am) 0

are finite, and if f is ip-integrable on A, then (7)

12.7.2. If v(A) = f

(A) f f dip = E (Am)

f

f dip.

and if A = SA where the A. are disjoint and the o(Am) 0

are finite, then in order that f be cp-integrable on A, it is necessary and sufficient, that f be co-integrable on every A. and that at least one of the two sums E X+(A and E X-(Am) be finite. 0 The proof is the same as for 12.2.5.

Now it can be seen without any difficulty that the theorems 12.1.1, 12.1.3, 12.1.31; 12.1.32, 12.1.4, 12.1.51, 12.1.52, 12.1.53, 12.1.6, 12.1.62, and 12.1.63 are valid also for the improper integrals. Theorem 12.1.521 is also valid; only the part of this theorem referring to "necessary" has to be proved in particular: f g is true by 12.1.521 first on A m , and hence also on A = SA,. . Furthermore 12.1.7 is valid for the improper integrals. In order to see this, one has only to consider that ip (A) = 0 implies w (Am) = 0; thus by 12.1.7 (Am) f g dip and, since by (7) (A) f d.p = E (Am) f f dip and (A m) f f d p

f

(A) f

g dcp, we obtain also (A) f f d.p

5 (A)

f g dp.

From 12.7.2 there results that 12.1.9 holds also for our improper integrals; for if 97(A) = 0 and f >_ 0, then by 12.2.3 X7 (A.) = 0 for all m. It is evident that then 12.1.91, 12.1.92, 12.1.93, and 12.1.94 are also still valid.

Theorem 12.2.1 holds also for improper integrals. For if (A) f f, dm and (A) f f2 d.p are not infinite of different signs, then by 3.1.2 the same is true as for

every ip-measurable subset of A.

Thus (2) holds for every A.. Now we set

(Am) f fi dp = X1(A,), (A.) f fs d*p = x2(Am), and (Am) f (fi + fs) 4 = X(Am)

178

[CHAP. IV

INTEGRATION

(=)L,(A..) + X2(Am)). It is impossible that either E X (A..) and E X=(Am) M

are infinite simultaneously or E X (A.,) and E Xt(Am) are infinite simultam

m

neously, since then by 12.7.1 and

12.7.2

a,(A,.) = (A) f f, dip and

E X2(Am) _ (A) f f, dp would be infinite of different signs, contrary to the m

assumption. Therefore by 12.7.2 either E X (A..) and EX (Am) are finite or ..

m

E ai (Am) and E X (Am) are finite. Thus since by 3.4.5 E X+(Am) 5 M

ra

M

E (Ai (A,.) + ),_ (Am)) and E X- (A.) s E (Xj (Am) + X (Am)), it follows that m

m

,n

at least one of the two sums E X+(Am) and F, X-(Am) is finite. m

Hence by 12.7.2

f, + fp is p-integrable on A. By 12.7.1 and (2) [for the Am] we now have: (A) f (f1 '+' fs) dip =

L. (Am) f U, + f:) d t (Am) f f1 dip + E (A..) f f: dso = (A) f fi dso + (A) f f,dsp.

Replacing f, by -f3 in 12.2.1 and using 12.1.62 (for c = -1) one obtains also 12.2.11 for improper integrals. Now we can show that 12.1.8 holds also for improper integrals. The proof of 12.1.8 cannot immediately be generalized. For improper integrals one can come to the very end of this proof, namely to the inequality A(M) ? psp(M) ; but from this it does not follow that X(M) z c'yp(M) if q,(M) = + eo and c' = 0 (and hence

p < 0). Yet this inequality is trivially satisfied for c' = 0 if all f, ? 0 on A and hence a O.--In order to prove the general case, we set f = f+ - f-, where f '(x) = max (f(x), 0) and f -(x) = max (- f(:x), 0), and analogously f, = f; - f-.,

where f; (x) - max (f,(x), 0) and f, (x) = max (-f,(x), 0). Then 0 5 fi <

f; 5 f+ and 0 5 f- 5 f: S fi .

Thus since (A) f fi dip >= 0, the proof of 12.1.8

can be generalized to improper integrals for the functions f+ and f; .

Hence

.f+ is .p-integrabl.e on A and (7.1)

(A) ff+rip1im(A)ff+dsp.

Set A- = A(f,() 4 0]; then from (A) f fi dto > - oo it results by 12.1.62 and 3.1.2 that (A) f fi dqp = - (A-) f fi d* < + ao and thus the proof of 12.1.8

can be generalized to improper integrals for the functions f- and f,-. f - is saintegrable on A and

Hence

I,EBESGIIE AND RIEMANN BUMS

§13]

(A) f

(7.2)

dip - lim (A)

179

f f,- dip.

Since f - f+ -- f and f, = f; - f, and since by 12.1.7 0 S (A)f

F dt 5

(A) f f,- dip s (A) f fi duo < + ac, the full generalization of 12.1.8 now follows from (7.1), (7.2), and 12.2.11.

Replacing p by -,p in 12.1.8 we obtain by 12.1.63 also the generalization of 12.1.81. Now 12.1.95, 12.1.951, 12.1.952 hold also for improper integrals.

We will not go further into the properties of improper integrals. We wish only to remark that 12.4.1 is also true for improper integrals. We have to show this in particular only for "sufficient": If A - SA,,, where the A. are disjoint and the f(A,,,) are finite, if X. designates the a--field A. 12(, and if Me 4C, then

we set M. = MA,,, and have by 12.4.1: X(M,) = (Mm) f f dip for every m; here

f is defined on every set M. and hence also on M. Since X is totall in A, by 3.4.32 X+ and a- are also totally additive; thus A+(M) =

additive

and X-(M) = L X-(Mm) and, since a(M) = X+(M) -- A-(M) (3.4.61), at most one of these two sums can be infinite. Therefore by 12.7.2f is qo-integrable

on M, and hence by 12.7.1 (M)

ffd.p = E A(M.) = A(M).

§13. Lebesgue and Riemann sums 1. Lebesgue sums. We again use the terminology introduced at the beginIn particular, we assume ,(A) to be finite (if the contrary is not stated), so that speaking about integrals we always mean proper integrals. ning of §12, 1..

+..

In the following we shall have. to employ sums of the form E ai where the a; are real numbers. Let I+ be the set of the indices i with ai k 0 and let I_ be

the set of the indices i with ai < 0. If now at least one of the two sums E a; +m

i.l +

and E ai is finite, then the sum L ai is considered to have meaning and its value is defined by

E ai m% E ai + E a, ; otherwise the sum E a; is considered to be meaningless. +40

finite,

ai I is also finite.

Thus if E ai is

f CHAP. IV

INTEGRATION

180

Let (z;) (i = 0, f 1, f2, ) be a set of real numbers with zi < zt+s, °o . Then we say the z; form a scale if the set lim z{ _ + co, and Em z; of the numbers z;+, - z; is bounded. If in particular z;+1 - zi < S for all i, then we say the z; form a b-scale. If now f is tp-measurable on A and {z{} is a scale, then we set

A[z; 6 f(t) < z:+il = A: , A [zs < f (t) S z;+11 = A{' A [f (x) = + oo l = A+m, A[f(I) = - oo l = Aand we consider the sums (1.1)

L'(f, , {z:}) _

(1.2)

L"(f,.p, {z:}) _

(+°o)c(A+.e) + (-c)sp(A-..),

(1.3)

L*(f, , {z:}) _

z:+ w(A.) + (+ co)io(A+W) + (- ao)v(A-,),

(1.4)

L"(f,

cc)-r(A+.o) + (- °°)so(A-.o),

{

+M

E

(+c)o(A+.) + (-

where, if ,p(A+,) = 0 or ,p(A,) = 0, the product (+ oc),p(A+.,) or (- co),p(A,), respectively, has to be replaced by 0. These sums are called Lebesgue sums associated with the integrand f, with the determining function .p, and with the Where it does not matter which of the four aforesaid Lebesgue sums is taken, we simply write L(f, sp, { z, }) " Such a Lebesgue sum has meaning if the infinite series appearing in its definition has meaning and if no two of the three terms in its definition are infinite of different signs. 13.1.1. If f is s--integrable on A and { z{} is a S-scale, then L(f, sp, { z; }) has meaning and scale {z;}.

.

(A) f f dip = L(f,,p, { z;)) + 0I*(A)

(-1 < 8 5 + 1).

We give the proof, say, for the sum (1.1). We consider the function g which

g f equals z{ on A; , is + - on A+m , and is -- °o on A,,,, ; then f - b Then g is sp-inteLet A= A' + A" be a decomposition §12 (1.1). on all of A. grable on A'; for if (A') f dip < +co, then this follows from 12.2.33, since

f

g g; f; if (A') f f dip = + co, and hence by 12.1.6 and 12.2.11

' If one wants to express also the set A on which the Lebesgue sums are formed, then one can write more explicitely: L(f, s,, Jz;I, A).

LEBESOUE AND RIEMANN SUMS

1131

181

(Al f (f_a)dy,> -- -, then it follows again from 12.2.33, since g i' f - S. Analogously one sees that g is 9-integrable on A". Now from f - 6 s g f we obtain by 12.1.7:

(A') f (f - 6) dip 6 (A') f g dips (Al f f dip.

(1.5)

f g dip is infinite, then (A') f g dip = (Al f f dip, and the same is true for (A") f g dip and (A") f f dtp. By 12.1.92 (Al f f dip and (A") f f d,p are not infinite of different signs; therefore the same is true also for (A') f g dip Thus if (A')

and (A") f g dip, and hence by 12.1.92 g is , -integrable on A. But then by 12.1.6 and 12.1.31 we have (A)

f g dp = L'(f, p, {z;}).

From (1.5) there results:

(A') f f die - &p(A') 5 (Al f g dsp 5 (Al f f dp, and analogously one gets:

f

f

(A") f g do 5 (A") f dcp - &p(A").

(A") f dco

Thus, since L' (f, w, { z, }) _ (A) f g d`p = (Al f g dip -f- (A") f g dp, the proposition follows by addition. From 13.1.1 there results immediately: 13.1.11. If f is pintegrabie on A, if { z {' } (v = 1, 2, ) is a 8,-scale, and if 8, -+ 0, then Jim

L(f,1p, { z :'' }) = (A) f f dip.

13.1.2. If (zt} is an arbitrary scale, then in order that the c-mceasurable function

f be p-integrable on A, it is necessary and sufficient that L(f, .p+, { z; }) and L(f, gyp`, {zr}) have meaning and are not infinite of the same sign.

NECESarrY: Let f be ,p-integrable. By 12.2.22 f is also p-integrable and

f

c -integrable and (A) f dp+, (A) f f dip- are not infinite of the same sign. Now the contention follows from 13.1.1. SumcrsNaT: We give the proof, say, for the sum (1.1).

We consider the function g which equals zion A i , is + co

on A+ , and is -won Ate, . Then by 12.1.6 g is,p+-integrable on every set A, , on. A.+,, , and on A ., .

Now we apply 12.2.5 to the ,p+-integral of g, substituting

[CHAP. IV

INTEGRATION

182

the sets A;(i - 0, f 1, f 2, ...),A+.. , Ate, for the sets designated thereby A. . Then for z; i? 0 X+(A;) there has to be replaced by z;p+(A;) and a-(A;) by 0; for z, < 0, \+(A;) has to be replaced by 0 and a-(A'j) by { z; I p+(A;); furthermore X+(A +..) has to be replaced by (+ ao )p+(A+.,), >.7(A+.) by 0,

by 0,

and )\-(A.) by (+ oo )p+(A,) . From the assumption that L' (f, p+, {z,) has meaning it follows that on account of those replacements at least one of the two sums F, and F, (A,.) of 12.2.5 becomes finite. Thus by 12.2.5 w

m

g is p+-integrable on A ; and in the same manner one sees that g is 0--integrable

on A. Since (A) f g dp+ = L'(f, p+, {zi}) and (A) I g d(p = L'(f, (P, {z;}), by

assumption (A) f g dp+ and (A) f g dp- are not infinite of the same sign, and hence by 12.2.22 g is p-integrable on A. Now we set

h= f -gon A'(i=0,±1,f2,...) and h = 0 on A+.e + A--.. Then h is bounded on A and hence by 12.3.52 Therefore by 12.2.1 f = g + h is p-integrable on A. The following theorem is contained in 13.1.2 as a particular case: 13.1.21. If { z; { is an arbitrary scale and (p is monotone, then in order that the ,p-measurable function f be p-integrable on A, it is necessary and sufficient that L(f, p, {z;}) have meaning. From this there results: 13.1.211. Let p be monotone and let f be p-measurable on A; if one of the four Lebesgue sums (1.1), (1.2), (1.3), (1.4) has meaning for a particular scale, then all `p-summable on A.

of them have meaning for every scale. 13.1.22. If { z; } is an arbitrary scale, then in order that the 9-measurable function f be p-summable on A, it is necessary and sufficient that

L(f, p+, {z;}) and L(f, w ,

{zi})

have meaning and be finite.

NEcassITY: By 12.3.2 f is both p+-summable and jp -summable. Hence by 13.1.1 L(f, p+, (z{}) and L(f, v7, {z;{) have meaning and are finite. SVFPWU NCY: By 13.1.21 f is p+-integrable and p`-integrable and by 13.1.1

(A) f f dp+ and (A) f f dp- are finite. Hence according to 12.3.2 f is p summable. The following theorem is contained in 13.1.22 as a particular case: 13.1.23. If {z1 } is an arbitrary scale and p is nwnotone, then in order that the p-measurable function f be p-summable on A, it is necessary and sufficient that L(f, p, { z; }) have meaning and be finite.

Finally we add a remark about the approximation of improper p-integrals Let A = SA, and A, Q A,+, , let p(A,) be finite

Q 12, 7) by Lebeegue sums.

v

LEBESOTE AND RIEMANN SUMS

§131

183

and p(A) = t co . We choose S, > 0 with S, --' 0, such that S,ya(A,) -- 0. Let f be rp-integrable on A, let {zi'1 be a S, scale, and let L(f, gyp, {z(')), A,) be it Lebesgue sum formed on A,. Then according to 13.1.1:

0, S +1).

(-1

(A,) f f p = L(f, cv, {z(j')}, A,) + 8.8.o(A,)

Letting v -- . co we obtain from it because of 12.1.31 and 3.2.2: (A) f f dtp

(1.0)

= lim ,

L(f, ro, {z;')), A.).

BSsUoo APHY: H. LEBEsoun, J. RADON, M. FatcHEr, H. HAHN, as in §12, 1.

2. Riemann sums. Now by A = SA i let a decomposition Z of A into countably many disjoint ",p-measurable subsets be given. on A and xi s Ai , then we call'0

If f is a function V -defined

S(f, p, Z) _ E f(xi)io(Aj) a Rlemann sum associated with the decomposition Z (if this sum has meaning)" The value of such a R.iemann sum still depends on the choice of the elements

xi in A; . We call the decomposition Z a d-decomposition for f ifs' sup f(x) - inf f(x) < sCAi

:eAi

for all i. 13.2.1. In order that to every S > 0 there be a S-decomposition for f, it is necessary and sufficient that f be (p-measurable on A. o Here "disjoint" can be replaced by "disjoint ford" (14,2) without any essential change of the following. For let A - SA i be a decomposition Z of A into subsets disjoint for P; i

then we can relace Z by a decomposition Z" of A into disjoint subsets in the following manner: setAi -A,,Ai a.4=- A,. ,A!-Ai- (A,+A,+...+-Ai_,);now A = SA * and the At are disjoint. Moreover, A f A j for all i. i

All the following theorems of Nos. 2-4 remain valid without any modification. Only in the proofs of 13.2.1 ("necessity"), 18.2.2, 18.2.3 ("necessity"), 13.2.4, anti 13.4.1 [always at the beginning on the occasion of the definition of the auxiliary function f, or g j, the given decomposition of .4 into subsets disjoint for ,p has to be replaced by a corresponding decomposition of .4 into disjoint subsets. 40 In the following sum, if f(xi) - f oo and to(Ai) - 0, the corresponding term has again to be replaced by 0. Thus sets A i with o(A i ) - 0 do not give any contribution to this sum. e, If one wants to express also the set A on which the Riemann sum is formed, then one can write more explicitly: S(f, c, Z, A). It If sup f (z) - inf f (x) - + ac or - co, then the difference has to be replaced by 0; this x*Ai

yeA{

has to be observed further on.

[CHAP. IV

INTEGRATION

184

NECESSITY : Let 6, --- 0 and let A = SA,{ be a b.-decomposition for f.

If

now x,i a A,, and we define f, by: f, = f(x,,) on A,,(i = 1, 2, - ), then by 9.1.51 f, is 0-measurable and I f,(x) - f (x) I < 6, on A. Thus f = lim f, wherever f is defined on A, and hence by 10.2.4f is c-measurable on A. SUFFICIENCY: If f is gyp-measurable on A, then we set

f1,-2,...), [aasf(x) <(i +1)61 =A1 [f(:) _ -ao1 a Ate, [f(x) _ +-] = and we designate by A* the set of all x for which f is not defined. Then

A= is a s-decomposition for f. 13.2.2. If f is rp-integrable on A and `.iJ is a 8-decomposition for f, then for every Riemann sum S(f, (p, Z) :

(-1 s 8 s +1}.

(A) f f dip = S(f, ip, Z) + Basp(A)

Let A = SA; be a a-decomposition Z for f and let x{ e A{ ; we define a function i

g on A by: g = f(xi) on A; . Then I g(x) - f(x) I < 6, and hence by 12.3.52 g - f is c-summable on A. Therefore by 12.2.1 g = f + (g - f) is q;,-integrable f(x,),p(A;) = S(f, 4p, )) has meaning. on A, and thus the sum E (A;) f g dip = Moreover, by 12.2.1 S(f, p, Z) = (A) f f dw ing to12.2.311I(A) f

-

(A) f (g - f) d*p, where accord-

(g-f)dp1;560(A).

From 13.2.2 there results immediately: 13.2.21. If f is qp-integrable on A, if D, is a a,-decomposition for f, and of E, --* 0, then

lim S(f,,p, Z,) - (A) f f dip. Let J, be the decomposition n-A = SA,1. We say the sequence ((a),)) of

decompositions is adapted to the function f if for every 6 > 0 the sum B, of all

those A,, on which sup f(x) - inf f(z) z 6 satisfies the condition rp(B,) -- 0. 13.2.3. In order that there be a sequence of decomposition adapted to the function f, it is necessary and sufficient that f be 0-measurable on A.

NECESSITY: Let A = SA,; be a sequence of decompositions adapted to the i

function f and let x,j e A,c ; we define a function f, on A by: f, = f(x,,) on

185

LEBESGUE AND EUEMANN SUMS

$131

A,{(i = 1, 2,

). Then by 9.1.51 f, is cp-measurable. For every 8 > 0 let C,

be the set of all the x e A at which I f,(x) - f (x) ` z 8; then C, r- B, with cp(B,) -- 0. Thus by 10.3.1 the sequence ((f,)) is asymptotically convergent to f, and hence by 10.3.64 f is p-measurable on A. SvrFICIEI3cY: This is con-

tained in 13.2.1. 13.2.4. If f is a bounded48 y>-measurable function on A and ((v,)) is a sequence of decompositions of A adapted to the function f, then for all Riemann sums associated with the decomposition Sl), we have (A) f f dcp.

S(f, o, Z,)

Let :D, be the decomposition A = SA,; ; then S(f, (p, Z,) _

f(x,,)w(A.c)

We define f, by f, = f(x,;) on A,c(i = 1, 2,

); then by

where x,; a A,c .

12.1.6, 9.1.51, and 12.3.52 S(f, cp, il,) = (A) f and 12.2.11 (A) f f dip = (A) f f, dip + (A) f

f, dcp.

Thus, since by 12.3.52

(f - f,) 4, we have only to show:'

(A) f (f - f,) dcp -' 0. Let 8 > 0 be arbitrarily given. We designate by A.' the bum of those A,c on which sup f(x) - inf f(x) < 8, and by A'' the sum of soA,t

m.A,{

the other A,, .

Then (A) f

(f - f.) dip =

(A,) f (f - f,) d,, + (A:') f (f - f.) dip.

On A; j f (x) - f,(x) I < 6, and hence by 12.2.311 1 (A;) f (f - f,) dip 1 5 60(A;) 5 6p(A) for all v. Since ((Z,)) is adapted to f, so(A;) --*0 and, since f is bounded, there is a k, such that I f(x) - f,(x) 1 S k on A for all P. Thus

1(A;') f (f - f,) dcp 16 ko(A.), and hence (A") f all v.

(f - f,) dcp - 0; therefore j (A,)f

So we have I (A) f

(f - f,) dcp I < 8 for almost

(f - f.) dcp I < 6(1 + cp(A)) for almost all v, and hence

(A) f (f - f,) dcp -' 0, as contended. u This condition is essential. Example in R, with ev - pi : Let A - 10, 1 ], f - 2- for

x 1 and f - 0 otherwise; let Z, (v

1, 2,

2,

) designate the sequence of decompositions,

( a

adapted to f, A m SA.c where A,c = I

then S(f, µ , t',) k

23` . 1

.

1 ,

2r]

(i = 1, 2,

, 2'). We choose x,i - Zly

2', and hence S(f, s, , ca.) -+ +to ; but (A) J f dµ, - 0.

;

(CRAP. Iv

INTEGRATION

186

3. Distinguished sequences of decomposition. Now we assume A to be a point set of a metric space E. Then we say: the decomposition A = SAi has the norm p if for all A; the diameters (12, 3) d(A1) S p. If p, -- 0 and Sl), is a decomposition of A into yo,-measurable sets of the norm p, , then ((`+E),)) is called a distinguished sequence of decomposition of A. 13.3.1. In order that there be a distinguished sequence of decompositions of A, it is necessary and, if the open sets are cp-measurable, also sufficient that A be separable

(§2, 2). NECESSITY: Let

), be the decomposition A = SA,i and let ((),)) be a i

distinguished sequence of decompositions. Then if x,i e A,i, the countable set of points x,i (v, i = 1, 2, ) is dense in A." SUFFICIENCY: To every a e A 1 we form the set S. of all x t A with xa < ; then every Ss has a diameter 2 . P

By 2.2.2 the system of the S. contains countably many S(i) , S(:)

,

. , SO)

which cover A. We set

Al = S(i

,

A2 = S(2) - So) , ... , At = S(i) - (S(,) -f ... + S(i_,)), .. .

then the A, are disjoint, A = SAi , and the diameter of Ai is not greater than i

v

thus the decomposition A = SAi has the norm 2 . Repeating this for is = s i 1, 2, we obtain a distinguished sequence of decompositions of A. 13.3.2. If f is bounded" and 9-conhinuous ($11, 2)41 on A and if the open sets are ip-measurable, then for every distinguished sequence ((Z,)) of decompositions and alt associated Riemann sums (3)

lim S(f,

to,

),) _ (A) f f drp.

y 11.2.3 f is fp-measurable. If (3) is not valid for every distinguished seBy quence of decomposition, then by 13.2.4 there is a distinguished sequence ((Z,)) of decompositions which is not adapted to f. If now Z. is the decomposition A = SA,i , then there must be a b > 0, such that, provided B, means the sum of those A,i on which sup f (x) -- inf f(x) std,4

E, p(B,) does not converge

to 0. But then lim p(B,) > 0, and hence by 3.5.11(p(Lim B,) > 0 also. Thus, since f cannot be continuous in any point of Lim B, , f is not u-continuous,

A converse of this theorem is contained in: 13.3.21. If A is separable, if the open sets are sp-measurable, and if f is bounded "Cf. footnote 5, p. 7. 4' Footnote 43, p. 185, shows that this condition is essential. *Theorem 13.3.21 shows that this condition is essential.

LEBESGUE AND RIEMANN SUMS

§13J

187

on A, then in order that for etc/ distinguished sequence ((Z.)) of decompositions there exists lim S(f, gyp, i),), however the Riemann sum S(f, gyp, Z.) associated with

), may be chosen, it is necessary that f be continuous. Since f is bounded, there is a p, such that 1 f y 5 p on A. Suppose f not to be go-continuous on A; that is, for the set B of all points of discontinuity of f we have s(B) > 0. According to 3.4.71 we make the decomposition: B = B+ + B-, such that go (B+) = 0 and p+(B-) = 0; then O(B) = ip+(B+) + v7-(B-), and hence either o+(B+) > 0 or 97(B-) > 0. We assume, say, so+(B+) > 0. Let

I and f be the upper and lower limit function of f, respectively.' We set

B+ [J(t) - f (l) >

mJ

= Bm ; then B= Lim B,,. and, according to 11.2.5 and on

Thus since p+(B+) > 0, by 3.2.2 we have also rp+(Bm) > 0 for almost all m. We choose m such that P+(B.) > 0, and let p > 0 be arbitrarily given. By 13.3.1 there is a decomposition B. = SCk 9.2.2, Bm(m = 1, 2,

of the norm

2.

) is yo-measurable.

Since/go (B,,,) _ E p+(Ch), there is a q, such that

o+(Ci) +---+ So+(C,,) >

4p+(B,).

, 9); we designate by S; the set of all x e A with xbi < a. Let bi a Ci (i = 1, 2, For a -- 0 we obtai n , q) are disjoint. For a sufficiently small a the S; (i = 1, 2, ,p (Si) - go( (bi 1) by 3.2.21 (since p(A) is finite) ; and from bit B+ and go (B+) = 0 it follows that go ((bi 1) = 0. Hence for or -- 0 we have go (Si) -+0(i = 1, 2, . , 9).

+ w (SQ) < 8pm c+(Bm). We set We choose a so small that go(S,) + U= Si + S2 + . + SQ and Ci = (Ci - U) + Si (i = 1, 2, . , q). Then , CQ are disjoint and, since d(S;) 5 2a, by §2 (3.21) d(Ci) C; , Cs ,

d(Ci) + 2a; thus

(since d(Ci)

S 2) we can choose a so small that

Since Si Q C, and bi a Ci(cBm), there are two , q). d(Ci) < p(i = 1, 2, points x' and x*i * in C;, such that f(x') - f(x'*) > m. As a result of

C,+C,+...+C.aCI+.CQ+...+Ca +* This set B certainly is , -measurable. For B is an F,-set in A (for instance, cf. H. Hahn [11, p. 199, or H. Hahn [2], theorem 26.4.1) and thus *-measurable, since by assumption the open sets are .p-measurable. 18 The upper and lower limit functions I and f are defined in the following way: For a e A consider the set of all those numbers y e A, to which there is a sequence ((a.)) in A with a, -+ a and f (a,) --' y. One sees immediately that the set of all those numbers y is closed. Now we define f(a) a max y and f(a) - min y. Using the sequence ((a,)) with a. - a we see that f (a) is one of the numbers y; hence: f (a) f(a) I (a). Thus f is continuous at a if and only if 7(a) - f(a).-One can easily show that f is continuous above and f is continuous below on A (according to R. Baire, Paris These 1899 $ Annali di mat. (3)3(1899), p. 6; cf. also H. Hahn [2], theorem 36.3.1).

188

(CHAP. IV

INTEGP.&TION

we obtain +{Ci) + qd+(Cti) + ... + c+(CC)

'+(C) + v+(Cs) + ... + p+(C,,) >

+(B,.)

Since C{ - Si Q B+, we f(xi*)) V+(Ci) > 2m have 0-(&j - Si) = 0, thus (C{) = yv (S;), and hence (Ci) + (P (CC) + ... + hence

cp+(B.).

q

c (Sr) + ,-(S,) +

+ (P (Sa) < 8pm p+(B.). Therefore, since

f(xi) - f(x**) we obtain E(f (x*)

- f (x{ *)) .g

2p,

(C;) < 4m m+(B,.1), and hence

(f(xi) - f(xi*))'SO(Ci) > 4mp+(Bw) By 13.3.1 there is a decomposition of A - (C; +

+ Cq) of the norm p, say:

+ &,) = SMk ; then A = Ci +

+ C, + SMk is a decom-

+

A

k

k

position aU of A of the norm p. If moreover xk erMk , then

S*(f, 0, Z) = } f(x*i)v(Ci) + E kf(xk)Sp(Mj) and S**(f, 0' Z) = Ef(xi*),p(C',) + Ef(xk),p(M,,) are two Riemann sums associated with the decomposition Z and {3.1)

S*(f, gyp, Z) - S**(f, gyp, )) > 4m rp+(Bm)

Thus we have the result: to every p > 0 there is a decomposition Z of the norm p

with which two Riemann sums satisfying the inequality (3.1) are associated.

Heace if we choose p = 1,

,

,

,

then we obtain a distinguished

sequence ((Z,)) of decompositions, such r that with every Z), two Riemann sums are associated which satisfy the inequality S*(f, ,, Z,) - S**(f, rp, ),) > But then, by means of the S* and S**, one can form a sequence 4 `p+(B") (> 0). ((S(f,,p. Z,))) possessing no limit. Thus if f is not gyp-continuous, (3) cannot be valid for all distinguished sequences

of decompositions and all associated Riemann sums. But we still have: 13.3.22. If rp is content-like (§7, 4), if f is Sp-summable on A, and if ((a),)) is a distinguished sequence of decomposition, then there are associated Riemann sums satisfying (3).

First we show: if e > 0 is arbitrarily given, then there is a R,iemann sum

LEBESQUE AND RIEMANN SUMS

§131

189

S(f,.p, T),) associated with Z., such that I S(f, ip,

(3.2)

`f1,) - (A) f f d*p I < e for almost all

P.

By 12.3.7, to every 6 > 0 there is an n > 0, such that ya(A - B) < n implies

(A - B) f I f I do < 5.

(3.3)

By 12.3.11 we have [f(t) = +oo] + [f(t) _ - oo] [I f(.f) I

A; thus, setting

5 k] = Ak ,

we obtain: SAk =,, A, and hence p(Ak) -* O(A); therefore k can be chosen, such k

that p(A - Ak) < 2 . Now by 11.3.1 there is a closed subset B of Ak, such that O(Ak - B) < 2 and Blf is continuous.

Then f is continuous and bounded

on B and rp(A - B) < n; so (3.3) is valid. Now let Z, be the decomposition A = SA,i . We designate by I' the set of all those i for which A,iB $ A and by i

P the set of the other i; furthermore we set A, = S A,i and A,' = S A,i . ia' i.l"

Since ((I,)) is a distinguished sequence of decompositions, there is a sequence of positive numbers ((p,)), such that p, is the norm of r, and p, -- 0. Let B, be the set of all x e A with xB S p, ; then A', C B.. But since DB, = B, we have .p(B,) - + O(B), and hence q(A' , - B) -+ 0. If i e I', that is, A, B * A, then there

is an a,i a A,iB; and since f is bounded on B, there is a q, such that I f(a,r) 15 q for all i e F. Now we set: S'(f, rp, Z,) = E f(a,i)(p(A,i) Then

S'(f, ip, Z,) _

f(a,,)m(A,,B) + E f(a.i),p(A,i - B).

Since S (A,, - B) = A; - B, p(Ar' - B) --i 0, and I f(a,i) I q for i e 1', we Oil have now: E f(a,,)co(A,i - B) --p 0. Moreover, since f is continuous and i.t'

bounded on B, we have by 13.3.2: E f (a,i)ip(A,,B) --i (B) f f d p. Therefore 01'

we obtain: S'(f, (p, ),) --+ (B) f f dp, and hence by (3.3) and 12.2.31: (3.4)

I S'(f, po, Z,) - (A) f f dw i < a for almost all P.

Now let i e P. We choose an a.i a A,i , such that I f(ai) 15 inf I f(x) I + n; m.A,t then by 12.1.32 1 f(a,r),p(A,i) 1

(A,i) f (I f I + n) dye. Thus if we set

INTEGRATION

190

ECHAP. IV

S"(f, sp, Z.) = Eiez, f(a.;)c(A.:), then I S"(f, cp, Z,) I is (A')

f I f I dip + ##(A' ).

Hence from A; S A - B,

.p(A - B) < n, and (3.3) it follows that I S"(f, yo, 2),) I < 6 +

.

But this

together with (3.4) furnishes (3.2).-Now applying (3.2) for e - (k = 1, 2,

we obtain 1tiemann sums Sk(f, ,, Z,), such that

)

k

Sk(f, v, Z,) - (A) f f dip < k I

vk can be assumed.

;,,%,

If we now set S(f, rp,

),) _

Sk(f, q p, W for Y,. s v < vk+1 , then there results that S(f, p, Z,) -4 (A) f/d10, and 13.3.22 is proved.

4. Darboux sums. Now for the sake of simplicity we assume the determining

function p to be monotone increasing; the function f is not supposed to be ,p-measurable.

Let Z be the decomposition A = SA; ; we set:

g; = sup f (x) and h{ = inf f (x) zcAj

z.Ai

and form the sums: (4)

.a R(f, p, Z) _ E g. (A{) and 5'(f, ,p, t) _ E h,Yv(A;)

If they have meaning, then we call them the Darboux sums associated with the decomposition Z, and in particular we call the first one the upper sum and the second one the lower sum. Because of 12.2.5 and §12(5) we have (always if the Darboux sums under consideration have meaning) : (4.1)

S(f, jp, )) s_ (A) f f dip s_ (A) f f 4 s SU, -P, Z)

13.4.1. If (A) f

f dp > - w (or (A) f f dip < + co), then all sums

Z)

(or S(f, ,, Z) have meaning.

By 12.5.2 there is a 9-integrable function h*, such that f Z o h* and (A) f f dso = (A) f h* d,. We define a function g on A by g = gc on A; . As a result of f

h* and g ? f we have g Z, h*; hence by 12.2.33 g is ,p-integrable.

4' If one wants to express also the set A on which these sums are formed, then one can write more explicitly: S(f, w, Z, A) and S(f, c, Z, A).

LEBESGUE AND RIEMANN SUMS

§131

Thus since (A)

191

f g dip = E (Ai) f g d,p = F gv(Ai)

gyp,

i

)), the sum

ft v, Z) has meaning.

f

13.4.2. If (A) f dip > - ao (or (A) f f dip < + ao), then there is a sequence ((),)) of decomposition, such that S(J, w, Z.) -+ (A) f f dip (or

t,) --), (A) f f gyp).

According to 12.5.2 and footnote 32, p. 171, there is a function g* ?. f, such

that (A) f f 4 = (A) f g* dip. Then since (A) f f dip > - ao, we have A[g*(&) = - oo1 -, A. Thus we can assume g* > - ao. Now we set

A,i=A[P 5g*()

<+1](i-0,f1,f2,...)and

1.

v

Let Z, be the decomposition A = SA,i + A.. Then by (4) and (4.1) : a

(A)f f dips S(f, 9, D,)

S(g*,,p, Z,)

(A) f g* d,p +

! c(A) _ (A)

l

f d,p + y m(A) .

But from this it follows that S(f, V, Z,) -p (A)f f d,p. From 13.4.2 and (4.1) there results immediately:

f

13.4.21. If (A) f dyo > - ao (or (A) f f 4 < + so), them

(A) f f d p = of S(f, p,

))

(or (A) f f dco = sup SU, V, Z) ),

where Z runs through all possible decomposition of A. And from this we obtain because of §12 (5.1): 13.4.22. If f is rp-summable on A, then

(A) f f dco = of S(f, p, )) = up SU, cv, Z) Supplementing 13.3.2 we have: 13.4.3. If f is boundedi0 and continuous on A and if the open sets are -measurable, then for every distinguished sequence ((Z,)) of doeompositions

S(f,,

,

V -, (A) f f 4 and

S(f, w, `),) -- (A) f f div.

If Z, is the decomposition A = SA,i and g,, = sup f(x), then there is an z,A,i

10 The example of footnote 43, p. 186, shows that this condition is essential.

[CHAP. IV

INTEGRATION

192

a.; a Apt , such that g,c -

< f(a,:) 5 g,i . Thus if we set S(f,

f(a,j)cp(A,;), we obtain I S(f, ;p, Z,) - S(f, c, Z,) I < Vcp(A). By 13.3.2

f

8(f, p, ).) -> (A) f f dip, and hence S(f, gyp, st),) -+ (A) f drp also. Theorem 13.4.3 is a particular case of : 13.4.31. If f i8 boundeaa' and continuous above (or below) (§11, 2) on A and if the open sets are gyp-measurable, then for every distinguished sequence ((a),)) of decomposition

S(f, cv, Z,) -(A)f f dyo (or 8(f, to, ;D,) -, (A) f f dco) Let B be the set of all x e A at which f is continuous above on A; then f is continuous above on $ and by assumption A -- B =, A. According to 111, 3 (p. 146) there is a monotone decreasing sequence ((fk)) of continuous functions on B, such that ft -a f on B; since f is bounded, the fl, can be assumed to be also bounded. Let SD, be the decomposition A = SA,i ; we set g.i = sup f (x) i

and gk,i =

sup

,.A,,g

fk(x). Let e > 0 be arbitrarily given; we designate by Ck,

the sum of those *"'i'sA,i for which g,i > gk,i + e, and show: if a e B and k is given, then a e Ck, is possible only for finitely many P. If at B, then f is continuous above on A at a, and hence, by footnote 48, p. 144, there is a neighborhood U. of a, such that f(x) < f(a) + a for all x e U.A. Since ((`a).)) is a distinguished sequence of decompositions, there is a v*, such that, for iv is s' , a e A.i implies A.; a U4A; but then, for v v", a e A,.4 implies also g,i 5 f(a) + e Since ((fk))

is monotone decreasing, fk(a) k f(a), and hence a e A,i implies gk,; :' f(a). v", a e A,i implies g,i 5 gk,i + e; therefore a e Ck, is possible only for v < v*, that is, only for finitely many v, as contended. So we have shown: Lira Ch. G A - B. Now from A - B =, A it follows that Uin- Ck, _, A also, Thus, for v

and hence by 3.5.11,p(Ck,) --+ D.

f

Because of 11.1.1, 12.3.52, 12.1.8, and 12.1.53

we have (since B -,A): (A) fk dso -* (A)

f f dip.

Thus there is a k*, such

f

that (A) l fk. dsp < (A) f dco -{- e. Since fk. is continuous on B and B =, A,

we have by 13.4.3 and 12.1.53: S(fk.,

is,

f), , B) --> (A) f fk. dSp;6° thus

(fk., co, Z. , B) < (A) f f dp + a for almost all P. Setting sup f (x) = g we have, ..A

according to the definition of Ck, :

S(f, p, ).) 5 S(fk., N, D. , B) + &p(A) + gm(Cn.) I" On B the decomposition 1), means: B - SA,4B,

i

LEBEBOUR AND RIEMANN SUMS

§131

Hence as a result of .p(Ck.,) - 0 we obtain 8(f, ,p, SD,) < (A)

198

f f do + e(2 -+» v (A))

for almost all v and, since certainly 3(f, y,, Z,) g (A) f f dip, it follows that S(f, w, m,) --+ (A) f f dso.

A converse of this theorem is contained in: 13.4.32. If A is separable, if f is bounded and .p-measurable on A, and if the open sets are yo-measurable, then in order that for every distinguished sequence ((a,))

of decomposition there exist lim S(f, .p, ),) (or lim S(f, .p, D,)), it is necessary and sufficient that f be .p-continuous above (or below) on A.

NEcEssrrr: Suppose f is not {p-continuous above on A. If B is the set of all x e A at which f is not continuous above, then B v A. Let 1 be the upper limit function of f (cf. footnote 48, p. 187), which by 11.2.5 is .p-measurable on A ;

then B = A f j'(f) - f(s) > 0]. Thus if we set B. =A CJ(t) - A -f) >

, we have

B = Lim B., and hence by 3.2.2 B 0, A implies B. 0, A for almost mI all m. Let m be chosen such that .p(B.) > 0 and let p > 0 be arbitrarily given. To every a e B. we form the set U. of all points x e B. with xa < p; every U. has a diameter not greater than 2p. many,

Vi = U; -- (U1 +

+ U"),

By 2.2.2, among the U. there are countably

We set V1=U1,V,

; then the V, are disjoint, B. = 8Vi ,

every Vi has a diameter not greater than 2p, and every Vi (since it either is open in B or is the difference of two sets open in Be,) is an F,-set in B,.. From

B,;, = SVi it follows that E .p(Vi) k .,(B.); thus there is a q, such that i

i

.p(Vi) + + 9,(V,) k j.p(B.). Since Vi is an F, in B. , V. contains a set Fi closed in B., such that .p(F1) Jp(Vi); then p(Fi) + ... + v(F,) z Jv(B,.).

Since Fi a Vi and the Vi are disjoint, the Fi are also disjoint, and hence6' there are disjoint sets Gi Q F. (i = 1, 2, , q) which are open in A; since d(Fi)

2p, we can assume d(G.) S 3p. By 13.3.1 there is a decomposition of

A -- (G, + Then A = G1 +

+ Go) of the norm 3p, say A - (Gi + ... + Gq) = SAk k

.

+ G,, + SAk is a decomposition of A of the norm 3p; we k

designate it by Z. Moreover, we designate by )' the decomposition

A=F1+..._}..F.+(G1-Fl)+...+(Ge-FQ)+SAk; k

V has also the norm 3p. Then Xf,.p, `.+l)) - S(f, .p, V') 51 For instance, cf. H. Hahn [2], theorems 141.]1 and 143.4.

CHAP. IV

INTEGRATION

194

L (sup f(x) .,(G) - sup f(x) s(Fj) zeQj

j-1

sup zeoj-11j

X011;

(sup f(x) -- sup f(x)) ,p(F'j) z011j ,-1 zeoj

g.

+L (sup f(x) i-1 zeQj

f(x)',p(Gg - Fj)) =

sup f(x)) e(Gj - F,).

zeQj-11j

But sup f(x) S sup f(x) and" sup f(x) = sup j(x); hence z.Qj

zeoj-11j

2601

Sao 1

f(x)) se(F) (sup j(x) - sup z.11j

,S(f, c', Z) - &f, ip, Z')

j-1 zeoj

Furthermore, since F; C; B. , on F; we have j > f + - , and thus sup j(x) sup j(x) zf11i

sup f(x) + ze11j

Therefore we have:

in

(f,

13 VI So,

Now we choose p

cp, z1)

M i-1

f (F1) `-

4m

v(Bm)

then we obtain two distinguished

2

sequences ((:,)) and ((T;)) of decompositions, such that, for every

Sf,. '',

&J,

ie,

Z.) z

m

p(Bm).

Then Z1 , Zi , Z2, Zs,

.

p,

is

also a distinguished sequence of decompositions and, since p(Bm) > 0, the sequence &f, ,p, Z1), S(f,,p, Ti'), S(, {p, X/2), S(, gyp, Z2), FICIENCY: This is contained in 13.4.31. Bi

has no limit. Sue-

iomtArwr to Nos. 2-4: B. RIEMANN, 100. Cit. (cf. §12, 1); G. DARBOIIX, Ann. ]to.

Norm. (2) 4 ;7875), p. 64, and also the further bibliography in Encykiop. d. Math. Wiss. II C9b (P. i\IONTEL-A. ROSENTHAL), p. 1037; H. LEBEBGUE, Paris These 1902, p. 24 Annali di mat. (3) 7 (1902), p. 254; [1], p. 29; [2], p. 29, 135; Ann. Fac. so. Toulouse (8) 1 (1909), p. 30; G. VITAL1, Rendic. let. Lombardo (2) 37 (1904), p. 72; W. H. YOUNG, Proc. Royal Soc. London 73 (1904), p. 445; Philos. Trans. Royal Soc. London A 204 (1905), p. 221; J. PIERPONT, Trans. Amer. Math. Soc. 6 (1905), p. 423; [1], vol. I, p. 519; vol. II, p. 1, 371; J. RADON, Sitzungsberichte Akad. Wiss. Wien 122 (1913), p. 1322; H. HAHN, Sitzungsberichte Akad. Wiss. Wien 123 (1914), p. 713; M. Fa cHET, Bull. Soc. math. France 43 (1915), p. 253; C. CARATHAOLORY 111, p. 453; E. TOHNIER [11, p. 53; H. NA%ANO, Proc. Phys.-Math.

Soc. Japan (3) 25 (1943), p. 279.

§14. Mean value theorems and inequalities 1. The first mean value theorem. If u, and v; (i = 1, 2, real numbers satisfying the inequalities: 82 For f(x)

3(z) implies sup f(x) zeQj

value 3(z) there is a sequence x, sup 3(x) is impossible. sec,

, k) are (finite)

sup J(z). But since (by definition of 7) to every zeoj

x in G'j with f(x,) -. J(z), the inequality sup f(x) < zeQj

195

MEAN VALUE THEOREMS AND INEQUALITIES

§141

vi i' 0

p;S u, Aq,

(i = 1, 2,

, k; p, q finite),

then

pv; i-1

k

k

(1)

L.

;-1

u,v;

qEv;. C-1

f

Now we want to obtain an inequality for integrals (A) fg dp which is analo-

gous to this inequality (1) for the sum

i-1

uivi. First we show:

14.1.1. If f is (p-aummable on A and if g is bounded" and 9-measurable on A, then f g is also r -8ummable on A.

Since f is so-summable on A, by 12.3.11 A [f (t) = + oo j =, A and A (f (t) _ - co = o A; thus since g is finite, f g is on A and hence by By assumption there is a finite c, such that I g I S c. By 12.3.4 of is o-summable; thus since I fg I S I cf I (wherever fg is defined), by 12.3.51 fg is also 9-summable. 9.2.3 also gyp-measurable.

Now we prove the inequality for integrals analogous to the inequality (1): 14.1.2. If f is and non-negative on A, if (p is monotone increasing, and if the 0-measurable function g satisfies the inequality p on A, then

f

(1.1)

p. (A) f dip

g

q (p, q finite)

(A)f fg drp S q- (A) f f dip.

fg 5 of

This follows from 14.1.1, 12.3.11, 12.1.7, and 12.3.4, since pf (wherever these terms are defined).

Herein, in particular, we can set p = inf g(x) and q = sup g(x). Then ins.A equality (1.1) says that there is a "mean value" m lying between inf g(x) and 21A

su

sup g(x), such that ¢.A

(A) f fg dp = m- (A) f f dcp.

(1.2)

Therefore 14.1.2 is also called the first mean value theorem for integrals.

2. The second mean value theorem. Again let u; and v; (i = 1, 2, (finite) real numbers; we set s1 = u1 , s, = u1+u1, . . , 8k = ul + us +

, k) be

+ uk

and assume: (2)

p$- sr;5

Since u1 = si , u, = 82 - si , ... , Uk = 8k - 8k-1 , we have: "Or (because of 14.1.58): v-bounded (i9, 1).-That this assumption is essential, follows from 14.4.31.

[CHAP. Iv

INTEGRATION

196 k_

21,4>J = Slvi + (as - 83)V2 + ... + (ak - $k-1)vk E i-1 = s1(v1 - v2) + .

+ sk-1(vk-i - Vk) + skvk

.

If we apply to this the inequality (1), in which u; has to be replaced by sc and vk-i , vk have to be replaced by vi - v2 ,

vi

, vk-1 - vk , vk , then we obtain:

k

pvi 5vi 5gvi.

(2.1)

Now from (2.1) we derive an analogous inequality for integrals (A) f fg dyo. In the following discussion p, q, r, s are to mean finite numbers. 14.2.1. Let f be rp-measurable and 0 S f S son A, l.-t g be rp-summable on A,

y])f g dip 5 q for 0 5 y S s; then

and let p S (A[f(i-)

p8 S (A) f fg dip 5 qs.

(2.2)

We set k k A[v{

t

8 = v; (i = 1, 2,

, k); furthermore, A[f(() k vi] = Ai

f ( i ) < v;-a) = A;(i = 2, 3, ... , k), and (A{) f gd* = u{. Then since

on As I f -- vi 15

, we have by 14.1.1, 12.3.4, 12.2.11, 12.2.31, and 12.1.7

(A.) f f 9 d i - avi I = I (A:) f (f - vs)g d i p I s

(A.) f 19 I dp,

k

and thus, since A = S A; : (l-1

(A) f fg d v -

u x.

Herein, by 12.3.21, (A) f I g I dq' is finite.

-k8-

pk

6

(A) f 19 I do.

Hence by (2.1) we obtain:

(A) f 19145 (A) f fgdcp s qk

k

'a+""

(A) f IgI do,

ao. and there results (2.2) by letting k The assumption f ? 0 in 14.2.1 can be disposed of as follows: 14.2.11. Let f be p-measurable and r 5 f son A, let g be q,-summable on A,

and let p' S_ r (A[ f(t) < y]) f g dW + s (A[f(x) z y)) f g dv 5 q' for r S y then

(2.3)

p' S (A) f f9 dyv

q'.

a;

MEAN VALUE THEOREMS AND INEQUALJTIES

§14]

197

We apply 14.2.1 to J (f - r)g d4p; if we set (2.31)

p = rly; inf

(A[f(x) ? y)) f 9 ct
and q - sup (A[f(E) Z y]) f 9 dp, riy4&

then by 12.3.12 p and q are finite and we obtain:

f

p(s -r) S (A) (f -

dp ;5 q(e- r).

Thus: (2.32)

Ps + 7( 1) f g d;p _- p) 5 (A) f fn dp f- qs + r((A) f 14

-q

By (2.31) there is a sequence ((yr)) of numbers in [r, el, such that y,J) f Y. dcp + r[ (,4) f

gd'p - (A[f(l)

y.]) f 9',' --> ps + :((A) f g dip - p

),

and hence

f

f

s. (A[f(I) k y,]) g dp + r- (A[f(t) < y,]) f g dip -9 Ps + rl (A) g dp -- p1. Thus, by definition of p', we have p'

f

ps + r((A) g d - - p). and analogously

48 + r((A) f g dcp - q). So from (2.32) there rerulte (2.3). If in particular A is an interval fa, b] of R, , if (p is the onedimensional measure pi, and if we assume f(x) to b: monotone, then we obtain from 14.2.1 and 14.2.11 the two following theorems kflown together as the second mean value theorem for integrals: 14.2.2. If f is finite, monotone decreasing aid non-negative in [a, b] and if g is q,

jursummable in [a, b], then there is a C in [a, b], such that

f

a

b

f(x)g(x) dx = f(a)

f

E

g(x) dx.

6

Since 0 5 f(i) 5 f(a), in 14.2.1 we can choose s - f(a).

Since here every

set A[ f(t) z y] is an interval [a, x] or [a, x), we can choose p = inf

and q = 5sa6 sup af

:

a$x;S b

f g(t) dt a

g(t) dt; by 12.3.12 p and q are finite. Because of (2.2) there

is a mean value m between p and q, such that

e

f f(x)g(x) dx = mf(a). a

But

ICHAP. IV

INTEGRATION

198

since by 12.3.7 f g(t) dt is a continuous function of x, there is a k in [a, b], such t

0

that m = f g(t) dt. a

In the same way one obtains from 14.2.11: 14.2.21. If f is finite and monotone and if g is µ1-summable in [a, b], then there is a >: in [a, b], such that a

b f(x)g(x) dx = f(a) I g(x) dx + - f(b) f g(x) dx. J BIBLIOGRAPHY to Nos. I and 2: The second mean value theorem in the form 14.9.2 is due to 0. BONNET, Journ. de math. 14 (1849), p. 249; Mom. Acad. Berg. 23 (1850), p. 8, while in

the form 14.2.21 it was first published by P. DU Bois-REYMOND, Journ. f. Math. 89 (1868), p. 83 (cf. also 79 (1875), p. 42). Theorem 14.2.11 is due to H. I.sBnsavm, Ann. Ee. Norm. (3) 27 (1910), p. 443. Moreover, of. the bibliography to Nos. I and 2 given by A. PatNOSHEIM, Sitsungsberichte Bayer. Akad. Wiss. 30 (1900), p. 209, and in Encyklop. d.

Math. Wise. II A2 (A. Voss), No. 34 and 35; II C9b (P.

RosKNTaAL), No. 84.

3. The Holder inequality. Let x = (x1 , x2 , be a point of R. . For , any real number p we set: px = (pxt , px2 , , px ). if x' = (xi , x2 , , x*) is a second point of R. , then we set x + x' _ (.Ti + xi , x2 + xx , , x, + x;.) and x - x' = (x1 - xi , x2 - xs , , x - x;,). In R. we introduce a "convex -

metric", first attaching to every point x a finite distance D(x) from the origin , 0) with the following properties: (0, 0, 0, and in particular D(x) = 0 if and only if x = (0, 0, 1.) D(x)

, 0). 2.) D(px) = I p I D(x) 3.) D(x + x') 5 D(x) + D(x'). Then we define the distance xx' of any two points by xx' = D(x - x'). In this way a metric space (§2, 3) is made of R w If in particular we choose D(x) =

1/x; + x; + .. + x2 , then we obtain the usual Euclidean metric (12, 3). If x' and x" are two different points of R. , then the set of the points x' + p(x" - x') with 0 5 p S 1 is called the line segment [x', x"). A set M of R. is called convex if with any two points x' e M and x" e M always all points of the line segment [x', x"] belong also to M. The set M of all points x with D(x) 5 1 is convex. For let x' e M and X" E M; then the line segment [x', x") consists of the points

x = px" -h (1 - p)x' (0 S p 5 1) and D(s) S PD(x") + (1 - p)D(x'), and hence from D(x) 5 1 and D(x") 5 1 it follows that D(x) 5 1 also; that is, x e M. 11 That the condition 1., of 12, 3 is satisfied, is evident. The condition 2., of 12, 3 is

also satisfied, since D(a - c) = D((a - b) + (b - c)) S D(a - b) + D(c - b).

MEAN VALUE THEOREMS AND INEQUALITIES

§14]

Now we designate by u = (u1 , us ,

199

the linear form

, U1x1+UXI+...

and in the set L of the linear forms we introduce a convex metric A(u) by (3)

A(u) = sup (uixl + u2x2 + ... + unx.) = suP I u1x1 + u3x2 + ... + D W-1

D (z)-1

1,

where x = (x1 , x2 , , x e R with D(x) = 1. One sees immediately that A(u), analogously to D(x), has the following properties: 1.) A(u) z 0, and in particular A(u) = 0 if and only if u = (0, 0, , 0). 2.) A(pu) = I p I A(u)

3.) A(u + u') 5 A(u) + A(u').

+

Herein pu means the linear form p(uixl +

linear form (ui + ui)xi +

u + u' means the

+ (u. + u' )x,, . We call A(u) the metric

polar to D(x).

=D(x),then (3)can

(Ixt1,Ix2l

If we set 1xl be replaced" by:

A(u) = sup (I u1 x1 I + Iu2 xs I + ... + I u* x* I)

(3.1)

D (z)-1

If x is an arbitrary point of R., different from (0, 0, . , 0), and if we set x* = Xx) , then by 2.) D(x*) = 1, thus i ulxi + + uxn 1 5 A(u), and hence

B 7(X-)

+ ux I S A(u). So we obtain the inequality, valid

I u1x1 +

for all x e R. (also for x = (0, 0, (3.2)

0)), 1 uu'x1 + uix2 + . .. + ,

1 5 A(u) . D(x),

We now give some examples.

+ I x 1. We contend: the polar

Example 1: D(x) = I x1 I + I x2 I + metric is given by (3.3)

=max(Iuil,Iuel,...,Iun1).

A(u)

Since certainly I u1x1 + u2x2 + ... +

I S 1 uix1 I + 1 U2 x2 1 + ... + I U. X. I

(3.31)

we have A(u) 5 max max (I u1 I , I u s ) ,

x;

,

(I u1

I u2

I u 1).

,

On the other hand let

I u 1 ) = I u; I > 0 and let x be the point 21 - 0 (j 0 i),

= sgn u; 5B, then D(z) = 1 and

u If we set x,' = (u,x,) (cf. 56) for Y = 1, 2, , n, then u.x: - I u.x, I. Thus ... + u x 5 I u,z, j + ... + I u,z, and, since - (v,xi + ... + I x I and D(x) = 1, we have also D(x') = 1. Therefore, indeed, we obtain (3.1). x' j 66 sgn a (i.e., "signum a") is defined by sgn a = 1, 0, -1 if a > 0, - 0, <0, respectively. I

[CHAP. rv

INTEGRATION

200

u121+ .. +UH2. = Iu;I = max and hence a(u) max (I ul 1, I us I, Here inequality (3.2) reduces to (3.31). Example 2: D(x) = max (I xl 1, 1 x: I,

,

I u. I) also. Thus (3.3) is proved.,

i x. 1).

We contend:

0(u)=lull+Iu91+...+Iu.I.

(3.32)

From I ujx1 +

...+u.x.I < (Iu,I

+... + 1u.1)-max (I x11

,..., IX.I)

it follows that A(u) < I u, I -}- . + I u,, 1. On the other hand, if , n), then + I u. > 0 and we set x; = sgn u4 (i - 1, 2, I it, 1 + + u. I, and hence A(u) D(x) = 1 and u1x1 + - - + u.x, = j u1 I + .

+ I u.

u1 I +

also.

Thus (3.32) is proved.

Example 3:D(x) =Ix1-xil+lx:- x8+...+Ix.-.1-XNI+Ia1

We contend: if we set ul = s1 , u1 + u 2 = 3 2 , then the polar metric is given by

+ u = sn,

A(u) = max (1 811,1 sa 1,

(3.33) Since

... +L U.Z. = 81(x1 - x:) + ... + 8._1(x._1 - x.) + 8.xn , D(x), and hence A(u) 6 we have I ulxl + + ux. max (I S1 On theother band, ifmax(Is,I,-.-,Is.1) = Is;I>0, u1x1 +

= 0; now D(x) = 1 + u.x. = I s; I = max (I 811, - , I s 1), and hence A(u) z , I s. 1) also. Thus (3.33) is proved.--The inequality (8.2)

then we set x1 = .. = x; = sgn s; , x;+1 = xj+t = and u1x1 + max (I a, I , gives here: (334)

.

utx1+...+u.x. <max(Is,f,...,1s.I)

+ Example 4: D(x) _ (I x1IP+ 1r2 1n + (p > 1). First we prove that this metric satisfies the condition 3.) ; it suffices to show that for positive x; and y; : ((x1 + y1)P + ... + (x. + y.)P)l/P (3.4)

= (xi + ... + XP-)"P + (yi + ... +

The inequality (3.4) is called the Minkowski inequality. We show this by induction. Thus first we prove the contention for n = 2. For this purpose we set:

J(y) = (xl + 42 )' P + (y2 + Then

yD)1'P

- ((x1 + yl)p + (xa + y)P)hJP (y > 0).

- (x! + y)' ((x1 + y)P + (x2 + y)P

y)(1-9)/P

f '(v,) = 8 1(yl +

P

Y-P)IP

= ((yl) + Since1

201

MEAN VALUE THEOREMS AND INEQUALITIES

§14)

P

p<0and ' +Y'>? fory>

X1

But since f( x!')

and hence f(y) >

,

XI

1\U-Y)IP

?.1

11

x'

f'(y) < 0 for 0 < y <

+

',x'+y'=y'fory=N',x1+y' X% + y x2 + y

< Nor 0 < y < y , we have f'(y) > 0 for y > y

+yy)P

- ((x!

\x1

' , f'(y) &+ 0 for y =

/ for all positive y f(xy')

0, we obtain f(y) > 0 for `all

/opsitive

x1

',

x'Y'

.

X,

and thus

y

substituting again y, for y: (3.41)

((XI + Y OP + (x1 +

y,)')'l9

-5 (XI + X?)"' ++(YIP + yi )11P

This is the contended inequality (3.4) for n = 2 (and we obtain the supplement: the equality sign occurs here if and only if y1 : ys = z' : x2). Now we assume (3.4) to be proved for a value of n and show that then (3.4) is true also for n + 1. By assumption we have (x1 + yl)P

+`

+ (XI, + y.)P + (xR+1 + ' + ... + x*)111 + (11' + ... + YO/"'), + (xn+, + yn+I)P. Yn+l)P

((xy

If we apply to the two terms of the right side the inequality (3.41), proved /already, we obtain:

(x' + y1)P + . + (x" + yr.)" + (xa+1 + y.+1)'

+ x* + xn}1)1/P + (yl +

5 ((xl +

+ 11 * + yw+i'IP)P

and this is the inequality for n + 1 we had to demonstrate. Thus (3.4) is proved. Now we show that the metric polar to D(x) is given by + I u2 I9/cP-1) + ... + u,. fP>'cP.a))cP-s)iP. (3.42) A(u) _ (I u1

According to (3), Au = sup (u1x1 + oW-1

+

here the set D(x) = 1

is given by I x1 IP + I x, IP + + I x,. IP -- 1 = 0. Since this is a bounded and closed set, and hence a self-compact set of R. , there is" a greatest value of + u,,x on the set D(x) = 1; this greatest value is A(u) and it can u'x1 +

be found according to the rule for constrained maxima (maximum of u4x1 +

+ u,,x,. under the constraining relation

+ .. +

ix'I., + IX.,

For the point x' _ (xi ,

xs

,

,

Ix.IP

- 1 = 0).

xA) which furnishes this maximum, one

obtains: "" For instance, of. H. Hahn [21, theorem 25.7.21.

CHAP. IV

INTEGRATION

202

(i = 1, 2, ... , n).

u; - ap. sgn x * . I xi IP-1 = 0

(3.43)

The "multiplier" X can be determined by the fact that x* has to belong to the set D(x) = 1: w

(3.431)

P10-1)

{

E t-1 1 Xp

w

d 1 E I u; IPI(P-1)`

= 1; that is, Xp

(P-1)/P

\,-t

J If one multiplies (3.43) by x` , adds, and considers D(x*) = 1, then one obtains for the maximum A(u) of u1x1 + + unxw :

ulxi + ... + uwxw = ?gyp. Thus because of the second formula (3.431), in which obviously the positive sign has to be chosen (the negative sign furnishes the minimum), (3.42) follows. From (3.2) the Holder inequality results: Iulxt+...+Uwxw I g

(3.5) (I u1 I

P;(P-1)

+ .. + I uw P/(P-1) (r1)/P

(I X1 I

Ih P .....i.. I xw Ip)(p > 1).

In particular for p = 2 one obtains the Cauchy inequality: (XI .}.... } x*)1. utxl + ... + U.X. i s (ui + ... (3.51) u*)+.

4. The Holder inequality for integrals. The following inequality corresponds to (3.31) for integrals

(A) f I

(4)

Id*v

sup I g(x) I s.a

(A) f If I do.

It is valid if f is gyp-measurable, g is bounded and gyp-measurable, and rp is monotone increasing. For if f is rp-summable, (4) is contained in (1.1), while otherwise

12.1.9 has to be considered. Now we derive the Holder inequality for integrals which corresponds to (3.5).14.4.1. If f and g are (o-measurable on A and if p is monotone increasing, then

for p> 11 (4.1)

(A)

4)I/P

f I f g I dw s ((A) f I f IP

((A) f

do)(P-1)/P

Ig

IPnP-n

Replacing I f I by f and I g I by g, we see that it suffices to prove the inequality (4.11)

(A) f fg dco 5

(A) f fP

under the assumption f k 0 and g

1/P

dip)

,((A)

f

gm,l) 4)(P-1)jp

0.-We define f, and g, by f,

2'

1

68 Here as well as in the following, a product has to be replaced by 0 if one factor equals

0 and the other equals ± - .

MEAN VALUE THEOREMS AND INEQUALITIES

§141

on A Ls

2.

1

f(1) < Z,] and g, =

A[' v

1 21

s

2.

1

201

on

2, ...

5 g(t) <

,

on A[f(t) i' vJ and g, = v on A[g}(t) z P]. Then by 12.1.61 and 12.1.8

(A) f fg, dp - (A) f fg do, (A) f f' dp -> (A) f fp do, and (A) f g, dip -3. (A) f gp dp.

Hence it suffices to prove (4.11) for the functions f, and g, , that is& for functions assuming only finitely many finite values.-Let a,, as , . , ak be the values off and let b, , b, , , b1 be the values of g; furthermore set A;= A[f(.t) = ail and B1 = A[g(x) = biJ. Then by 12.1.61 k

1

k

1

(A) f f9 o= Ei_1Ei_1aibi,o (A.B,) _ E E Now from (3.5) it follows that

(A) f fg 4 <

( a' 1

Herein, since A; = S A;B, and B, =

bfRCP-n Sk

\

p(A:B;)) 1

A;B,, we have E ,p(A;B i) = p(A j)

and E ,p(A;B,) _ p(B). Hence by 12.1.61 E a,' E r(A;B,) = (A) f fp duo i..l

i_1

i_I

and 2, bJ'JCp-1j E,p(AB,) = (A) f gpicp-1j dp; thus (4.11) is proved. i_1 irl For p = 2 we obtain from (4.1) the Schwarz inequality: (4.12)

(A) f 1 fg 1 d,o 5 A// (A) f fs d
As the inequality (3.5) was a particular case of the general inequality (3.2), so the inequality (4.1) can be regarded as a particular case of a general inequality analogous to (3.2) for integrals. In order to obtain this inequality we introduce

a "convex metric" for functions by attaching to every ip-measurable function f on A a number D(f) with the following properties:

1.) D(f) z 0, and in particular D(f) = 0 if and only if f =, 0. 2.) D(pf) _ I p I D(f)

3.) D(f1 + f2) s D(fj) + D(f2), provided that f1 + fs is 9-defined. Moreover, for the sake of simplicity we will restrict ourselves to the case where the following conditions are also satisfied:

4.) D(f) = D(If 1).

[CHAP. IV

INTEGRATION

204

5.) If f = 1 on A, then D(f) is finite. The determining function p is assumed to be monotone increasing.

If A is a zero-set for p, then because of 1.) for every f we have D(f) = 0. Therefore we assume further on: p(A) * 0. Now, analogously to (3.1), we define for every p-measurable function p on A1°:

A(g) = sup (A)

(4.2)

D(1)-1

f I fg I dc,

where f runs through all gyp-measurable functions f on A with D(f) = 1, and we call A(g) the metric polar to D(f). One sees easily that A(g) possesses the properties analogous to 1.), 2.), 3.) (and 4.)).

Now if f *, 0, and hence by 1.) D(f) $ 0, furthermore if D(f) < + co, and if we set f* = D(f) f, then by 2.) D(J*) = 1; thus (A) f I f*g I dcp 9 A(g), and hence (A) f I fg I d

Since this inequality obviously is also

true for f =, 0 and for D(f) _ + -60, we have: 14.4.2. If rp is monotone increasing, if D(f) is a convex metric satisfying the conditions. l.), 2.), 3.), 4.), 5.), and if A(g) is the corresponding polar metric, then for all ,p-measurable functions f and g on A (A) f I fg I d4p 5 D(J) . A(g)

(4.21)

Example 1: D ( f ) = (A) f I f 14. We designate by p the "y,-supremum" of g 1; that is, p is the infimum of all y e R, for which A[I g(t) I > yj =, A. We contend: A(g) = p. First we show: All g(t) I > p] A. This is trivial if p = + w ; hence let p < + oo . T h e n All g(t) I > p] = SAt i g(1) I > p + ,m]; by definition of p rr

we have A[I

L

g(t) I > p +

1

J

m =, A, and thus by 4.1.32 A[I g(i) I > p) _,, A

also, as contended.-From this it follows because of 12.1.53, 12.1.7, and 12.1.62

immediately that (A) f I fg I dfp

and by (4.2) there results that

A(g) S p. In order to prove the reverse inequality, let ((p,)) be a sequence of numbers with p, < p, f, and p. ---+ p. We set A [I g(1) I > A. ; then A. +, A, that is,,p(A,) > 0. Moreover, we set f. - 1 on A, and otherwise

w(A.) r Since v(A) * 0, by 5.), 1.), 2.) there is a w-measurable function f with D(f) - 1; thus, according to 9.2.8 and 12.1.9, the right side of (4.2) always has meaning.-For ,(A) .. 0 one could not form the right side of (4.2) at all (since then always D(f) - 0); in this owe one could define for every g: 0(g) - 0. 00 Cf. footnote 58, p. 202.

f.

205

MEAN VALUE TUBOREMS AND INEQUALITIES

§141

0; then

D(f,) = (A) f

if. I dcp - 1 and (A) f I f g I

=

sp(A,)-'A,) f I g I dp 9; p,.

p, and hence by (4.2) A(g) ?. p also. So it is proved

Thus sup (A) f I f.9 ( dip r

that,4(g) = p. The inequality (4.21) gives a trivial improvement of (4). Bxampie 2: D(f) = V-supremum of I f I. The polar metric is given by A(g) = (A) f I g (dip.

(A) f I g I dp.

f I fg I dip s D(f) (A) f I g (dsv, we have A(g) S If we set f = 1, then D(f) = 1 and (A) f I fg I dw = (A) f I g I dco, For since (A)

and hence also t(g) k (A) f I g I d,p. So the contention is proved.--The inequality (4.21) gives the same result as in example 1.

f

Example S: D(f) = ((A) I f 1P drp)1ap (p > 1). First we prove that this metric satisfies condition 3.), that is, that (4.22)

((A) f I fi Ip 4)'V p + ((A) f Ifs Ir dcp)up.

((A) f I f1 + f: Ip d,p)up

The inequality (4.22) is called the Minkowski inequality for integrals. For the proof we obviously can restrict ourselves to the case f1 0 and f2 it 0, and, as in the proof of 14.4.1, one sees that it suffices to furnish the proof for those functions fi and f2 which assume only finitely many finite values. Let , bj be the values of fs ; a1 , a2, , at, be the values of fi and let b1 i b2, moreover, set A, - Ajf1(t) = a;) and B1 = A[fs(&) = bi]. Then by 12.1.61

(ai + bi)psp(A:B

(A) f (f1 + fa)p dP =

k

Z

E (a,(p(A.B,))" + b,(o(A.Bt))11')p, !_t j_1 and hence by (3.4) :

((A)fii1 + fOp d,p)

1/p

k

l/p

Z

t E E ar (ABi)

!k

Z

\lip

k

i

Since E ,p(A;B,) _ ,p(AJ and E p(A;B;) _ P(O j), the right side equals ,_1

((A)ffi'

\llp dip J

i-1

+ ((A)! fi dp)

lip

,

and thus (4.22) is proved.

Now we show that the metric polar to D(f) is given by (4.23)

Q(g) _

(t.i)f I g ip:(p-1) 41

(p-1); p

.

206

INTEGRATION

[CRAP. IV

From (4.1) and (4.2) it follows that

L(9) 5

(4.231)

((A)f 19 Ip/(P-n (gyp/

rp

On the other hand, if

0<(A)f

(4.232)

IgIP(P-ndp<+ao,

we set

f dsp)(p--n/p

fI

(A) f I g

lP/(P-l)

2

and hence also (4.233)

(p^I) lP

A(g) Z (A) f 10 P/(P_1)

Thus (4.23) is proved under the assumption (4.232). For (A) f 1 g jP/(P--'u d(p = 0, (4.23) is contained in (4.231).

Finally if (A) f I g

dip = + oo, then we set

gm = g on (A) [I g(x) 1 S m] and gm = 0 otherwise. According to 12.2.311 and the above proof, A(gm) = ((A)f I gm IP/(p_t)

(p-1)/P

dso1

; now we let m -- + co

;

then by 12.1.8 and (4.2) we obtain again (4.23). Inequality (4.21) becomes here the Holder inequality (4.1). For the above three examples the metric D(f) satisfies not only the conditions 1.), 2.), 3.), 4.), and 5.) of p. 203-204, but also6' the condition: 6.) If I f, J converges monotone increasing to f 1, then D(f,) ---+ D(f). Now we prove: 14.4.3. If rp is monotone increasing, if the metric D(f) satisfies the conditions 1.), 2.), 3.), 4.), 5.), and 6.), and if t(g) is the corresponding polar metric, then in order that fg be (p-summable on A for all f tti.th finite D(f), it is necessary and sufficient that A(g) be finite. NECESSITY: Let A(g) _ + co. Then there is a sequence ((f,)), such that M This results for the first and third examples because of 12.1.8. We see it for the second

example in the following way: We set A l1 f (1) 1 > yl - A, and A 11 f,(4) 1 > y) - A,,, . Then since A,,,,, Q A,,,, Q Ay Cu < P), we have r(A y,,,) I c(A,,,,) S , (A,), and hence D(f,,)

D(f). Now if lim D(f,) < D(f), then there would be a y with v(A,,,) . 0 ) and po(A y) * 0. But this is impossible, since by 3.2.2 Lim A,,, n A y (y . 1, 2, D(f,)

implies,p(A,,,) - c(Aw). Hence D(f,) - D(f).

§14]

207

MEAN VALUE THEOREMS AND INEQUALITIES

D(f,) = 1 and (A) f I fg I dsp --p + oo. Taking a subsequence if it is necessary, we can assume: (A) f I fg I dip ? v'; moreover, replacing f, by If, I, we can "'

1

1

assume: f, z 0. Now we set fm =y_1 2:v-2 f, and f = E -vsf, ; then by condition 2.) and 3.):

5 ,..1 F vz D(f,), and hence D(fw',) ,5'1v1 1

2

.

Thus since the f,'.

converge monotone increasing to f, by condition 6.) D(f) 5 E V212 , and hence D(f) is finite.

By 12.1.8 (A) f

g I dp --> (A)

f f I g 14. But according to

12.2.1 and 12.1.62 (A) f ff I g I dp =

i (A) f f, I g I dsp and thus, since by ,1 v assumption (A) f f, I g I duo z v', we have (A) f I g I dip m. Therefore (A) f f I g I d,p = + -, and hence by 12.3.2 1 fg is not -summable.

SUFFICIENCY:

This follows from 14.4.2 and 12.3.21. In particular, on account of our three examples, 14.4.3 furnishes the following theorems: 14.4.31. If (P is monotone increasing, then in order that fg be (p-summable on A for all sp-summable f, it is necessary and sufficient that g be gyp-bounded (§9, 1) and .p-measurable on A. 14.4.32. If p is monotone increasing, then in order that fg be (p-summable on A for all gyp-bounded and sp-measurable f, it is necessary and sufficient, that g be fp-summable on A.

14.4.33. If sp is monotone increasing, then in order that fg be s*-summable on A

for all f with finite (A) f I f I n dip (p > 1), it is necessary and sufficient that (A)

f

Ig

Ip/(P-1)

dsp

be finite.

In particular one obtains from 14.4.33 for p = 2: 14.4.331. If p is monotone increasing, then in order that fg be (p-summable on A for all f with finite (A) f f'- 4, it is necessary and sufficient that (A) f g2 duo be finite.

BIBLIOGRAPHY to Nos. 3 and 4: The theory presented here in Nos. 3 and 4 is due to E. HELLY, Monatahefte f. Math. u. Phys. 31 (1921), p. 60, who started from certain developments of H. MIN%OWSKI (cf. [I], p. 102; Math. Annalen 57 (1903), p. 447 = Gea. Abhand-

lungen 2, p. 230, [also p. 132, 144]); subsequent to E. HELLY: H. HARN, Jahresberichte

[CHAP. IV

INTEGRATION

208

Deutsch. Math: Vereiniguug 30 (1921), p. 94 (italics); Monatshefte f. Math. u. Phys. 32 (1922), p. 3.-In particular, the inequality (3.51) is already due to A. L. CAUCUT 11 ], p. 456 Oeuvres (2) 3, p. 373, and, moreover, is already contained in an identity of J. L. LAGRANGE,

Nouv. MOm. Acad. Berlin 1773 - Oeuvres 3, p. 662; therefore (3.51) is mostly called the "Cauchy" or "Lagrange-Cauchy Inequality"; the corresponding inequality (4.12) for integrals is found in H. A. SCHWARZ, Acta Soc. Sc. Fennieae 15 (1885), p. 344 - Ges. Abhandlungen 1, p. 251; but already earlier in V. BUNIASOwasY, MOm. Acad. St. Peterebourg (7) 1 (1859), No. 9, p. 4. The inequality (3.5) is due to O. HOLDER, Nachrichten Ges. Wise. Gottingen 1889, p. 44; one owes inequality (3.4) to li. MINKOWSEI 111, p. 115; as to (4.1) and (4.22) we refer to F. R.IESZ, Math. Annalen 69 (1910), p. 450; Bolletino Unione.Mat. Ital. 7(1928), p. 77. Cf. also G. H. HARDY, J. E. L?rrLEwooD, G. P6LYA [1].

§15. Theorems on convergence ) 1. Mean convergence. Let ce(A) again be finite and let f and f,(v = 1, 2, be.,--measurable on A; we designate the absolute-function of p again by (P (§3,4);

and let p be a positive number. Now if (for all v) f, ' is (p-summable on A and if

(A)f If,

(1)

-fI'do-,0,

then we say: the sequence ((f,)) is convergent in the mean of order p to f.°! First we prove the following lemma: 15.1.1. If f, and f2 are,,,p-measurable and if I f, I' and I f2 I' are (p-summable on A,

then I f, + f2 I' and I fi - f2 I' are also .e-summable on A and

f (A) f I fi - f2 I' do 5 2' (A) f I f i I' do + - 2- (A) f I f2 I' dq.

(A) f I fi + fJ I- do 5 2'. (A) I fI I° d$ + 2' (A) f I f2 I' do,

Replacing f2 by -f2 we see that it suffices to prove the contention for fi + f2 . Since I f, I' and I f2 I' are 9-summable, by 12.3.11 f, f2 is be-defined and hence by 02.6 cp-measurable on A. Set f(x) max (I f,(x) I, 1 f:(x) 1). By 12.3.41

f' is o-summable on A. Since

f1

f2

5 f, we have if, + f2 I' S 2'f', and

2 hence by 12.3.51 1 f, + f2 I' is also p-summ able on A and by 12.1.7 (A)

f I f -i- f2 I' do S 2". (A) f f dq 5 V. (A) f I f, I' dye + 2' (A) f I fe I' do.

15.1.11. If ((f,)) is convergent in the mean of order p to f, then I f I' is c-summable.

By assumption I f, I' is s, summable for all v and I f, - f I' is `o-summable for almost all v. Thus since f f , - If, - f), by 15.1.1 I f I' is also ,P-summable. 0 If p - 2, then one often says simply "convergent In the mean."

THEOREMS ON CONVERGENCE

§151

209

15.1.2. If the sequence ((f,)) is convergent in the mean of order p to f, then ((f,)) is also asymptotically convergent to f.

By 15.1.11 and 12.3.11 f is p-finite. If ((f.)) was not asymptotically convergent to f, then by 10.3.1 there would be a q > 0, a p > 0, and infinitely many v, such that p([1 f,(t) - f(l;) 1 z q1) > p. But for these v we would have (A) f 1 f. - f P do z q1 p, contrary to (A) i f. - f I' do --' 0. The converse of 15.1.2 is not true in general. Example: Let A = [0, 11,

f

f, = v "FPin CO,

, f, = 0 otherwise, and o = µi . Then ((f,)) is asymptotically

convergent, but not convergent in the mean of order p to 0.-Yet we have: 15.1.21. If the f, are ep-measurable, 1 f 1' and the 1 f, I' are cp-summable, and ((f,)) is asymptotically convergent to f on A", then in order that ((f,)) be convergent in the mean of order p to f, it is necessary and sufficient that to every e > 0 there be an rt > 0, such that for every qp-measurable subset M of A with q,(M) < ,l and

for all v": (M) f 1 f, 1' dip < e. NEc wm: By 12.3.7 there is an -j, such that (p(M) < n implies

(M) f IfI'dp <2:+i. Since ((f,)) is convergent in the mean of order p to f, (A) f 1 f, - f 1' dip -- 0;

thus there is a v, such that for v > v*: (A) f If. -- f 1' de < 2' , and hence

(M)f If. - f I' dq, < 2n+, also. Therefore since f, = f + (f, - f), by 15.1.1 for v > v*: (M) fi f,1' dip < e. By 12.3.7 ,t can also be chosen such that .p(M) < 0 for v = 1, 2,

v implies (M) f 1 f,1' do < e. Thus this inequality is valid for O(M) < , and for all v, as contended. SumcmNCY: Let e > 0 be ,

arbitrarily given. By assumption there is an q, such that o(m) < 0 implies (M) f 1 f, I' dp < e for all P. By 12.3.7 q can also be chosen such that O(M) <

,1

implies (M) f 1 f I' dip < e. Then by 15.1.1 (M) f 1 f. - f 1' dye < 2'+'e for

all v.-Now we set A[1 f,(i) - f(x) 1 z el = A,. Since ((f,)) is asymptotically convergent to f, by 12.3.11 and 10.3.11 there is a v*, such that .p(A,) < ,' for v is v*. Then (A,) f If, -- f 1' dip < 2'+'e for v Z v*. Since on A - A, 14 Then by 10.3.84 f is also,-measurable on A. 41 Here "all v" can be replaced by "almost all ,", as the proof shows.

[CHAP. IV

INTEGRATION

210

we have (f, - f ( < e, by 12.1.32 (A - A,) f If, - f (' dg, 5 e'rp(A). Thus (A) f If, - f (' dip < 2'+1e + Ap(A) for v Z v*. Therefore since e > 0 was arbitrary, (A) f

(f, - f (' dip -+ 0; that is, ((f,)) is convergent in the mean of

order p to f. 15.1.22. If the f, are rp-measurable, the (f, (' are co-summable, and ((f,)) is asymptotically convergent to the rp-finite85 function f on A's, if, moreover, to every e > 0 there is an rf > 0, such that for every (p-measurable subset M of A with P(M) < ,i

and for almost all v: (M) f

(f,

- f (' dip < e, then (f (' is also gyp-summable and

((f,)) is convergent in the mean of order p to f.

Let e > 0 be arbitrarily given; we set A[( f,(:t) - f(t) ( ? e] = A,. Since ((f,)) is asymptotically convergent to f, by 10.3.11 p(A,) -- 0; thus, for almost all v, p(A,) < 71, and hence by assumption (A,) f I f,

- f (' dp < e. Since on

A - A, we have (f,-f( < e, by 12.3.52 and 12.1.32 (A-A,) f(f,-fI'do E',p(A).

5

Thus (A) f If, - f (' dq' < e + e'p(A) for almost all v; and hence

(A) f ( f, - f (' dip - 0; that is, ((f,)) is convergent in the mean of order p to f. Then by 15.1.11 (f i' is ip-summable. 15.1.23. If the f, are gy-measurable, if ((f,)) is asymptotically convergent to fB°,

and if there is a g, such that (g i' is p-summable and (f, ( 5 , (g ( for all v, then

((f,)) is convergent in the mean of order p to f. Since f f, I S,, ( g (, by 12.3.51 (f, I' is co-summable for all v. Moreover, by 10.3.51 and 10.3.33 ( f ( 5 ,, ( g (, and hence by 12.3.51 ( f (p is also w-summable.

Now by 12.3.7 there is an 71, such that p(M) < n implies (M) f ( g (' dip < e. Then since I f, ( 5 ,, ( g (,

(Al')

f (f, (' dp < e for all Y. Hence the contention

follows from 15.1.21.

From the definition of the mean convergence of order p, it follows by 12.1.52 immediately: 15.1.3. If ((f,)) is convergent in the mean of order p to f and if f, g, , then ((g,)) is also convergent in the mean of order p to f. 15.1.31. If ((f,)) is convergent in the mean of order p to f and if f g, then ((f,)) is also convergent in the mean of order p to g. 15.1.32. If ((f,)) is convergent in the mean of order p both to f and to g, then

f =v g11 This condition is essential. Example: A - 10, 1); ,(M) - 1 + , 1(M) if 0 e M, and

'p(M) aµ,(M)if O-eM;f,-f-0for x$O,f,- randf - +- forz=0. " Then by 10.3.64 f is also r-measurable on A.

THEOREMS ON CONVERG NCZ

§15]

211

This follows from 15.1.2 and 10.3.32.

15.1.4. If ((f,)) is convergent in the mean of order p to f, then ((-f,)) is convergent in the mean of order p to -f.

15.1.41. If ((f,)) is convergent in the mean of order p to f, then ((If, I)) is convergent in the mean of order p to I f I. This follows from 12.3.51 and 12.1.7, since f, I - I f I I s- I f, - f I

15.1.42. If ((f,)) and ((g,)) are convergent in the mean of order p to f and g, respectively, then ((max (f,(x), g,(x)))) and ((min (f,(x), g,(x)))) are convergent in the mean of order p to max (f(x), g(x)) and min (f(x), g(x)), respectively. By 12.3.41 max (I f,(x) I', I g,(x) I') is gyp-summable; thus since

I max (f,(x), g,(x)) I' S max (I fr(x) I" by 12.3.51 1 max (f,(x), g,(x)) I' is also q-summable.

I

I'),

Furthermore we have :61

I max (f,(x), g,(x)) - max (f(x), g(x)) I' s I f,(x) - f(x) I' + I g,(x) - g(x) and from this the contention follows by (1), 12.2.1, and 12.1.7. 15.1.43. If ((f,)) and ((g,)) are convergent in the mean of order p to f and g, respectively, then ((f, + g,)) and ((f, - g,)) are convergent in the mean of order p to f + g and f - g, respectively. For by 15.1.1 1 f, + g, I' is tp-summable and

(A)f I(f +g,) - (f+g)Ip4

s 2'.(A) f Iff - fI'do +

f I g, - gI'do.

We call the sequence ((f,)) convergent in the mean of order p if there is a function f to which ((f,)) is convergent in the mean of order p. Then we have: 15.1.5. If the f, are (p-measurable and the If, I' are ip-summable on A, then in order that ((f,)) be convergent in the mean of order p, it is necessary and sufficient

that to every e > 0 there be a vo , such that (A) f

I f,. - f, I' do < e for v z vo

and v' Z vo. NrssrrY: If ((f,)) is convergent in the mean of order p to f, then

(A)f If,

- fI'd ->0.

Hence there is a vo , such that (A) f I f,

- f I' dip < 2v+i+, for v ? vo ;then E

r (A)J I fe - J I' dq' < 2P+1 a If, say, f.

if,- g

.1

f.-g-' 0.

>_

g, and g ? f, and hence g, - g

f, - g S f, - f, then

I f. - fI provided that f, - g ? 0, and if,- gI

19,-91 provided that

[CHAP. IV

INTEGRATION

212

for v' 9 vo also. Thus since f,, - f, = (f.- - f) - (f, - f), by 15.1.1:

(A)f 1f,,-f,Ipdo<e SUFFICIENCY: First we show that ((f,)) is asymptotically convergent to a ,-finite function f. By 12.3.11 the f, are p-finite on A.

for v ? vo and v'

vo.

q] = A,.'(q). If ((f,)) was not asymptotically convergent to a So-finite function f, then by 10.3.611 there would be a k > 0, a q > 0, and sequences ((vi)) and ((v{)) of indices with v{ --* + co and v{ --p + cc, such that o(A,4,, (q)) := k for all i. But then by 12.1.32 we would also get

We set: A[I f,,() - f.(:) I

q'k for all i, contrary to the assumption.-Now by dp (A) f and by 10.3.63 10.3.62 ((f,)) contains a p-convergent subsequence f,,, --4 f; according to 10.3.64 f is ,p-measurable on A. Then by assumption (A) f

dq' < e for r Z vo and F. z vo . Since

If,. -- f. I' ->r I f - flip, w e obtain by 12.1.952 and 12.1.53: (A)f

I f - f, I' do S e for v

vo .

Thus

((f.)) is convergent in the mean of order p to f.

15.1.6. If f is immeasurable and If I' is sp-summable on A and if 0 < q < p, then I f I' is also p-summable on A.

We set A' = All f(I) I z 1] and A" - All f(c) I < 1). On A' I f I ' s I f 1', and hence by 12.3.51 1 f I ' is v-summable on A'.

On A" I f I' is qt"-summable by

Thus according to 12.3.3 I f I' is p-summable also on A = A' + A". 15.1.61. If ((f,)) is convergent in the mean of order p to f and if 0 < q < p,

12.3.52.

then ((f,)) is also convergent in the mean of order q to f.

Let e > 0 be arbitrarily given; we set A[I f,(x) - f(t) I e) = A,. On A - A. we have I f. -- f I < e, on A, we have I f. - f 1' e" I f, - f I-, and hence (A) f If. - f I' dp --9 e''(A - A,) + e1--n. (A,) f If, -- f I' do. By assumption (A) f

If. - f I' do -- 0, and thus (A,) f If,

therefore (A,) f I f, - f I' drp < e' for almost all Y.

- f I' d,p --+ 0 also;

Hence (A) f I f, - f I' dp <

e'(,p(A) + 1) for almost all is and, since herein e > 0 was arbitrary, (A) f

If, - f I' do -- 0.

Since, moreover, by 15.1.6 the I f, I' are rp-summable, ((f,)) is convergent in the mean of order q to f.

THEOREMS ON CONVERGENCE

§15]

213

BrsuoGaArvy: Preliminary formulations are already found in A. HARNACK, Math. Annalen 17 (1880), p. 126,128; F. Rinse, 0. R. Paris 143 (1906), p. 740. For e - µ, and p - 2,

the notion of the "mean convergence" and the notation as well as theorem 15.1.5 are due to E. Flsonnn, C. R. Paris 144 (1907), p. 1022. Subsequent to him: F. R1asz, C. R. Paris 148 (1909), p.1303 (for (p - pt and an arbitrary p > 0); Math. Annalen 69 (1910), p. 468; H. WsTL, Math. Annalen 67 (1909), p. 243; M. PLANCHEREL, Rendic. Cire. mat. Palermo 30 (1910), p. 292; Bull. Sc. math. [581 . ] (2) 47r (1923), p. 195; W. H. and G. C. YOUNG, Quart. Journ. 44 (1912), p. 49; E. W. HansoN, Journ. London Math. Soc. 1 (1926), p. 211; A. ZYGMUND [1), p. 72; L. M. GawvEs [11, p. 236.--Generalisations in: Pre NALLI, Rendic. Cire. mat. Palermo 38 (1914), p. 306, 320; A. HAAR, Acta Univ. Szeged 1 (1923), p.167; P. Ltvr, Bull. so. math. (60r -) (2) 491 (1925), p. 344, 374; Sr. KAczMutz-L. NimaBoac, Fund. math. 11 (1928), p. 161; J. C. BURHILL, Proc. London Math. Soc. (2) 28 (19'28), p. 493; Z. W. BIRNBAUM-W. Oumcz, Studis math. 2 (1930), p. 197; Z. W. BrRNBAUY, Nachrichten Gee. Wiss, Gottingen 1980, p. 338; S. Ihoxr, Jap. Journ. of Math. 7 (1930), p. 27; C. CAxATuAoaosy, Abhandlungen Math. Sem. d. Hansiachen Univ. 14 (1941), p. $61.

2. Integrable and completely integrable sequences. Let ((f,)) be a sequence

of ip-integrable functions on A. The fact that ((f,)) converges on A to the re-integrable function f does not imply that (A) f

ple: Let A = [0, 11, , = it, , f,

f. dc,--a (A) f f drp also. Exam-

in (0, 11, and f, = 0 otherwise. Then f, -- 0 \\

v

in 10, 11, yet f 1 f, dx = 1, and hence we do not obtain the result that f1 f, dx---> 0.

Let ((f,)) be a sequence of rp-integrable functions which is asymptotically convergent to f on A;n then the sequence ((f,)) is called se-integrable if f is also qs-integrable and (A) f f, dip -+ (A)

f f dip.

In order to express f, we say

then also, ((f,)) converges in a g,-integrable manner to f. From this definition there results immediately: 15.2.1. If ((f,)) converges in a cp-integrable manner to the So-summable function f on A, then almost all f, are also p-summable on A.

The fact that the sequence ((f,)) converges in a 9-integrable manner to f on A

does not imply that for every so-measurable subset M of A we have(011), also: (M) f f, drp --n (M)

in

f f d,,.

Example: Let A

f

, f.

= v in 1-v

0), and f, = 0 otherwise. Then f, - 0 in [-1, 1) and, since

{1 1

[--1,1], ,p = µ1

f f, dx --> 0; yet J f, dx - 1, and hence we do 1

f, dz - 0, we have also

not obtain the result that

1

1

f f, dx -4 0. 1

0

If the f, area-integrable on A and if f, -+. f, then by 10.3.2 ((f.)) is also asymptotically oonvergent to f on A.

CHAP. IV

INTEGRATION

214

Let ((f,)) be a sequence of V-integrable functions which is asymptotically convergent to f on A;89 then the sequence ((f,)) is called completely Antegrable if f is also l -integrable and for every (p-measurable subset M of A: (M) f f, dp -4

(M) f f dp. In order to express f, we say then also, ((f,)) converges in a completely cp-integrable manner to f. 15.2.2. Let ((f,)) be a sequence of So-integrable functions which is asymptotically convergent to a ,p-summable70 function f on A. Then in order that ((f,)) be com-

pletely (p-integrable, it is necessary and sufficient that (A) f J f, - f I dq, - 0.71

NE,cEsSITY: We assume that ((f,)) converges in a completely (p-integrable

manner to f, but that (A) f I f, - f I dip -, 0 is not true, and we show that this leads to a contradiction. By 15.2.1, 12.3.42, and 15.1.21 there exists an e > 0 with the following property: to every sequence ((gi)) of positive numbers there is a sequence ((M;)) of co-measurable subsets of A and a sequence ((v{)) of indices, such that s9 It is possible that we have for every so-measurable subset M of A: (M) f f, dip -+ (M)

f f d,p, without ((f,)) being asymptotically convergent to f. in

12i 2i+ 1\ 2v , 2v

+1 -1 in 2i2;_ I

Example: Let A - (0, 11,

2i + 2\ 1

2i

1 i : 0,1, . . , v - 1), and

e

f,(1) - 0. Then for every interval [a, b] C [0, 1] we have f f,dp, - 0. If M is a p,-measure

able subset of 10, 1], then by 8.2.521 there are finitely many disjoint intervals [aj , bj] Q [0,11 k

(j - 1, 2,

, k), such that, if we set B = S [aj, br], we obtain t-,

p, (B - M) < e and p, (M - B) < e. Thus

I

(M) f

f, dp, - (B) f f, dp, I < 2e. Therefore, since (B)ff. dpi -s 0, we have for

almost all v: I (M)

f f, dp, 1 < So; and hence (M) f f, dpi -., 0 for every it,-measurable subset

of 10, 1), while ((f,)) is not asymptotically convergent to 0.

70 For "8ufcient" we can here replace ",p-summable" by ",p-integrable". For if, say, (M)

f f dip - +m, then a proof is needed only for those f, for which (M) f f, 4 r+oo;

but for those f, the proof given here is valid.-Yet for "necessary" the o-eummability of f is essential. Example: f, = v and f = +eo on A; ,p (A) ,-+` 0. Then ((f,)) is completely ,p-integrable, but (A)

f I f, - f I drp _ + -.

71 If the f, are assumed to be ,p-summable, then this condition can be formulated also in the following way: "that ((f,)) be convergent in the mean of order 1 to f".

THEOREMS ON CONVERGENCE

§151

215

o(Mi) < n; and (Mi) f If,, I dip z e. Then from 12.2.3 it follows that there is a 4p-measurable subset Bi of Mi for which

and

p(Bj)
(2)

I

(B;) f f,; d p z e

Moreover, from 12.3.7 it results that ni -> 0 implies vi - +oo. Furthermore by 12.3.7 there is an n > 0, such that O(M) < n implies (2.1)

I

(M) f fdcl <8;

since the sequence ((ni)) was arbitrary, we can assume:

ni < n. Now we

can pick out from the sequence ((Bi)) a subsequence ((Bi,)), beginning with Bi1 = B1, according to the following rules: If it , is , , i, are already chosen, then i,+1 has to be chosen such that 1.)

2.)

(B1, + B;, + ... +B,,) f frill+l dip I <

I

8'

(B+.+...) f f.,K d(p < $.

I

This certainly is possible: 1.) can be satisfied; for since O(B1) < ni and E in < n,

we have by 3.1.41 (p(Bi, + Bi, + . + Big) < n, and hence by (2.1)

(B;, + B,, + ... + Bit) f f dp


therefore, because of the complete p-integrability of ((Iif,)), we have also (Bi1+B4,+... +Bi,)

f f,dipI

<8

for almost all P. 2.) can also be satisfied; for by 15.2.1 and 12.3.7 there is a

> 0, such that (p(M) < r implies (M) f f,;. do < and, since lp(B4) < n: and E n; < n, we have for almost all is lp(B{) + p(B;+1) + < r; thus i,+1 can be chosen so great that E p(B) < t, and then by 3.2.3 certainly .p(Bi.+1 + B4+2 + ...) < .. I

Now we set B = SB4, .

Then since p(B1) < n; and E ni < n, by 3.2.3 ,p(B) < n,

and hence according to (2.1): (2.2)

1 (B)

f f dp

I

<

g.

CHAP. IV

INTEGRATION

216

Moreover, (B)ffi.dp

(Bi) f A. dip

(Be, + ... + Bi _,)

f

f,{R dip I

I

f

f,,R dv I,

and hence by (2) and the conditions 1.), 2.) : (B) f f,i, dp > 4 .

Thus,

- -I (Bi:", + I

I

according to (2.2), (B) f f, dp --> (B) f f dp cannot be valid, and so we have derived a contradiction to the assumption that ((f,)) be convergent to f in a completely V-integrable manner. SuFFIcmNCY: For M C A we have (M)

f If. - fI do 5 (A) f If. - fId,

and hence (A) f I f, - f I dp --, 0 implies (M) f I f, - f I dip -4 0. But by 12.3.21 and 12.2.31

(M) f (f, - f) d
(M) f (f, - f) duo -- 0 also; that is, by 12.2.11: (M) f f, do - 4 (M) f f dcp. From 15.2.2 there results because of 15.1.11, 15.1.2, 12.3.51, and 15.1.61: 15.2.21. If ((f,)) is convergent in the mean of order p z 1 to f, then ((f,)) converges also in a completely cp-integrable manner to f.

Because of 15.2.1, 12.3.42, and 15.1.21, theorem 15.2.2 can be formulated also in the following manner: 15.2.22. Let ((f,)) be a sequence of 9-integrablen functions which is asymptotically convergent to a cp-summablen function f on A. Then in order that ((f,)) be completely S -integrable, it is necessary and sufficient that to every e > 0 there be an ,1 > 0, such that for every 9,-measurable subset M of A with O(M) < vl and

for all v": (M) f I f, I do < e. " Here also for "sufficient" the f, may be assumed to be only ,,.integrable. For the f, the result that

proof of "sufficiency" in 15.1.21 furnishes also for such (A)

f If, - fIdip - 0.

" That the condition of the y,-summability of f is essential for "necessary", is shown by the example of footnote 70, p. 214; for there (M)

f I f, I dp a vrp(M).-As to "sufflcient",

theorem 15.2.23 provides a supplement. " Because of footnote 64, p. 209, "all v" can here be replaced by "almost all v".

THEOREMS ON CONVERGENCE

§15]

15.2.221. In theorem 15.2.22 the inequality (M)

217

f f, I dip < e can be replaced

by I (M) f f, dip I < E.

NEcESSiTy: By 12.2.31 (M)

f

f, do < e implies (M) f f. dp < I

I

SuFFicIElccY: From 12.2.3 there results: if (M) f f, dp I < e for all 141 E 1 with I

-p(M) < rt, then (M) f If, I dip < 2e. Because of 12.1.31 and according to §5, 8, theorem 15.2.221 can also be formulated in the following manner: 15.2.222. Let ((f,)) be a sequence of c*-integrabie functions which is asymptotically convergent to a (p-summable function f on A. Then in order that ((f,)) be completely

So-integrable, it is necessary and sufficient that the sequence

(((wf f. duo ))

be

equi-9-continuous in W.

15.2.223. If ((f,)) converges on A in a completely 9integrable manner to the rp-summable function f, then I f I and almost all J f, I are also cp-summable on A and ((I f. j))76 converges on A in a completely cp-integrable manner to I f

By 15.2.1 almost all f, are p-summable on A; thus by 12.3.42 (f I and almost all If, I are also p-summable on A. According to 15.2.22, to every e > 0 there

is an n > 0, such that for every p-measurable subset M of A with O(M) <

and for all v: (M) f I f,14 < e. By 10.3.51 ((I f, I)) is asymptotically convergent to If I; hence again by 15.2.22, ((I f, I)) converges in a completely .p-integrable manner to I f I. 15.2.23. If cp is atomless76, if the sequence ((f,)) of rp-measurable functions is asymptotically convergent to f on A, and if to every e > 0 there is an p > 0, such that for every ..p-measurable subset M of A with (p(M) < *] and for all v:

(M) f If. I do <

,

then f and all f, are cp-summable and ((f,)) converges in a completely .p-integrable manner to f. By 10.3.64 f is also gyp-measurable on A. Let k > 0; we set f;k' = f,

on All f,(:r) 15 k} and f;k) = k on A(I f,(t) I > kl; analogously we set f (k)

=f

7' Here we mean the sequence of those I f. I which are c-integrable; thus, if it is necessary, finitely many beginning terms of the sequence ((I f, 1)) have to be omitted. 76 This condition is essential. Example; Let a e A, b e A, and M C, A; lot 9(M) - 0 if a - e M and b - a M, c(M) = I if either a e M, b- e M or a. e M, b e M, and p(H)w2

if a e M and b e M; let f,(x) = 0 for z t (A - (a + b)), f,(a) - v, and f,(b) according to 12.1.1 f = lim f, is not even p-integrable on A. ,

- -v. Then

218

[CHAP. IV

INTEGRATION

on A[1 f(x) 15 k] and f«' = k on A[I f(t) I > k]. Then by 10.3.51 If,"' I is asymptotically convergent to I f(k) I on A. According to 12.3.52 the I f;k' 1)) converges in a comand I f(' I are q-summable, and hence by 15.2.22 ((1 pletely Fp-integrable manner to I f ck' on A. Therefore (A) f I f (k) I dip = I

lim (A) f I f;k' 1 dip 5 Jim (A) f I f, I dip (since I f ,(A:) 15 I f, I and by 12.1.9 and

Let k --> + oo; then I P) I converges monotone increasing to I f 1, and

12.1.7).

hence by 12.1.8 (A) f I f I dip 5 lim (A)

f 1 f, j do also.-Now let e > 0 be

arbitrarily given. Since V is atomless by 5.4.31 and 5.6.11 there is a decomposition A = M, + M2 + - - + M, in finitely many disjoint sets of 91, such that O(Mi) < i

(i = 1, 2,

, q); thus by assumption (Mi) f I f, 1 dip < e, and hence

(A) f If,1 for all

P.

t(Mi) = i-I

f If,1do
Therefore (A) f If I dip 5 lim (A) f If. 14 r

qe also.

Thus by

12.3.21 f and all f, are (p-summable, and hence by 15.2.22 ((f,)) converges in a completely integrable manner to f. 15.2.231. In theorem 15.2.23 the inequality (Al) f 1 f, I dip < e can be replaced

by I (M) f f, dp I < e. Proof as in 15.2.221 ("sufficiency"). Because of 12.1.31 and according to §5, 8, theorem 15.2.231 can also be formulated in the following way: 15.2.232. If ip is atomless, if the sequence ((f,)) of *-measurable functions is asymptotically convergent to f on A, and if the sequence

(((1w)! f, dp)) is equi-fp-

continuous in W, then f and all f, are co-summable and ((f,)) converges in a completely (p-integrable manner to f.

15.2.3. If the sequence ((f,)) of 9-measurable functions is asymptotically convergent to f and if there are So-summable functions g and h, such that g 5 f, h"

for all v, then f and all f, are to-summable and ((f,)) converges in a completely p-integrable manner to f.

By 10.3.33 g 5 f 5 h; thus, by 10.3.64 and 12.3.5, f and all f, are co-summable. According to 12.3.42 1 g I and I h I are rp-summable, and hence by 12.3.41 f*(x) = max (I g(x) 1, 1 h(x) 1) is also cp-summable. Thus by 12.3.7 to every

s > 0 there is an n > 0, such that (p( M) < i implies (M) f f* dip < e. But 77 Because of 12.1.53, g

f,

h can be replaced by g ;5 . f,

0 h.

219

TREOREMS ON CONVERGENCE

§151

g 5 f , S h implies I f, 15 f*, and hence we have by 12.1.7 (M) f I f, I dip' < e Therefore according to 15.2.22 ((f,)) converges in a completely (F-integrable manner to f. 15.2.31. If the sequence ((f,)) of
pletely rp-integrable manner to f.

f, - f is gyp-summable by 10.3.64 and 12.3.51 and

finite by 12.3.11; thus by

12.2.11 f = f, - (f, - f) is y-integrable. According to 10.3.53 ((f, - f)) is asymptotically convergent to 0, and hence by 15.2.3 ((f, - f)) converges also in a completely p-integrable manner to 0; that is, for every M e A we have:

(M) f (f, - f) dp -- 0. Adding (M) f f d1p to both sides we obtain by 12.2.1: (M) f f, dp -> (M) f f dcp, as contended. 15.2.311. If, under the assumptions of 15.2.31, the f, are (p-summable, then f is also

summable.

In the proof of 15.2.31 it was shown that f, - f is fp-summable. Thus by 12.3.43 f = f, - (Jr - f) is also p-summable. As we saw above, the q -integrability does not imply the complete (p-integrabil-

ity. Now we give a condition (theorem 15.2.41) under which from the rp-integrability the complete 9-integrability can be derived. First we prove: 15.2.4. If the sequence ((f,)) of q -integrable functions is asymptotically convergent to the gyp-summable function f on A and if to every e > 0 there is an n > 0, such

that for every immeasurable subset N of A with ip(N) < n and for almost all v: (N) f f, dip < e (or > - e), then for every gyp-measurable subset M of A we have:

(M) f f dcp z lim (M) f f, dp (or 5 1im (M) f f, dip). r

According to §12 (1.1) we form the decomposition: A = A' + A" with A'A" = A, c (A') = 0, and,p+(A") = 0. We write MA' = M' and MA" = M"; moreover, we set M'[f(t) > f,(t) - e1 = M' and M"[f(.f) < f,(1) + e1 = M;' . Then, because of the asymptotic convergence, by 12.3.11 and 10.3.11 we have O(M' - M;) -> 0 and ip(M" - M,') --* 0. Thus by assumption

f

(M' - M;) f f, dip < e and (M" - M;') f,4 p < e

f

for almost all v, and according to 12.3.7 (M' - M) f dcp > -e and (M" - Mf',') f f d,p > -e for almost all v.

(M:) f f dip ? (M.) ff dp

-

Furthermore, by 12.1.7,

v(M") at (M.) f f, dip - &p(W) and

78 Here "all" can be replaced by "almost all".

220

[CHAP. IV

INTEGRATION

(M:) f f, dip + ecp(M;') z (M:) f f, dcp + ep(M" ).

(M:') f f d j,

Hence it follows that

f

(M') f f dip = (M;) f f 4 + (M' - M") f f dip > (M:) f. dp - e,e(M') - e and (M") f f dip = (M,) f f dw + (M"

- M;') f f dip > (Al",') f f. dip + ev(M") - e

for almost all P. Now herein by 3.1.117° we have

(M") f f. dp = (M') f f, dv -- (M' - M;) f f. dip > (M') f f, dcp for almost all v and analogously (,W,) f f. dip > (M") f f, dcp - e for almost all P. So we obtain: (M') f f dsp > (M') f f, drp

- erp(M') - 2e and

(M") f f d50 > (M") f fr dv + ero(M") - 2e for almost all P. Hence by addition:

(M) f f dip > (M) f f, the - e(sc(M') -- o (M") + 4) for almost all v, and thus (M) f f drp z lien (M)

f f. dip.

Theorem 15.2.4 contains 12.3.81 as a special case, under the assumption that there f is c*-summable.8° 15.2.41. If the sequence ((f,)) converges in a cp-in4rable manner to the fp-summable function f on A and if to every e > 0 there is an +7 > 0, such that for every 9-meas-

urable subset M of A with O(M) < rl and for almost all v: (M)f

f. dy7 < t (or > -e),

then ((f,)) converges in a completely rp-integrable manner to f on A.

f

Here 3.1.11 can be applied only if (M' - M.)rff. dp is finite. But if this value is infinite, then (since < e) it is - co; by 3.1.2 also (M')

f

f

dip = - oo, and hence the inequality

(Mo f f, d p ? (M') f, 4 - e is valid also in this case. 80 For by 12.2.33 the f, are so-integrable. Moreover, by 12.3.7 to every e > 0 there is an

> 0, such that for every N e PC with O(N) < 17: (N) 12.1.7 also (N)

f f, d -p > - e for all v.

f p do > - e.

But since f,

Thus the conditions of 15.2.4 are satisfied.

g, by

THEOREMS ON CONVERGENCE

§151

221

Let MeY1;by 15.2.4 we have:

(M) f f dip

ffm- (M) f f, d p and

(2.3)

(A -M) f f dcp z lin (A -- M) f f, dcp. If in the first of these inequalities we have the symbol >, then by addition it would follow that (A) f f dip > lim (A) f f, dp, contrary to the assumption that (A) f f, dcp --> (A) f f dcp. ii-m- (M) f f, dip

Hence: (M) f f dp = 9-m (M) f f, dcp. Now if ,

f

> lim (M) f, dip, then there would be a sequence ((v{)) of

indices, such that lim (M) f f,; dip < (M) f f 4; and since by (2.3)

1m (A - M) f f,; dcp s (A - M) f f dce, we would again by addition obtain lim (A) f f,{ d p < (A) f f dip, contrary to the assumption that (A) f f, dip -+ (A) f f dip. Thus l ym

(M) f Jr coo

° lim (31) f .f, dp = (M) f .f ds;

that is, (M) f f, dcp -- (M) f f drp, as contended. 15.2.5. Let p be monotone81. If the sequence ((f,)) of ce-measurable functions is asymptotically convergent to f and if there are two sequences ((g,)) and ((h.)) of cc summable functions which converge on A in a ipintegrable manner to the (p-summahle functions g and h, respectively, such that g. < f, 5 h,82 for all v, then f and all f, are also (p-summable on A and ((f,)) converges on A in a rp-integrable manner to f.

We assume that c is, say, monotone increasing. By 10.3.33, g, S f, h, implies g 5, f ;5. h, and hence it follows from 10.3.64 and 12.3.5 that f and all f, are,p-summable. Moreover, by 12.3.11 all functions appearing here are p-finite. "' This condition is essential. Example: Let A = [-1, 1]; ,(M) - )S1(M) for if ..C (0, 11, ,v(M) _ -k, (M) for M C 1-1, 0J; h, = Yin

/

(0,!)andin(_

, 0 !, otherwise h_O; g, = -

Y

1

in 1 0, 1) and in

0), otherwise g, : 0; f, ° Y in

0,

Y

-fin - - , 0), otherwise

f, = 0. Then ((h,)) and ((g,)) converge on A in a V-integrable manner to h - 0 and g w 0, respectively. But f, -- f - 0, while (A)

f f, dv a 2 for all v and (A) f f dq, - 0.

82 g, ;g f, ;g h, can here be replaced by g, 5 , f, 5 , h,.

[CHAP. IV

INTEGRATION

222

0 and by 10.3.53 ((f, - g,)) is asymptotically convergent to lim (A) f (f, - 9). dip. - 9> we have, according to 12.3.81, (A) f (f - g) dip ;-

Since f, - g.

f

Analogously (A) f

(h - f) dcp S Lm (A) f (h, - f.) dip.

;f

f in one of these two

inequalities we had the symbol <, then by addition, according to 12.2.1, we would obtain: (A) f

(h - g) dip < lim (A) f

tions it follows, because of 12.2.11, that (A) f

g.) dp, while from our assump-

(h, - g,) dip - (A) f (h - g) dp.

Thus we have:

(A) f (f - g) dcp = Lm (A)

f (f, - g.) dy

(2.4)

and If we had, say, lim (A) f

(A) f (h - f) dip = lim (A)

f (h. - f,) dv.

(f, - g.) dip < lim (A) f (f, - g,) dip, then there would

be a sequence ((vi)) of indices, such that

lim (A) f (f,; - g,;) dp > (A) f (f - g) dip, and, since by (2.4) lira (A) f

(h,, - f,{) dip z (A) f (h - f) dip, we would obtain

by addition: lim (A) f (h,; (A)

dip > (A) f (h - g) dip, while by assumption

f (h,{ - g,;) dip -+ (A) f (h - g) dip.

Thus lim (A)

f (f. - g.) dip =

1im (A) f (f - g) dp and from (2.4) it results that

(A) f (f. - g,) dip--* (A) f (f -- g) dp.

f

f

If we add to this the relation (A) g, d4p --* (A) g dip, valid by assumption, then by 12.2.1 we obtain: (A) f

f, dvp -* (A) f f dip, as contended.

15.2.51. Theorem 15.2.5 holds also if always p-integrability is replaced by complete p-integrabitity; then the assumption that rp is monotone is superfluous.

According to §12 (1.1) we form the decomposition: A = A' + A" with A'A" = A, 9 -(Al) = 0, and p+(A") = 0. The sequences ((g,)) and ((h,)) converge also on A' and A" in a completely rp-integrable manner to g and h, respectively. Thus because of 12.3.3 it suffices to prove the contention for a

MULTIPLICATION OF SET FUNCTIONS

§16]

223

monotone (o. But for a monotone rp one has only to apply 15.2.5 to every ,p-measurable subset M of A. BIBLIOGRAPHY: The notion of complete integrability and the theorems 15.2.22-15.2.232 (fore - va) are due to G. VITALI, Rendic. Circ. mat. Palermo 23 (1907), p. 137; subsequent to him: C. Da LA VALL.B POUSSIN, Trans. Amer. Math. Soc. 16 (1915), p. 445; M. NAOUMO,

Jap. Journ. of Math. 5 (1928), p. 97; 6 (1929), p. 173; R. L. JEFFERY, Trans. Amer. Math. Soc. 33 (1031), p. 433; 41 (1937), p. 171; T. H. HILDEBRANDT, ibid. 33 (1931), p. 441; G. FlcsaRA, Portugaliae Math. 4 (1943), p. 1. Theorem 15.2.2 is due to H. HAHN, Sitaungeberichte Akad. Wiss. Wien Us, 127 (1918), p. 1774.--One owes the theorems 15.2.3 and 15.2.31 essentially to H. LESEsous, Paris These 1902, p. 29 : Annali di mat. (3) 7 (1902), p. 259; (1], p. 114; [21, p. 125, 131; Bull. Soc. math. France 36 (1908), p. 12; before H. LEassGuE, the most extensive result in this direction was obtained by C. AssuLJ, Rendic. Aecad. Lincei (4) 1 (1885), p. 537; Memorie Iet. Bologna (5) 8 (1899/1900), p. 723.-As to 15.2.5: W. H. YOUNG, Proc. London Math. Soc. (2) 9 (1910/1911), p. 315.-Moreover, cf. to No. 2: E. W. HoasoN [11, vol. II, p. 289, and Encyklop. d. Math. Wise. II C9b (P. MONTELA. RosENTMAL), No. $7.

§16. Multiplication of set functions 1. Product of monotone set functions. Let A and B be any two sets; then we designate by A X B the set of all pairs (a, b) with a e A and b e B, and we call A X B the product of the two sets A and B (cf. §8, 1). The following formula is obvious:

(AI X BI) -- (A2 X B2) (1)

= ((AI -- A2) X (BI - B2)) + ((AI - A2) X BIB2) 4 (AiA, X (BI - B,)); the three terms of the right hand side of (1) are disjoint. Furthermore we have the distributive laws: (1.1)

(S A,) X B - S (Ak X B) and ket

kelr

(D Ak) X B = D (AA X B), keE

keR

where K is a given set of indices k.

Now let E' and E" be any two sets; let 11' and a" be o-fields consisting of subsets of E' and E", respectively, with E' e a' and E" a s"; and let tp and gyp" be monotone increasing, totally additive set functions in 5' and a", respectively; moreover, let a' be complete for jpI and let j5" be complete for sp" (14, 4).°i a If the systems ' and 3" are only field. (instead of a-fields) and if r' and *" are mono-

tone increasing and totally additive in jV' and in s", respectively, then, according to 16, 5, gyp' and p" can be extended to monotone increasing, totally additive set functions in the complete cover W' and _g" (with respect to and V"), where W' and $" are a-fields com-

plete ford and ip", respectively. For this reason it is no restriction to assume from the first $r' and )f" to be o-fields complete for gyp' and c", respectively.-Moreover, the assumptions that 13' and a" are a-fields and that E' a 13' and E" e 6" will not be used before 16.1.2. Furthermore, in the theorems 16.1.1-16.1.122 the only property of and t " used is that they are additive (instead of totally additive) in the fields 13' and a", respectively; and under the same assumptions it also follows immediately that the set function,., defined by (1.5)

in $+ . is additive in ql-f .

M4

[CHAP. IV

INTEGRATION

We consider the product E = E' X E" (which now plays the role of the space)

and in E the system $ of the product sets M = M' X M" where M' ,c 15' and M" a a". We seta` (1.2)

then hereby a set function 4' is defined in J3 and we ask whether it is possible to find a or-field 0 a 3, consisting of sets of E, and a totally additive set function 9 in S2 which in 3 coincides with the function 4, defined by (1.2). In order to satisfy a first preliminary condition for the possibility of this question, we show that the function 4', defined by (1.2) in $, is additive. For this purpose we start from the following particular case: + PA;, where P; = Pi X P"(i = 1, 2, , k), 16.1.1. If P = P1 + P2 + P" c a", and the P; are disjoint sets of j5'eb, then ,(P) = 1G(PI) + G(PO) + ... + 4,(Pk).

(1.3)

+ P,y = P', then P e a' and by (1.1) P = P' X P", . If we set P1 + Pi + . and hence P e'3. Because of the additivity of rp' in a' we have ,/(P') _

'(P') +

+ V'(Pk); thus since

¢(P) = so'(P'),,p"(P") and 4'(P,) = we obtain (1.3). 16.1.11. 4, is additive in 13.

We have to show: If P e $ and P = P1 + P: +

+ Pk where the

Pi (i = 1, 2, , k) are disjoint sets of 43, then (1.3) holds. Set P = P' X P" and P. - P, X P;' (with Pea, P" e a"; P, a ar', P'i'e a'). Since empty terms may , k). be omitted forthwith, we can assume: Pi * A and P;' $ A (i = 1, 2, We prove the contention by induction. It is trivial for k = 1; thus we assume it to be proved for less than k terms and show that it then holds also for k terms.

Here we can assume that we have not for all is P.' = P", since otherwise the contention would be already proved by 16.1.1; hence we assume, say, P;' C 1'"'

that is, P" - P' * A. Since P = (P' X Pk) + (P' X (P" - P')), we have by 16.1.1:

4'(P' X (P" - A)).

'P(P) x 4,{P' X P') +

(1.31) k

k

88.

Since P = S P, and P" = S P'i', we have i_1

i-I

8' We assume also here that a product one of whose factors is 0 has the value 0, even if the

other factor is infinite. 88 An analogous proposition holds, if Pi = P' X P; (i are disjoint sets of 15".

1, 2, . k

/f k

88 For by (1.1) we have P' X P"x - (S P; I X Pk

\inl

, k), P' e jy', and the Pi k

S (P; X Pk) D S (P; X PiPF).

i_1

iv1

On the other hand, since P X Pig Q P X P" a P - S Pi , to every a e (P' X Pf) there is i-l

MULTIPLICATION OF SET FUNCTIONS

§161

225

k

and

P' X Pk = S (P' X P" Pk) :_1

(1.32)

k

P' X (P" - Pk) = S (P; X (P:' - PI Z)) We show that in each of these two sums (1.32) at least one term is empty. For Pk = A. As to the first sum, the second sum it is the k`h term, since Pk -

there is an a' e Pk and an a" e (P" - Pk), since Pt' * A and P" - Pk' * A.

Then (a', a") a (P - Ps), and hence there is exactly one i < k, such that (a', a") e Pi

But thep 11"i Pi, = A (and therefore in the first sum (1.32) . the iu` term is empty) ; for if b" e P: Pk , then a' a Pi, and b" e Pk would imply (a', b") a Pk ; and (a', a") e P: implies a' a P; , and hence from b" a P"it would follow that (a', b") a Pi also, contrary to the assumption that PiPk = A. Thus since in each of the two sums (1.32) there is an empty term and such a term can be omitted, these sumR consist of less than k terms, and hence we have by assumption: k

4'(P' X Pb) _ i-1 E ow, X P{P;;) and k

1P(P' X (P" - Pk)) =i_1 E op; X (P'{ - P"')), (where the terms originating from the empty sets in (1.32) are aero).

Hence

k

by (1.31) : # (P) = E (#(P; X Pi'p') + 4, (P; X (Pi -- Pk )). But since P{' i_1

P'P, + (P' - PP'), we have here by 16.1.1 $'(P; X P:'Pk) + 4'(Pc X (P;' - P't')) = P(pi), and so 16.1.11 is proved. In general, the system '3 is not a field 87 But now we form the system c3+ of all sets which are sums of finitely many sets of 3, and we prove: 16.1.12. '43+ is afield. We have only to show that Ql e 93+ and QE a SJ3+ imply Ql - Q2 a 93+. Thus k

1

let Q, = S P1,i (with P1,i a ,3) and Q2 = S P2,i (with Ps,, a i_i

i_1

3).

Then

Q1-Q2 =i_lS (P1,i- SPs,i1 j..1 an index i, such that a e Pi = P; X Pi ; thus we have also a e (Pi X PiiPZ), that is, P' X Pk k

S (Pi X P;PZ).-The second equation (1.32) may be handled analogously. 87 Example: Let a' consist of the sets .4, B, A + B, A (with A 0 A, B ,E A, and AB = A). Then (A X A) + (B X B) ti e 13, and hence $ is not a field.

226

[CHAP. lv

INTEGRATION

and, since 13+ is closed with respect to addition, it suffices to show that 1

P1,, - S P2. a

(1.4)

;_1

+

We prove (1.4) by induction. According to (1) the relation (1.4) is true for 1 = 1. Thus we assume (1.4) to be valid for l and show that then (1.4) holds also for l + 1. For 1 (1.4) says: Pi.i - S P2.! = S P, (with P, a -3). Now f`I

;_1

I+1

let P2,1+1 a I$; then P1,i - S P2,; = S (P, - P2,1+1) and this sum belongs to i_l

, 1

V+, since by (1) every term P, - P21 1+1 a 3+ . 16.1.121. Every set Q e $+ can be represented as a sum of finitely many disjoint sets of $. 0

Let Q = S Pi (with Pi e V. For m = 1 the contention is trivial. We k

assume the proposition to be true for m, that is, Q = S P* where the P* e i-1 are disjoint. Then we prove that the contention holds also for m + 1. Thus k

,w+1

k

let Pm+1 a 13; then S Pi = S Pi + (P- +1 - S P i i_1 k

\

).

Therefore it suffices to

i.1

show that Pm+1 - S P* is a sum of finitely many disjoint sets of 13. By (1) i-1

this is true for k = 1. We assume it for k and show that it then holds also fork + 1. Thus let P.+1 - S pi = S P*t where the P** a 3 are disjoint, and 1

It

i_1

k+1

let P +1 a

i-1 1

Then P.+1 - S P* = S (P** - P +1). In the latter sum all

till i-i terms are disjoint and by (1) every difference P** - Pk+l is a sum of disjoint sets k+1

of 13; hence P,+1 -- S P` is also a sum of finitely many disjoint sets of 13. i.-l

So the proposition is proved by induction. We call the field 13+ also the product field ar of a' and a", and we write:

V+ = c5 = a, X W'.

(1.41)

Now we will extend the set function 4i, which is defined in $, to 13+, sucb that 4, is additive also in 13+ .

Then if Q e $+ is represented as a sum of finitely

many disjoint sets of ¶3 in the form: Q = PI + P2 + . + Pi, we must have 4'(Q) = 4'(P1) + 41(P2) + . + 4 (Pk). This is possible only if the value of the right side is the same for all representations of Q as a sum of finitely many disjoint sets of 63.

Therefore we first prove: 16.1.122. Let QsW - $+ ;if

Q=Pl+P2+...+Pk and Q-

A+P2+...+gI

MULTIPLICATION OF SET FUNCTIONS

6161

227

are two representations of Q as sums of finitely many disjoint sets of V, then

4(P1) + 4(P3) + ... + 41(P,) = 4'(P1) + CA:) + ... + 4'(P,).

Obviously Q - S S P;Pi. If Pi = Ps X P" (P; a ', i_I 1_i

a a") and

P, = Pi X Pr (Pre ', Pi' a a"), then P;P1 = P;Pi X P"P", and thus, since P'P'E and P; Pi a", we have c P;Pi a' . Hence the PiP; are disjoint sets of .93. Since Pi = Si_iP;P3 and k

k

I

Pi = S PiPi, we have by 16.1.11 4'(Pi) _ E 4'(PiP,) and ¢(P) k

k

E ,4(PiPi),

1

t

and thus E ¢(P:) - E E P(P,P1) Now we extend the definition of the function ¢ from to ¶+ by the stipula+ Pk is a representation of Q as a sum tion: if Q e 93+ and Q - Pt + P: + of finitely many disjoint set of j3, then let 4'(Q) = CPO) + 4'(P2) + ... + 4'(P2). We will show that 4, is totally additive in 10+ . For that purpose we first strengthen 16.1.11 to : 16.1.13. 4' is totally additive in 3. (1.5)

We have to show: If P e J3 and P = SPi where the Pi are countably many i

disjoint sets of 13, then ¢(P) = E ,'(Pi). Since 3+ is a field and P e i P, + . + P k a $+ , we have also P - (Pi + . + Pk) a $+ ; and since every set of 3.,. is a sum of finitely many disjoint sets of J3, we obtain P = P2 + ... + Pk + P; + ... + P1 , P, are disjoint sets of 1$. Thus by 16.1.11 , Pk , P1 , ,k(P) _ 4 ,(PI) + . + $(P,) + 4'(Pi) + + ¢(Pl) ; therefore 1G(P) 4(Pi) + . + *(Pk), and hence:

where the P1

(1.51)

¢(P) ? ; (Pi).

Let P = P' X P"(P a a', P" e s") and Pi We set gi(x') _ v'(P') for x' a P'i and gi(x')

P; X Pi'(Pi e '',. Pi o f"). 0 for x' a (P' - Pi); more-

over, let fk = gx + g' +

. + gk. We foam the sets D,, a $'(s = 1, 2, .

which result from PIP,'

Pt by replacing 0, 1, 2, zk

,

-, 2k)

k factors P; by P -. P'i

in all possible ways; then P - S D. On Dt. 'the function fk is constant; we designate its value on Dk, by zk,, setting zk, 0 for Dt, = A. Then

4'(P1) + ON + ... + 4'(Pk) = E 0-1 zk,o (D,.). For a fixed x' a P', let P'i. (a = 1, 2, ) be those P,' for which z' 4 Pi. (1.52)

[CHAP. IV

JNTEORATION

228

Then the P'{, are disjoint (since the P; are disjoint) and (since P = P' X P" SP;) we have P" =SP;; . Thus ,p"(P") E fP"(P'i,,) _ E g:,(x') and, since a i g;(x'); that is, for all the other i g;(x') = 0, we obtain: p" (P")

lim f, (x') = P" (P") for every x' e P'.

(1.53)

k

Now let z be any number smallor than e(P"). If DA': a ['designates the sum of all Dk, on which fir. > z, then it follows from (1.52) that

Y'(Pi) + G(Ps) + ... + +P(P*)

(1.54)

z,p'(Dk).

But since z < co"(P"), it results from (1.53) that P' = SDk, and thus, since (P') = lim w (Dk). Therefore, according to (1.54), we

Dk Q Dk+1, by 3.2.2:

k

zSo'(P') and, since this holds for every z < ,p"(P"):

have: E P(Ph)

/

Ek 0(Pk) ? IP (P ). /'(1'") _ 40)-

(1.55)

Now the contention follows from (1.51) and (1.55). 16.1.14. 4, is totally additive in a = $+. We have to show: If Q 93+ and Q = SQ; where the Q; are disjoint sets of 3+ ,

then ¢(Q) _ E *(Q;). Since Q e $+, we have Q = P, + P2 + - - - + Pk r analogously Q; = P,1 + Pa + - + P;k; where the Pi are disjoint sets of -

where the P;1 are disjoint sets of

.

By (1.5) +.(Q) _

f-1

>#(P,); since Pi Q Q k{

and Q = SQ;, we have a l s o (for j = 1, 2, - - - , k) P f = SP jQ; = S S P;Pi1 . i

i

k

Thus by 16.1.13 ,'(P,) _

4'(P,P;1), and hence P(Q) = i-1

k

k

7-1

k,

i

k

But since Q; = 8 P3Q; = S S P;P. , we have herein by (1.5) : E Therefore '(Q)

1-1

k

A:

1-1

G(PP;1},

k;

4,(P At)

'(Q,), as contended.

Now let (5 be the system of all subsets of E = E' X E". Because of 16.1.14 and according to 6.5.2 and 6.5.22, i, can be extended to a regular measure function,p defined in (5 (which in $+ = a = a' X l" coincides with ¢). Moreover, according to §6, 5 we designate the complete cover of a _ a' X " with respect to ,p (that is, the a--field of the ,p-measurable sets) by a. By 6.5.23, a is a a--field complete for io, and p is totally additive in . Now we call this totally additive set function ,p in to the product ,p' X cc" of the totally additive set functions ,p' and ,p", defined in a' and a", respectively.* Thus we obtain the following theorem strengthening 16.1.14: a If, according to 16, 5, one extends the totally additive set functions v' and 9,' defined in `:}'r' and s", respectively, to the measure functions defined in (9' and 49" and associated with m' and v ', respectively, and if one designates these measure functions also by and;o ', then the measure function a itself, defined in. (1, can also be designated by X p".

MULTIPLICATION OF SET FUNCTIONS

§161

16.1.15. If gyp' and rp' are monotone increasing and totally additive in a' and a",

respectively, and if a = a' X a", then P = c X gyp" is monotone increasing and totally additive in . By 6.5.21 we have, moreover: (1.6)

tea

Now before we discuss the set function 4p = w X rp" for arbitrary M e , we consider rp first for the sets of !a, . If the set M C_- E, then we designate by M. (for x e E') the set of all y e E" for which (x, y) a M, and by My (for y e E") the set of all x e E' for which (x, y) a M.

16.1.2. Let M e a, ; then M. a $" for all x e E' and M a a' for all y e E"; moreover, p" (M.) is a 0-measurable function of x on E' and urable function of y on E".

is a rp"-meas-

Since M e & , we have M = SQ; whereQ; e 6 and Q; a Qj}1. Thus M=Q1+(Q,-Q,)+...+(Qi-Q,--1)+...;

since a is a field, (Q; - Q f..,) e a _ $+, and hence by 16.1.121 Q, - Q$_1 is # sum of finitely many disjoint sets of $3. Thus M = SP, where the P. are disjoint sets of $3.

Let P, = P; X P;' (P; a' and P;' a"). Since M = SP., we

have M. = SP' where the vi are those v for which x e P; . Thus since P;' a a" i

and $" is a o-field, we see that M. e a".-Now we define a function g,(x) on E' by g, = w"(P';) on P; and g. = 0 on E' - P; . Since P , ' a a' a n d E' a a', the + g, function g, is 9-measurable on E', and hence by 9.2.2 f, = pi + g2 + and by 10.1.21 lim f, are also .p'-measurable on P. According to the definition

g.,(x) _ E g.(x) = lim f,(x), and

of g,, we have cp"(M:) _ hence w"(Mz) is -measurable on F. Now we assume 82

E' = SE,' where the Ek a 9' are disjoint and the e(Et) are finite; k

(17)

E" = SET where the Ei' a a" are disjoint and the P'(Ei') are finite. e

16.1.21. If (1.7) holds and if M e a, , then (1.8)

' X v"(M) = (E')

f ip'(M.) duo

= (E") f v'(Mv) d4".

First we assume 4 (E') to be finite; then the first integral in (1.8) is proper. We use the same terminology as in the proof of 16.1.2; then if we again set a, If (1.7) is satisfied, then, since {I' and " are a-fields, E' a a' and E" e ll" hold auto-

matically.-Moreover, E - E' X E" - S Ek X El and (Ek X Ei) - so k,1

thus

ordering the E. X E1 into a simple sequence we obtain: If (1.7) holds, then for B the assumption §6 (5.2) is satisfied.

e1

[CHAP. IV

INTEGRATION

it follows from M = SP. by 16.1.15 and (1.6) that

X

F,rp(P,), and hence rp(P,) + ... + ,p(P.) -- v(M). Now because of ,&(P,,) = o (Pa) sv (tea) = (Pa) f ip (Pa) drp - (E) f gx do', and

12.1.6

hence by 12.2.1 co(P,) -l-

-}- yo(P,) _ (E') f f. dip'. But since f,(x) ---> cp"(M.)

f

and ((f,)) is monotone increasing, we have according to 12.1.8: (E') f. d,' -->

(E') f c'(M.) dip'. Thus p(M) - gyp' X p"(M) = (E') f o"(M:) dip', as con-

tended.-Now let yo'(E') = + ao ; then the first integral in (1.8) is improper (§12,7). We set M(EE X E") = M. ; the Mk are disjoint. Since Ek X E" a j,3, we have Ek X E" e a, , and hence by 1.2.3 Mk a {f, also. According to (1.1) M = SM,,; thus by 16.1.15 and (1.6):,p(M) = E p(Mk). But since according r

to (1.7) 0'(E),) is finite, we have, as has been already proved: rp(Mk) _ (Ek) f cP"(Mk.) d,' _ (Ek) f so"(M.) V. Thus P(M) - I (.RA',) f q/'(M:) dy,', and hence by 12.7.2 and 12.7.1: -r(M)

X ,'(M)

(Elf cP"(M.) dip',

as contended. 16.1.22. If (1.7) )olds and if M e &*, then M. a a" for all x e E' and My for all y e E"; moreover, is yo'-meaasurable on E' and F'(M.) is c"-measurable on B". Since M e &s, we have M = DM, , where M. and M, M,+1. Thus s

M. - DM,,. and, since by 16.1.2 M. a W" and j" is a a-field, because of 1.2.4 M-- e a" also.-Now first let O"(E") be finite. Then by 3.1.21 c"(M,.) is also

finite and from M. = DM,. and M,.

M,+1, it follows by 3.2.21 that

w'(M,J -- v"(M.). Thus since by 16.1.2

is w'-measurable, according to 10.1.21 co"(Ms) is also gyp'-measurable: Now let V"(E") _ + co. If we set Ms(Ei' + E ; ' + +E ') = N, , then N, Q N,+i and M. = SN, , and hence 1

by 3.2.2 co"(N,) --> p"(M.). But since according to (1.7) sp"(E7' + ... + Ej) is finite, p"(N,) is c'-measurable, as has been already proved; and thus by 10.1.21 v"(M,) is also r'-measurable. 16.1.221. If (1.7) holds and if M e D,1 , then (1.8) holds. We use the same terminology as in the proof of 16.1.22. First let 4o'(E') and V"(E") be finite. /Setting again c X p" = (p we obtain from M, a , by 16.1.21: These integrals are finite, since v"(M..) rp(M,) = (E') f Now M,. M,+,. implies w"(M,,) z w"(M,+1=), and hence from

MULTIPLICATION OF SET FUNCTIONS

§161

it follows by 12.1.8 that (E') f tP"(M,,) do' --. (E")f tt"(M,) iW; thA is, ,p(M.) - (E') f c1'(M.) d41. Moreover, since M = DM., M.

M,+1, and

tp(M,) is finite, we have by 16.1.15 and 3.2.21 w(M,) --' v(M) also. Therefore

,p(M) _ (E') f 411(M.) dip', as contended.-Now let AE') be finite and

0;'(E") _ + oo. We set M(E' X (Ei' + A' + ... + E;')) - M1 ; then M, a ad and, since 9"(Et' + E;' +

-

Ei') is finite, we have, as it is already

proved: 1p(R1) - (E') f 9"(S1.) do'. Now M1. C M1+1. implies and, since SRI, ° M., we have by 3.2.2,"(ftu)

=i

Thus

1

by 12.1.8:

41'(M.) d4", and hence

(B') f ,e(M1)

(F) f ,'(M.) V.

But since SM1 - M and M1 a R1+1, we have by 16.1.15 and 3.2.2 also

so(M1) -+ ,(M) Thus ip(M) - (E')f ,p"(M.) d,p', as contended.-Then the case 4'($') - + eo is treated in the same way as in the proof of 16.1.21. Now we oonsider any set M belonging to the o-field ;.

16.1.23. If (1.7) holds, if M e 9, and if .p' X ,"(M) .. 0, they p"(M.) -. 0 on A' and

-,,,0onE".

By 6.5.421 there is a measure-cover B of M which belongs to $.e ; then m' X 0"(B) - 0. Thus by 16.1.221 (Y) f e;'(B.) 41 = 0, and hence by 12.3.6 is complete 4v"(B.) -., 0 on P. Now M Q B implies M. Q B. ; thus since for p", from ip"($) - 0 it follows that e,"(M.) _ 0; hence we have also

ye()f,) -., 0 on Jr. 16.1.24. If (1.7) holds"% and if M e , then the sd of all x e E' for which M. is

#"-meastemble =,, E' and the at of all y s E" for which M, is p'-measurable =,,, E"; moreover, rp"(M.) is p'-measurable on E' and 41(M5) is j1'-measurabk on E", and (1.8) holds. By 6.5.421 there is a measure-cover B of M which belongs to o.e . Since

B Q M, we have B. a M, , and B. = M. + (B - M).. By 16.1.22 B. is t'"-measurable for all x e E. Since by 6.2.3 and 6.3.3 gyp' X 9"(B - M) = 0, by 16.1.23 the set of all x for which (B -- M). is e;'-measurable -,, P. Thus the set of all x for which M. is gyp"-measurable -,, E' also, and for all those x we

have w"(B=) - p"(M.) + 4"((B - M).). But according to

16.1.23

e;'((B - M)s) _,. 0, and hence p"(M--) =,, o"(B.). Thus since by 16.1.22 4'(B.) is 4v -measurable on E', V'(M.) is also o -measurable on B1 and by 12.1.9 I" The condition (1.7) is essential here; see S. Sake [21, p. 87-88.

232

INTEGRATION

[CHAP. rq'

and 12.1.52 (Elf qo"(Mr) dv' = (Elf co"(B,) d,/. But according to 16.1.221

(Elf ,p'(B..) dp' =

X w"(B). Thus from ,s' X qo"(B) = or x VII(M) it

follows that ,p' X w"(M) =

(Elf w"(Ms) dw'.

Now let also a third space E"' be given, in E"' se-field {5"' (consisting of subsets of E"'), and a totally additive, monotone increasing set function w"' in a"'.90

The three products (E' X E") X E"', E' X (E" X E"'), and E' X E" X El" can be considered as identical. We designate by $3 the system of all sets M a E' X E" X E"' which have the form M = M' X M" X M" where We a', M" a 15", and M"' a a"'. Then we designate by' 3+ the system of all sets which

are sums of finitely many sets of 3. We have q$+ - (j5' X 15") X IS"' = 15' X (j)" X `j5"') Icf. (1.41)] and, applying 16.1.12 and 16.1.121 twice, we we that $3+ is a field and that every set Q e 13+ can be represented as a sum of finitely many disjoint sets of $3: Q = P: + P2 + ... -i- Pi, . Accordingly we call the field $+ the product field 15 of j5', $r", and 15', and we write:

+=

(1.9)

=$'XW"X9"'.

By means of the totally additive set functions w', qVI, and qp"f defined. in 15', a", and 15"', respectively, we can first form the set functions pf X qV' and so' X o"', which by (1.41) and 16.1.14 are totally additive in (15' X a-") and (15" X a"'), respectively. Then starting here (and applying §6, 5) we can form

the two measure functions (w' X w") X 0.. and w' X (w" X w"') in E' X E" X B"' 91 These two functions are totally additive in `j5 = 15' X 15" X 9,"' by 16.1.14 and also in Vr, by 16.1.15 and (1.6). Now we prove the associative law of the multiplication of set functions. 16.1.3. For all sets A C E' X E" X E"' we have: (1.91)

(w'Xse")X_qV

X

X gyp" = vi, vi X w"' = xl, and o' X c: = xs

For P e $3, that is, for P - P' X P" X P"/(P' a 15', P" a a", P"' a 15"'), we have

according to (1.2): ,pl(P' X P") -
.I(P) = OI(P' X 1 "),c (p"') = w'(P') ,c"(P") ,w",(P",) and X P",) = V(P) .,pi,(P") .p"(P'l; x:(P) = thus xi(P) = x:(P); that is, (1.91) holds for the sets P e $3.-If now Q * 15 _ 15' X a" X 15"', then Q - P, + P2 + . + P,, where the P; are disjoint sets M Cf. footnote 83, p. 223; accordingly it suffices here to assume `5', a", and a"' to be only

folds (instead of o-fields).

11 Of. footnote 88, p. 228.

§161

MULTIPLICATION OF SW FUNCTIONS

of 3. According to (1.5) xi(Q) = x,(P,) + xi(P2) + x2(Q) = x2(Pi) + x2(P2) +

.

233

+ xi(Pk) and

+ x2(Pk); thus from x,(P;) _ X2(P;) it follows

that XI(Q) = x2(Q); that is, (1.91) holds also for the sets Q e a.---Since (40' X v") X p " and So' X (p" X se"') are totally additive in &, it follows now that (1.91) holds also for all B e W. , and then from §6 (5.1) we obtain the E' X E" X E"'. result that (1.91) is also valid for all A Since therefore the two measure functions (rp' X w") X v"' and .e' X (cp" X "),

defined in E' X E" X E"', coincide, we can designate them simply by w X ,p" X c ". Now the complete cover to of a(= 5' X a" X s"') with respect to gyp' X fp" X p"' is also uniquely determined and by 6.5.23 yo' X c' X tP`" is

totally additive in a. As an example for the above theory, we consider the space R,.+,, as the product space R. X R and prove (cf. §8, 2) :

16.1.4. u.+.=/r XA.. The u-field of the M. X W.-measurable sets is complete and contains all closed intervals I of Rm+n , and for these obviously A. X p.(I) = pm+.(I). Thus by

8.2.4 and 8.2.41 every µ.+« measurable set M is also p.. X p.-measurable and A. X pn(M) = pm+n(M). In order to show conversely that every A. X An-measurable set M is also p,.+.-measurable, by 6.5.5 and (1.5) one has only to prove:

every set of the form P = P' X P"(P s R. and P" C R.), where P is then for these sets P we have pm+.(P) = p.(P') p.(P"). By 8.2.5, to P and P" there are p.-measurable and P" is p.-measurable, is

measure-covers B' and B" which are Gs-sets and measure-kernels C' and C"

which are F,-sets; then, according to 6.2.3 and 6.3.3, p.(B' - C') = 0, and p.(B" -- C") = 0. If we set (B' X B") - (Cl X C") = D and consider that

also D = ((B' - C') X B") + (B' X (B" - C")), then it results that Am X pn(D) = 0. Now D is a Borel set in R.+. thus by 8.2.21 D is p.+.-

'

0 implies A. X D, also p.+.(P - (C' X C")) = 0,

meas-urable, and hence, as it has already been shown,

p.+.(D) = 0. Thus since P - (C' X C")

and therefore, since C' X C" as an F,-set (cf. footnote 92) is p,.+.-measurable, P is also p.+,; measurable, as contended. Thus it is proved that the A.+. -measurable sets coincide with the µ.. X p, measurable sets and that for them p..+.(M) = p.. X µ.(M) . But, more generally, 16.1.4 holds also for the associated measure functions Am, An , and p.+. (§6, 5 and §8, 2). For by 8.2.1, 7.2.1, 6.5.23, and footnote 88, p. 228, p,.+. and P. X A. are regular measure functions (§6, 4); that is, for every A C R.+. we " First, since C' and C" are F,-seta in R. and R., respectively, C' X C' is also an F. in R.+.. For Cl m SA. and C" - SA: , where the A, and .4; are closed sets in R. and R. , respectively, with A' r. A;,+, and A: C A:+, ; then Cl X C" a S(A: X A,) where the

sets A' X A; obviously are closed. One sees analogously that B' X B" is a (Ja-,set in R.+.. Thus D - (B' )( B") - (Cl X C") is a Borel set in Re" (cf. p. 82), since theseBorel sets form a v-field containing the F,-seta and Go-sets.

234

[CHAP. IV

INTEGRATION

have p.+.(A) = inf p.+.(M) and µ. X M

A

inf A. X w,(M) where M

XaA

runs through the u.+..-measurable sets or, what amounts to the same thing as we have proved above, through the p. X p,.-measurable sets. Thus since for these sets M p.+.(M) = is. X p.,(M), we have also ,+..+.(A) - A. X p.,(A). BiaLIOanAPHY: H. HAHN, Annali Scuola Norm. di Piaa (2) (1933), p. 249; S. SAKS [11, p. 257; 121, p. 82; F. MAKDA, T6hoku Journ. 37 (1933), p. 446; Z. LOMNICKI and S. ULAM,

Fund. math. 23 (1934), p. 237; W. FELLER, Bull. internat. Acad. Yougoslave 28 (1934), p. 41; J. voN NeUMANN [i1, p. 105, 221; J. RIDDER, Fund. math. 24 (1935), p. 87, 112; Nieuw

Archief v. Wiskunde 19 (1936), p. 31; Proc. Nederl. Akad. Wetensch. 49 (1946), p. 172; 0. HAUPT [11, vol. III, p. 87; C. CARATH*oDoRY, Annali Scuola Norm. di Pisa (2) 8 (1939), p. 129; B. JESSEN, Mat. Tidsekrift B 1939, p. 7.

2. Product of non-monotone set functions. In No.1 the two totally additive set functions .p' and ,p" were assumed to be monotone increasing. Now we generalize that theory to the case where .p and re" are not monotone increasing. All the other assumptions of No. I are to be valid. According to 3.4.61 we have for M' t W' and M" a a": V..-(M"), P,+(M') - w (M') and P "(M") c

and, by 3.4.3 and 3.4.32, the cp'+,

and v r are monotone increasing and totally additive in a' and a", respectively, such that the theory of No. 1 can be v,+ applied to them. Thus the set functions 0'+ X X o" or- X 4'"'+, and ip' X (P I'- can be formed; each of them is monotone increasing and totally additive in a c-field containing 9- =a' X 5" [cf. (1.41) ]. We designate by D2 the intersection of these four c-fields; then 0 is also a c-field containing {F. Now we assume: Among the four numbers +(Li i) c i+(E"), cp' (E') . we" (E"),

(2)

_ ,?+(E?) .q/'-(E"),

and -lo'-(E') 9p"+(E")9S no two are infinite of different signs. This is the case if and only if one of the following conditions is satisfied:

9 (E') and p'(E") are finite. Both p and p' are monotone. (2c) At least one of the two functions a and se" is both finite and monotone. For it follows immediately from 3.4.13, 3.4.4, and 3.4.12 that each of the (2a) (2b)

conditions (2a), (2b), and (2c) implies (2). If conversely (2) is satisfied, then from 3.4.41 it results that 1) either ,p' is monotone or V"+(E") and are not + cc , and that 2) either rp" is monotone or rp'+(E') and ,p'-(E') are not + ao. Thus, because of 3.4.13, one of the conditions (2a), (2b), (2c) must be satisfied. a If in one of these products one factor is 0 and the other is .o, then we replace the

product by 0.

235

MULTIPLICATION OF SET FUNCTIONS

§161

Under the assumption (2) we can form the set function ((,'+ X o"-) + (0- X "+)) (2.1) q' X e' _ ((,p + X o"+) + (i X ,s"-)) --

for all M e ? and by 3.2.52 we obtain: X gyp" is totally additive in the o-field W. 16.2.1. Under the assumption (2) 16.2.2. Under the assumptions (2) and (1.7) we have for all M e 9:

XXc'+)+('P-Xso'l

(gyp

and

(2.2)

(cv XP') = (.P,+XP'-)+(cp-X0"+). First we discuss the first equality (2.2). 3.4.4 we have for M e 912: (2.21)

(cl X ce")+

Because of (2.1), 3.4.51, 3.4.5, and

(9'+ X c"+) + ('p'

X 0o9.

Thus according to §3 (4.1) we have only to show: there is an M'` a MID, such that i' X l"(M") = ce'+ X cp'+(M) +'P - X P"-(M). Now first let M e'3, and hence M = M' X M" with M' a j' and M" a a". By (2.22)

3.4.71 there are decompositions M' = M'+ + M'` and M" = M"+ + M" (with M'+ a a', M'- a a' and Ma s", M" a a"), such that M'+M' _= A, M"+M'r = A, rp'(M'+) = 0,+(M'), -c (M'-) = 9" (M'), ip"(M"+) = 0"+(M"),

and --cp'(M'9 = 9"-(M"). We set M' = (M'+ X M"l + (M'- X M"-).

qv(M,+)V'(M"+) Then by (1.5) : cp' X co'(M*) = = + +(M')o"+(M") + (M')9p'-(M") = ,p + X ip"+(M) + (p- X c r(M); thus 9 (2.22) is proved for M e 13.-Now let M e a,. Then (cf. the proof of 16.1.2) M = SP, where the P, are disjoint sets of J3. Thus by 16.2.1 and 3.4.32

(so X

X 'p')+(P,), and hence, as we have already proved:

W X "')}(M) _ E 01+ x c"+(PI) + E c

X 9"-(P,) P,+ X P"+(M) + e- X

gyp'.-(M).

So the first equality (2.2) is proved for M e &, and in the same way one proves also (2.22) for Me a.,-Now let M e fit with finite p' X ip"(M). Then by (2.1) X tp"-(M), v'+ X cp"-(M), and o' X ,p"+(M) are also finite. (P'+ X

Thus by 6.5.4 to every e > 0 there are sets B; a a, (i = 1, 2, 3, 4), such that

Bi 2 M and ,p'+ X o'+(BI) < o+ X 9"+(M) + e, o'- X ,p'- X v- (M) + e, vP + X cl' (Ba) < o + X o -(M) + e, v X

< <

0 X 'p"+(M) + e. If we set B1B,B3B4 - B, then by 1.2.3 we have also B e B;_D M, and (p'+

X V"+(B) < o + X o"+(M) + E, or- X 0"-(B) < or- X ,p"-(M) + e, rp+ X 91' (B) < V'+ X 9' (M) + E, 0 X V'+(B) < o!- X 4p"+(M) + e,

[CHAP. IV

INTEGRATION

236

and hence by 3.1.11 w + X gyp"+(B - M) < e, o- X so"-(B - M) < e, 0 + XP"-(B - M) < E, and pr X sp'+(B - M) < E,;r Thus by (2.1) for all A e (B - M) 1'1)2: I 'P X cp'(A) t < 2e. Since B e too , there is a B* E B112 :according to (2.22), as we have already shown, such that. Ip' X pp"(B*) o'+ X ' + (B) -{- rp'` X cp`(B) . Now we set M* = MB*; then since B* - M* = B*(B - Al), we have:

V' X :p'(M) _

X ac'(B*) - X o"(B* - M*) > 4' X so"(B*) - 2e = ,p'+ X V,+(B) +,p'- X ce" (B) - 2E; thus, since Af C If, also: X P"(df*) > fp'+ X ip"+(M) + X p' (M) - 2E. Therefore, since 111

Jl*, by §3 (4.1):

X ip ' (M). (cp X o")`(M) a ,o'+ X P"+(M) + This together with (2.21) furnishes the first equality (2.2) for M e 9 with finite Then from (2.1) we obtain immediately also the second equality oX X gyp" =_ (gyp' X p')+ - (cc X gyp') .-Finally let M be (2.2), since by 3.4.61 any set of V. The assumption (1.7) implies according to footnote 89, p. 229 that

Al = SM. where the Al, e 731 are disjoint and the c X cp"(M,) are finite. Then, as we have already proved, both equalities (2.2) hold for every M, , and

thus, since the functions appearing there are totally additive by 3.4.32 and 3.2.52, also for M. 16.2.21. Under the assumptions (2) and (1.7) the or-field 11)2 is complete (14,4) X cp'. for

We have to show: if M e T2 with ip' X rp'(M) = 0 and if A CA then A e 1J also.

By §3 (4.2) and 116.2.2: V'+ X c"+(111) = 0, gyp,- X tp"-(M) = 0,

+ X V'-(:LM) = 0, and p'- X p"+(M) = 0. But since D2 is complete for

,p,+ X p,.+, for

- X p'-, for c + X 0"-, and for rp` X gyp"+, A is p'+ X

measurable, c'- X p`-measurable, measurable; thus A E;.

rp'+

X cp"--measurable, and p'- X

(p"+-

16.2.3. L1,uier the assumption (2), theorem 16.1.24 holds for M e 51,l (instead of for Af e t) alRo in the case of non-monotone p' and V!'. Since M is + X (p'+-measLet O' be 0e set of all x e Efor which M. F. urable, by 16.1.74 C' =,- E'; analogously, since M is - X cp"+-measurable, C' =m.- E'; thus by 4.1.11 C` =,p E'. Moreover, by 16.1.24 cp"+(Mx) and. are both (p'{-measurable and cp'-"-measurable, and hence also cp'-measur;:able on E'. Thus, according to 3.4.61 and 9.2.2, cp'(Alx) is also c -measn"_

arable on E'." Finally by 16.1.24 rp'+ X p"+(M) _ (E') f p'+(M.) dtp'+,

X w ,-01) = (E,) f rp

(Mx) dcp'

p'

X

gyp"-(M)

= (V) f q," (M.) dp+,

11 Hence this first part of the proposition holds also without the assumption (2).

287

MULTIPLICATION OF SET FUNCTIONS

§16]

and oT X w'+(M) = .(P) f So"+(Mz) dip'-'. Thus by (2.1) because of (2), 12.2.11, 3.4.61, 12.2.22, and 12.2.23 we have:

so X "(M) (E')

f (O"+(M) - O`(M=)) dd'+

- (E') f (q'+(BIz)

-O'`.(M=))

= (E') f

dip'-

c"(M..) dyo'.

BIBLIOGRAPHY: The multiplication of non-monotone, but finite p' and q " was discussed by J. RIDDER, boc. Cit. (cf. No. I].

3. Geometric interpretation of the integral. Let 0 be totally additive in the complete for gyp, let A e a, and let O (A) be finite; moreover, let f be a nonnegative function .p-defined on A. We form the product R, X A of the points (z, a) (with z e R, , a e A). To every a e A with finite f(a) we form the set F(a) of all (z, a) with 0 ;s z s f(a), and to every a e A with f(a) = + ao we form the set F(a) of all (z, a) with z z 0. The sum of all these sets F(a) is called the ordinate set Zf of the function fonA. 16.3.1. If ,p is monotone96 and if f is So-defined and noon-negative on A, then in order that f be gyp-measurable on A, it is necessary and sufficient that the ordinate set Zf be µ, X c-measurable. NECF.ssITY: Let f be c-measurable on A. The contention is obviously true if f assumes only finitely many different values. But if f assumes infinitely many o-field

values, then we define AI

by f, =

a

on

f(i) <

,1

2m]

A[fl(d)

and f = m on

1

m].

(i = 1, 2, ... , rn 2')

't'hus ((f,,)) is monotone increasing and f -- f.

Since f, assumes only finitely many values, the ordinate set Z f,, is µ, X O-measurable. Therefore, since Zf = SZ f,, , Z f is also p, X gyp-measurable. SUFFICIENCY : EEO

Let Zf be p, X O-measurable. We designate by (Zf), the set of all z e A for which Cr, x) e Zf ; then (.Z,)z = A[f(x) z z).

Thus according to 9.1.21 we have

to show that (Z1), is c-measurable for all z e R, . By 1.6.1.24 the set of all z e R, for which (Z f), is c-measurable =,,, R, . Therefore to every z there is an z, such that (Zf),, is O-measurable. But increasing sequence ((z.)) with z, since (Zf),, = A[f(I) ? z,) and(Zf), = Aff(t) L> z], we have (Zf), = D(Zf),, , and hence (Zf), is also rc-measurable. ' The case in wbich v is monotone decreasing can immediately be reduced to the case of the monotone increasing -c; for by (2.1) and i3 (4) p, X qo : -(p, X (-0) -

238

(CHAP. IV

INTEGRATION

16.3.11. If f is bounded, then theorem 16.3.1 holds also for non-monotone p.

For then, according to (2c) and (2.1), one can form µI X V. Furthermore in the proof of 16.3.1 one has only to use 16.2.3 instead of 16.1.24. 16.3.2. If v is monotone and if f is c*-measurable and non-negative on A, then

(A) f f dip = pI X ,(Z,).

(3)

By 12.1.61 the contention is true, if f assumes only finitely many values. Now if f assumes infinitely many values, we again introduce the function f., , as in

the proof of 16.3.1; then we have (A) f f,. dip = /h X V(Z,, ). Since ((f..)) is monotone increasing and f,. -* f, by 12.1.8 and 12.1.81 (A) f f., dsp -* (A) f f dsp.

Since the sequence ((Z,.)) is monotone increasing and Zf = SZf,, , by 16.3.1 µI X cp(Zf) also. Thus (3) is proved. 16.3.21. If f is bounded, then theorem 16.3.2 holds also for non-monotone p.

and 3.2.2 it, X

For then, according to (2c) and (2.1), one can form IAI X sp. by 12.3.52, 12.2.23, 16.3.2, and (2.1) : (A) f f dcp = (A) f f dsp+ JAI X lp+(Zf)

Furthermore,

- (A) f f dp =

- µ1 X (P (Zf) = )At X IP(Zf)

The above theorems hold also for improper integrals (112, 7); that is, the assumption that p(A) be finite can here always be replaced by the more general assumption that A = SA., where the A are disjoint and the ip(A.,) are finite. 12

The proofs of 16.3.1 and 16.3.11 remain forthwith valid. The theorems 16.3.2 and 16.3.21 can be generalized by means of 12.7.2, 12.7.1, and 16.1.15; hereby for 16.3.21 the following has to be taken into consideration: if one forms the

decomposition A = A' + A" according to §12 (1.1), then by 3.1.23 only one of the two values p(A') and ip(A") can be infinite. BIBLIOGRAPHY: H. LEBESGUE, Paris These (1902), p. 18 - Annali di mat. (3) 7 (1902), p. 248; [11, p. 45, 116; C. CARATHAODORY [11, p. 420; J. voN NEuMANN [11, p. 197; J. RIDDER,

Fund. math. 24 (1935), p. 72; H. KEerELMAN [1], p. 113; A. J. WARD, Amer. Journ. of Math. 66 (1944), p. 144.

4. Fubini's theorem. Again let E' and E" be any two sets; let a' and J;" be two c-fields consisting of subsets of E' and E", respectively, and let E' a a' and E" a a"; let (p' be totally additive in j5' and let qp" be totally additive in a"; let ' be complete for and let j5" be complete for rp"; and let (p'(E') and rp"(E") be finite. Then the assumption (2) is satisfied and by (2.1) the totally additive set function q X so" can be formed in a c-field 'lR consisting of subsets of E' X E". Now we have Fubini's theorem:

MULTIPLICATION OF SET FUNCTIONS

239

16.4.1. If M is X p '-measurable and f is rp' X 'e '-integrable on M, then the set of all x e E' for which on M. f is V"-integrable =,. E' and the Bet of all y e E" for which on M, f is p'-integrable =,. E"; moreover, (M.) f f do" is ,p'-integrable

on E' and (M,) f f dip' is (p"-integrable on E", and (4)

(M) f f d(,p' X c') = (E')

f ((Mi) f f dco") dp' J

(E") f ((M.,) f f (w) do" First we assume cP and s' to be monotone increasing and f to be non-negative on M. Then if we designate by Zr the ordinate set of f on M, we have by 16.3.2:

(M) f f d(,p' X p') = A, X (gp' X go")(7.r); according to 16.1.3, 16.3.1, and 16.1.24, this can be written also in the following form:

(4.1) (M) f f d(,P X go") = (µ, X go) X p'(Zf) = (E") f u, X where (Z1), for every y e E" designates the set of all triples (z, x, y) a Zr (with z e R, and x e E') whose third element is y; that is, (Z1), is the ordinate set of f on M,. By 16.1.24 the set of all y e E" for which M,, is -measurable =,. E";

again by 16.1.24 the set of all y e E" for which (Z,), is µ, X p'-measurable =,. E"; thus by 16.3.1 and 12.1.9 the set of all y e E" for which f is g -integrable on M,,, also =,. E" and by 16.3.2 we have for these y: µ, X g ((Z,),) = (M,) f f dcp'. Substituting this into (4.1) one obtains the second equation

of (4). One may derive the first equation of (4) analogously.-Now let f a 0, but let g and v" be not monotone increasing. By 12.2.23

f

(M) f f d(v' X go") _ (M) f f d(se' X p')+ - (M) f d(co X gp'Y by 16.2.2

(to x rp")+ = (so+ X P"+) + (so' X e-)

and (rp' X v")- = (g'+ X tp"+) + (0

X (p"+); thus by 12.1.9 and 12.2.2

f

f

(M) f fd(so' X 011) _ (M) f d(go + X s'+) + (M) f d(tp'- X ql" )

-- (M) f f d(,p'+ Hence, according to the above proof,

X p") - (M) f f d(ce'' X

go"+).

(CHAP. IV

INTEGRATION

240

(M) fi d(d Xsp") _ (E") f

((M)ffd4

+ (E,") .l ((Mv)f f

a1

dp"-

d*p'-)

ii+

-

(E'") ( ((M)ffd') do

- (E") i (M.) f f dip') Therefore by 12.2.22, 12.2.23, and 12.2.11:

(M)! .f d(,p' X gyp') = (E")

f

dv"

- (E") f ((M)f f dW,

(4.2)

= (E") f ((I)! f dc'+ - (Mo f f f dp'+ --

Herein

f f dip __./

f

dsp'--)

) dip

k"

is o"-defined on E"; thus, except for

the y belonging to a zero-set for p", these two integrals exist and are not infinite of the same sign. Therefore by 12.2.22 the set of all y e E" for which on Mp f is ' integrable =,,. E", and hence by 12.2.23 we have for all these y:

f

(M.) f f 4'+ - (Mb) f di' _ (Mr) f f V. Substituting this into (4.2) one obtains the second equation of (4). One may derive the first equation of (4) analogously.-Finally let f be a X gyp'-integrable function of arbitrary sign. We set M1 = M(f (x) ? 01 and M2 = MUM < 01;

then (M) f f d(,p' X p") = (Ml) f f d(cp' X v") + (M2) f f d(cp X c "). Furthermore we set ,f, (x) = max (f (x), 0) and f2(x) = -min (f (x), 0) ; then (111,) f f d (,p' X p") = (M) f f, d(,p' X v") and -- (1112) f f d(,p' X p") 7= (M) f ,f2 d(v X sp'), and hence (M) f f d(,p' X p") =

(M) f fi d(.p X v") -- (M) f f22 d(jp' X P"). As we already have proved,

(M) f f; d(w X v") _ (E") f ((M)f fd') dip" (i = 1, 2), and thus by 12.2.11: (4.3)

(M) f f d(,p' X cP") = (E") f ((Mr) f f, dip' - (M.) f f2 dcp) dcp".

§161

Hence

MULTIPLICATION OF SET FUNCTIONS

241

((M)f f, dip' - (Mv) f f2 dcp') is i'-defined, and thus, again by 12.2.11,

we have, except for the y belonging to a zero-set for ip": (4.4)

(Mv) f ff dip' - (Mv) f fs di' = (M.,) f (fi - fs) (14/ _ (Mr) f

f V.

(This gives also the following result: the set of all y e E" for which on My f is ip'-integrable =,.. E".) Substituting (4.4) into (4.3) one obtains the second equation of (4). One may derive the first equation of (4) analogously. BIBLIOGRAPHY: Theorem 16.4.1, for p' s ,p' a pt , is due to G. FUBINI, Rendic. Accad. Lincei (5) 161 (1907), p. 608 (who started from H. LEBESauE, Paris These (1902), p. 46 Annali di mat. (3) 7 (1902), p. 276). As to corresponding previous investigations referring

to Riemann integrals and as to other proofs, valid also for c _ v' - pt (or for 'R' a p,,, and of Fubini'e theorem, cf. the bibliography in Encykiop. d. Math. Wise. IIA2 w' m (A. Voss), p. 104, and 11C9b (P. MoNTEI,-A. ROSENTHAL), p. 1117. For more general ,p' and yo" cf. the bibliography to No. 1.

5. Iterated integrals. The inverse of 16.1.24 does not hold: there are sets M

in E = E' X E", such that M. is ip"-measurable for every x e E' and My is ip'-measurable for every y e E", while M is not cp X 0"-measurable. An example of this is furnished by the following theorem: 16.5.1. There is a non-p2-measurable point set A in the interval I = [0, 0; 1, 1] of R2 which is met by every line parallel to the x-axis and by every line parallel to the y-axis in at most one point.

The set of all closed sets F C. I has the power K of the continuum" and the same is true for the set of all closed intervals I' Q I ; thus" the set J3 of all closed sets P C; I with pt(P) > 0 has also the power K. According to the well-ordering theorem K = bta , and hence there is a one-to-one mapping of all P e $ on the ordinal numbers < wa , where wa is the first number of the class Za"; thus all the sets P e J3 can be designated by Pt (with k < wa). Since the set of all points p e 1

has also the power K, these points p can be designated by pE (with Z < wa). Now we form a set A of points qE (with k < wa) by transfinite induction according to the following rule: 1.) Let qe be an arbitrary point of Po. 2.) Let qt = (xE, yE)

be defined for all t < y'(< c,) and let Q. be the set of these qE (for i; <

By

16.1.4 p! = pi X pI , and hence by 16.1.24 ps(PE.) = (RI) f ,1(P1.,) dA, . Thus

since ps(PE.) > 0, for the set X of all x e RI for which pt(Pe..) > 0 we have by 12.1.52: AI(X) > 0. Therefore, according to 7.2.3, 7.2.32, and 6.2.3, X contains " For instance, cf. H. Hahn [21, theorem 18.2.51, or F. Hausdorff 12), p. 128. 97 Cf. H. Hahn [2], theorem 8.2.1, or F. Hausdorff [2], p. 27. 91 E. Zermelo, Math. Annalen 69 (1904), p.514; 65 (1908), p. 107; cf. H. Hahn [2], p. 39, or F. Hausdorff [2], p. 56. 99 For instance, ef. H. Hahn [2], p. 43, or F. Hausdorff [2], p. 70.

242

[CHAP. Iv

INTEGRATION

a (non-empty) perfect set; but such a perfect set has the power i;`l,i0° and hence' X has also the power Dt = 1!t.. Since ¢* < We., the set Q. has a power smaller

than M., and thus there is an x different from all xE (with Z < *), such that µi(PE.x) > 0. Hence, for the same reason, PE.,, has the power K. , and thus there is a point (x, y) whose y is different from all ye (with t < f*). So in there is a point q E PE. which does not lie on the same line parallel to the x-axis or on the same line parallel to the y-axis as any one of the points qt (with l: < E*) ; we choose such a point as qt.. In this way the set A of the points qe (for < wa) is defined by transfinite induction. According to its definition, the set A is met

by every line parallel to the x-axis or to the y-axis in at most one point and APE * A for all < we, . We still have to show that A is non-µ2-measurable. Certainly µ2(A) = 0 is not possible; for then we would have µ2(I - A) = 1, and hence by 7.2.3 and 6.2.3 there would be a PE I - A, contrary to APE $ A. Now if A was µ2-measurable, then, since p,(A) > 0, again by 7.2.3 and 6.2.3 there would be a PE C A. This PE would be met by every line parallel to the y-axis in at most one point; thus pi(PEX) = 0 for all x e Ri , and hence

(Ri) f pi(PEx) dpi = 0; that is, by 16.1.24 we would have pt(PE) = 0, while p2(PE) > 0. Now we see that the inverse of 16.4.1 does not hold either: We again set I = (0, 0; 1, 1]. Let f = 1 on the set A of 16.5.1 and f = 0 otherwise. Since

f

every set A. and every set A,, consists of at most one point, we have (Ix) f dpi = i

r

f f(x, y) dy = 0 for all x e [0, 1) and

f f dpi = f f(x, y) dx = 0 for all

y e [0, 1], and hence

ki (ffx, y)

(5)

dy\

) dx = f i (f i f(x, y) dx) dy,

while f is non-p2-measurable on I, since A is non-p2-measurable. One can give also examples of functions f(x, y) for which (5) holds and which are prmeasurable on I, but not p2-integrable (cf. footnote 101, p. 243).

If f(x, y) is p2-integrable in I = [0, 0; 1, 1], then it follows from 16.4.1 that

the two integrals in (5) exist and are equal (specifically, both

= (I)

f f dµ2).

Now we give both an example of a p -measurable, but non-printegrable function on I for which the two integrals in (5) exist, but have different values, and an example of a P4-measurable, but non-t-integrable function on I for which one of the two integrals in (5) exists, but not the other.

Let f(x, y) = a, in , = 1 - 21i , 1 -

1

; 1 - 2' ,

i-

2'

,

f(x, y) = b,

inf.a 1- ,1- i;1-2.+i2.>,(v== L

1

21

1

1

1

1, 2, . . . ), and f(x, y)

G. Cantor, Acta math. 4 (1884), p. 381; of. H. Hahn [21, theorem 19.3.1.

0

243

MULTIPLICATION OF SET FUNCTIONS

First we determine the a, and 6, from at = 2, b, = -2a,, and f(x, y) dx = 0 for 0 5 y < 1 a,+, = 2'' - 2b, (v = 1, 2, ). Then otherwise.

JO

and f

1

x < 1, and hence f 1 (f 1 f (x, y) dx

f (x, y) dy = I for 0

dy = 0 and

ui f 1(f f(x, y) dy ) dx = 1.

Now we determine the a, and b, from a, = 2, b, -2a, , and a,+, = i (-1)'22r+' - 2b, (v = 1, 2, ). Then f f(x, y) dx = 0 for 0 y < land 0

flf(x,y)dy=(-1)'-'2'-'forI

Hence

f '(f f(x, y) dx) dy = 0, while by 12.1.4 and 12.1.3 f (f' f(x, y) dy) dx does not exist; for if we set

Si -

1-

2, 1 - 22 ') = A and

(f f (x, t1) dy) dx = 2 E (-1)21-2

B, then

(A) f

and

(B)f (f 1 Ax, y) dy) dx = 2

1=

_l.

(-1):,-i m

+ 00

-

.

Yet we have the following theorem, if for E', E", ', and io" again the assumptions of No. 4 are satisfied: 16.5.2. Let E' be a separable metric space and assume the open sets in E' to be i -measurable; let f(x, y) be defined on E' X E"; for every y e E" let f(x, y) be a bounded and .p'-continuous function of x on E' and for every x s E' let f(x, y) be a v"-measurable function of y on E"; moreover, let I f(x, y) 1 S g(y) where g(y) is e"-summable on E".

Then (E") f f (x, y) d,p" is also a bounded and rp'-continuous

function of x on E' and102: 104 Set f*(x, y) a f(z, y) + f(y, x) where f(x, y) is the function just defined. Then for f"(z, y) the equation (5) holds and f(x, y) is µ,-measurable on I, but non-14-integrable. The latter statement is shown to be true in the following way: As one proves by induction,

a, - 2'(2' - 1); moreover,,e(f,) - 2, and µ,(1') + 81,'- B, we have by 12.2.5: (A)

b, 2y+x- -E a,

fjdv: - 2 E a,

2z'

22;+1 .

Thus if we set Sf, - A and

22, 2' 1

+ ac and (B) f f' dpz -

Hence, by 12.1.4 and 12.1.5, f is non-g-integrable on I.

Theorem 16.5.2 still holds essentially, if one replaces x e E' and y e E' by z e(E' - N')

and ye(E' - N'), respectively, where N' and N' are sero-sets fore' and ye', respectively. For then the contention of the theorem first holds for E' - N' and E' - N' (instead of B' and E'); but, because of 12.1.53, we obtain (5.1) at once again for E' and E', while

(E')

f f (z, y) 4' is bounded and w'-continuous at least on E' - N'.

(5.1)

Let

[cHA.P. Iv

INTEGRATION

244

(E,) k Elf) f f dip") d ' = (E") f'((E') f f dp ) be a distinguished (sequence of decompositions (§13, 3) of P.

If

Z, is the decomposition E' = SEi , then we have by 13.3.2, however the point x,, a E;i may be chosen: (5.2)

lim E f(x,i , y),v'(E:;) = (E') Jf(x, y) d,p'. i P

Herein (5.21)

r

Ef(x,i, Or'(Eyi) = Jim E f(x,i, t_i k

Since by assumption f(x,i, y) is (p"-measurable on E" and I f(x,i, y) 15 g(y) where g(y) is a"-summable on E", by 12.3.51 f(x,i , y) is also v"-summable on E", k

and hence by 12.3.4 and 12.3.43 E f(x,i, y),p (E;i) is also (P"-summable on Elf. ;_t

Moreover, by §3 (4.21) and 3.4.31 (5.22)

E f (x,i, y)cp (E.i) 1 < g(y) E I p'(E.i)

5

Since by assumption ,p'(E') is finite, and hence by 3.4.14 ,p (E') is also finite, and since g(y) is p"-summable on E", according to 12.3.4 g(y),p (E') is also (p"-summable on E". Thus from (5.21) and (5.22) it follows by 15.2.3 that f(x,i , y),p (Eri) is also a v'-summable function on E" and that

(5.23)

(E") ( (Ef(xi.i, y),p (E:.)) dp' = Jim (E") f (±f(x. i J (E") f f(x,;,y) d")f

y)

/

From (5.21) and (5.22) it follows that (5.24)

f(x,i, y)4p (Eri) I S g(y)W(El)

and from (5.2) and (5.24) it follows by 15.2.3 that

lim (E") f (E f(x,i, y),p'(E.;)) hp l' =

f ((E') f f(x, y) dcv') dip',

and hence according to (5.23):

lim E

((E") ff(x, y) de")

, rp (Eoi)

(5.3)

_ (E")

f ((E') f f(x, y) dp ) do

§161

245

MULTIPLICATION OF SET FUNCTIONS

If we set (E") f f(x, y) dhp" = h(x), then from I f(x, y) 1 S g(y) it results by 12.3.51, 12.2.31, 12.1.7, and 12.3.21 that h(x) is bounded on E', and

((E") f f(x , y) dip"

.'(E,,) is a Riemann sum, associated with the decom-

position Z, , for h(x). Since (5.3) holds for every distinguished sequence ((l),)) of decompositions and for all Riemann sums S(h, cc', Z,) associated with Z,,

by 13.3.21 h(x) = (E") f f(x, y) dc" is a p'-continuous function of x on E' and according to 13.3.2 we have:

lim E ((E") f f(x,j,

y) dcp")

Substituting this into (5.3) we obtain (5.1).

f ((B") f

f(x, y) dp")

J

//

dcc'.

BIBLIOGRAPHY: Theorem 16.5.1 (together with a still farther reaching result) is due to (In connection with this, cf. also S. MAZUa:cEwicz, C. R. Soc. so. Vareovie 7 (1914), p. 382; A. ROSENTHAL, Sitzungsberichte Bayer. Aicad. d. Wise. 1922, p. 221; Sitzungsberichte Heidelberger Akad. d. Wise. 1934, No. 13.) As to the examples subsequent to 16.5.1 of. the bibliography in Encyklop. d. Math. Wise. II C 9b (P. MONTnL-A. ROSENTHAL), p. 1119, and furthermore: G. FICHTHNHOLZ, Fund. math. 6 (1924), p. 30; S. SAxs 111, p. 75. As to theorem 16.5.2 of. the bibliography in Eneyklop. d. Math. Wise., loc. cit., and in particular: L. LICHTENSTEIN, Nachrichten Gee. d. Wise. Gottingen 1910, p. 468; Sitzungsberichte Berliner Math. Gee. 10 (1911), p. 55; C. CA$ATR.oDORr 111, p. 638; furthermore: H. J. ETTLINGER, Annals of Math. (2) 28 (1926), p. 65; Amer. Journ. of Math. 48 (1926), p. 215; J. RiDDEn, Math. Zeitschrift 31 (1929), p. 141. W. SIERPI&TsKt, Fund. math. 1 (1920), p. 112.

CHAPTER V

DIFFERENTIATION

§17. Derivation of set functions 1. Derivates of a set function. Let E be a metric space and let be the system of all subsets of E; moreover, let Q be a o-field consisting of subsets of E with E e x)2. Let 0 and P be totally additive, finite set functions in V; and let 0 be complete for ¢. Now let a be a point of E. We designate by 9L a subsystem of T2, consisting of non-empty sets, such that to every p > 0 there is a set of M contained in the

; if 4'(M) = 0, sphere S,. For every set M e TL we form the quotient IP then this quotient has to be set = + oo, =0, or = - oo, according as p(M) > 0, =0, or <0, respectively. We shall say: the sequence ((31,)) of sets of P. converges to a if there is a sequence of positive p, with p, --b 0, such that M, c 5,,, . If in TL there is a sequence ((M,)) converging to a, such that the sequence

(()) converges to the (finite or infinite) limit d, then d is called a derivate of v with respect to 4' at the point a on P.. Thus if ((M,)) is any sequence of sets of fit, converging to a, then every value of accumulation of the sequence is a derivate of c with respect to 4' at the point a on P,

The set of values of accumulation of

qipwo`1) is obviously closed.

Thus

among all those derivates of v with respect to 4, at a on 9L there is a greatest one and a smallest one; they are called upper and lower derivate of p with respect to 4, at a on SD2, and are designated by D(a,,p, 4', TL) and D(a, gyp, 4', x)2,), respec-

tively; herein some of these arguments can be omitted if there is no ambiguity. We have always: D(a, gyp, 4', D2,) S D(a, gyp, y, x>2,). If D(a, (P, 4', xrl,) = D(a, p, 4', xrr,), then every derivate of p with respect to 1G at a on

2, has the same

value; this value is then called the derivative of (p with respect to 4' at a on 0, and is designated by D(a, p, 4', D2,). Thus such a derivative D(a, gyp, ,', TL) exists if and only if for all sequences ((M,)) of sets of 10L, converging to a, lim `P (M') exists.

4401.)

(Then this limit automatically has the same value for all these

sequences.)

There results immediately: 17.1.1. If the derivatives D(a, V1, 44, 9N,) and D(a, , 4', 9N,) exist and are not infinite of different (or equal) sign, then the derivative D(a, V, + ^, 4,, SD2,) (or

D(a, gyp, - V2, 0, x12,)) exists also and D(a, i + ,ps) = D(a, cpJ + D(a, 0 (orD(a,,pi -
§171

DERIVATION OF SET FUNCTIONS

247

A system C a V1 of non-empty seta will be called an indefinitely fine system if to every point a e E a subsystem C., a C is attached, such that to every p > 0 there is a set Q e Q. contained in S., . Then instead of D(a, gyp, ', C.), .P(a, 9,, J,, C.), and D(a, c, ', O.) we write: D(a, w, 4,, Li), D(a, gyp, 4,, Li), and D(a,,p, 4,, C), respectively. If hereby Li. consists of all sets Q e C which contain a, then C will be called an ordinary indefinitely fine system. Thus, in this case, to every a e E and to

every p> 0there is aset QeC with aeQandQ cS.,. 2. Vitali derivates. We call a system 113 a D1 of sets a Vitals system (for if it possesses the following properties: 1,) Every set V e Q3 is non-empty.

2.) To every a e E and every p > 0 there is a set V e 93, such that a e V and V C; S.,. 3,) If A is any subset of E and Y3' is a subsystem of Q3 which to every a e A

and to every p > 0 contains a set V with a e V and V C; S., , then there are countably many disjoint sets V, , V2 ,

,

V, ,

in 113', such that

A - SV, _,P A. 17.2.1. Every Vitali system for 4, is also a Vitali system for and vice versa. For, according to 4.1.12, the relations = . A and Z A are equivalent. 17.2.11. If x is #-continuous (§5, 7), then every Vitali system for ,, is also a Vitali system for X.

For, according to §5, 7, A - SV, = # A implies A - SV, =, A. Examples of Vitali systems will be discussed in No. 3 and No. 5. Let $3 be a Vitali system. We designate the system of all sets V e% containing a by $3.. Then $3 is an ordinary indefinitely fine system (No. 1) and, by No. 1,

at any point a e E we can form the derivates of p with respect to ¢ on Q3. , in particular the upper and the lower derivate b(a, gyp, , ., $3.) and D(a, 4o, #, 18.) ; we denote them by D(a, tp, J,, 113) and D(a, W, /', 13), respectively, and call them the upper and the lower Vitali derivate (on Q3) of (p with respect to '. A function which at every point x e- E equals a derivate of ,' with respect to ¢ on Q3, shall

be called a Vitali derivate (on 1) of a with respect to ' and is designated by D*(x, tp, 4,, $3). If at the point a D(a, p, 4,, 93) _ .Q(a, J,,!6),-then this value shall be called the Vitali derivative D(a, tp, y', 43) on $3 of ,p with respect to tiG at a.

We shall soon see (theorems 17.2.53 and 17.2.62) that under certain conditions every Vitali derivate of p with respect to *is 4t-measurable. First we show only: 17.2.2. If *3 is a Vitali system, then D(x, v, 4,, 113) and D(x, 'o, ', 13) are 4-measurable functions of x on E. According to 9.1.21 we have to prove: for every finite c the set A = [D(1) ? cl

is 4,-measurable, that is, A e 0. Let K > 0 be an integer; to every a e A there

248

[OBAP. V

DISIi'18RENTIATION

are sets V e Q3, such that a e V, V

Sow.), and

,? > c -

IKV)

!.

We designate the

system of all these sets V by 83. ; then 18. satisfies the conditions imposed on Q3' in 3.); thus there are countably many disjoint sets Vd , Va , , V. , in Q3,

,

such that setting SV,, = B. we have: A - B. _ # A. Since V. a 842, ,

we have also B. a V. We set B = DB. ; then by 1.2.4 also B e 842. Since by R

§1 (1.21) A -- B = A -- DB. = S(A - B.), by 4.1.32 A - B. _0 A implies A - B = i A.-Moreover, A B. For let b e B; then b e Nit = 1, 2, .. ) also, and hence there is a P., such that b e V.,,. According to the definition R

of $3R

> c - K , and for the diameter of V.,; we have d(V.,r) -- 0.

, we have

But from this it follows that D(b)

c, that is, b e A; hence b e B implies b e A,

that is, A a B. Therefore A = B + (A - B), and thus, since B e 842 and A - B = , A, we have also A e 841, as contended. Now, according to §6 (5.1), we extend the set function i, totally additive and monotone increasing in the a-field Dt, to a regular measure function in t, which we designate also by J (cf. 6.5.2 and 6.5.22). 17.2.3. If 4, is content-like (§7, 4), then in 3,) for every c > 0 the ads V, can be chosen such that also

(SV,-A) <e.

(2)

To every A e (1, according to 6.4.2, there is an M e 87t with M Z A and O(M) = (A). Since by assumption (No. 1) ¢(E) is finite, by 7.4.31 to every e > 0 there is an open set G Q M with '(G -- M) < e. Thus J(G - A) 5

1Z(G - M) + (M - A) < e.

Now one can choose a subsystem Q3" of Q3',

such that R3" satisfies the conditions imposed on $3' in 3,) and at the some time consists exclusively of sets V C G. Then for the sets V,(v - 1, 2, ), existing in $3" according to 3,), we obtain J(SV, - A) 9 (G - A) < s. 17.2.31. In theorem 17.2.3 one can replace (2) by

E (V.) < ;(A) + e.

(2.1)

By (2) and §6 (1.1) we have:

> (SV, -- A) r

J(SV,) - (A) _ E (V.) - (A) ,

Because of 4.1.1 and §6 (1.1) it follows from 3,) that '(A) 9 J'(SV,) _ (V.). Thus if y' is content-like, by (2.1) we can choose the sets V, of 31) such that (2.11)

(A) 9 i (SV,) _

1G(V,) < J(A) + e.

249

DERIVATION OF BET FUNCTIONS

§17]

17.2.4. If Q3 is a Vitali system and ' is content-like, then [D(.f , p, p, Q3) = + oo ] =,p A and [D(1, p, 4,, Q3)

+ A.

According to 3.4.71, we form the decomposition E _ E+ + E- where B+E- = A, ¢-(E+) = 0, and 4i+(E-) = 0. We have to show: E+[D(Z) = + -] _ # A and E-[D(t) _ + co ] =,r A. We prove, say, the first of these two equalities and we demonstrate it indirectly. We set E+[D(-*) _ + oo ] = C; by 17.2.2 and 9.1.2 we have C e P. Now suppose C 5,6 f A; then since C Q E+ and V (E+) = 0, we have ¢(C) > 0. Let p > 0 be a finite number and let it > 0 be an integer. If a e C, then, since D(a) _ + oo, there is a V e Q3, such that a e V, V C Sani.) , and -( V)

> p. If Q3 is the system

of all such V (for all a e C), then Z. satisfies the conditions imposed on Q3' in and hence, by 3,) and 17.2.3, for every e > 0 there are countably many disjoint , such that setting SV., = B. we have: C - B. = * A V., , Va , , V., ,

and (B. - C) < e. Thus letting x - + oo and at the same time a --1- 0 we obtain from y(B.) _ #(C) + 4,(B. - C) -*(C - B.) that #(B.) -P(C). Therefore ¢(B.) > 4 '(C) for almost all x. We designate by V' , those V., for which y'(V.,) > 0 and we set C. = SV; then 4(C.) z , .(B.), and hence also '(C.) But

> 44(C) for almost all x.

.o(C.) > p.(C.), and hence

(V

> p implies p(V;,) > p '(VL,); thus

90(C,) > 2 on for almost all K. Since herein p > 0 is arbitrary and ¢(C) > 0, it would follow

that ip is not bounded, contrary to 3.3.21. D(x), there results from 17.2.4: Since D(x) 17.2.41. If Q3 is a Vitali system and 4, is content-like, then the set of all x at E. which both D(x, ,p, ¢, Q3) and D(x, gyp, 0, Q3) are finite From this there results immediately: 17.2.411. If 83 is a Vitali system and is content-like, then the set of all x e E Q3) is finite = o E. at which a Vitali derivate D*(x, p, 17.2.5. If ' is content-like and is it'-continuous, if A e )2, and if for a Vitali derivate D*(x, .cp, f, 18) we have I D*(a, (p, 4,, Q3) - c I < Sat all points a e A, then

1p(A) - O(A) 5 SS(A).

(2.2)

Let x > 0 be an integer. To every a e A there are sets V e Q3, such that a e V, V Q Sa(l/.) , and

- cf

I

< S.

We designate the system of all such V (for all

v

at A) by Q3. ; it satisfies the conditions imposed on Q3' in 3,). Thus, by 3,) and

17.2.3, for every e > 0 there are countably many disjoint sets V,1 , Ve ,

,

250

V., ,

CHAP. V

DIFFERENTIATION

of Q3. ,

such that setting SV., = B. we have: A - B. = + A and

(B. - A) < e. Hence letting K -i, + ao and at the same time a --> 0, we obtain from (B.)

J (A) + (B. - A) - J (A - B.) and #(B.) = G(A) + 1.(B. - A) - #(A - B,) that (B.) --* i(A) and 4.(B,,) --+ (A). Since p is *-continuous, A - B. _, A implies cp(A - B.) = 0 and, by 5.7.2, (B. - A) -* 0 implies ta(B. - A) --+ 0. Thus it follows from cp(B.) = c(A) + ,p(B,, - A) - p(A - B.) that cp(B.) - q,(A) so(V°)

< 8, and hence p(V.,) = O(V.,) + e.,*(V.,) 1). Thus since cp(B.) f.(B,) ¢(V.,), and (with 10., (by §3 (4.21)) E I #(V,.) I s (B,), we have also cp(B.) = cq,(B.) + 6,8 (B,)

also. Now we had

c

I

(with 1#. I S 1), and hence so(A) = c#(A) + 88o(A) (with 10 15 1) ; so (2.2) is proved. 17.2.51. If 4, is content-like and

is 0-continuous', then for every M e 1')'2 and every upper (or lower) Vitali derivate S(x) (or Q(x)) of w with respect to 0: (2.3)

'(M) = (M) f D(x) d¢ (or p(M)

(M) f D(x)

According to 3.4.71, we form the decomposition M = M+/+ M- where = A, 4,-(M1 = 0, and ¢+(M-) = 0. Then it suffices to prove the

M+M_

following two formulas: (2.31)

cp(M+) = (M') f D(x)d# and cp(M-) _ (M-) f D(x) d¢.

We prove, say, the first. Let 8 > 0 and M; = M+((i - 1)8 S D(t) < i8]

(i = 0, f 1, f2,

); we set M* = S M; . By 17.2.2 M; e 9 and M* e 0;

by 17.2.41 M* = ,pM+. Thus since .p is #-continuous, M* =, M+also. Hence, because of 12.1.53, it suffices to prove

ce(M*) _ (M*) f D(x) dV,.

(2.32) According to 17.2.5:

(2.33)

(M,) _ (i

Since by assumption rp and

are finite,

i er I s 1.

e,a (M;),

(M;) I and E (M,) are also {--40

+00

finite, and hence by (2.33) E (i - 1)8#(Mi) has meaning. Thus since V(M+) = 0 and M* Q M+, 1 The condition that.r is V'-continuous is here essential; this follows from 17.2.68.

DERIVATION OF SET FUNCTIONS

§171

it follows from 13.1.21 and 17.2.2 that D(x) is P-integrable on M*.

251

Moreover,

we obtain from (2.33) by summation that .f.m

cp(M*) _ E (i - 1)84'(Mj + 04(M*) (with 1 01 S 1), and hence it results from 13.1.1 that p (M*) _ (M*)f D(x) d¢ + 21VS (M*) (with 10' 1 S 1). Since herein 6 > 0 was arbitrary, (2.32) and thus also the first formula (2.31) are proved. 17.2.52. If 4' is content-like and so is J,-continuous2, then the set of all x e E at which the Vitali derivative D(x, ip, 4, $3) exists = i E. For from 17.2.51 it follows by 12.1.521 that D(x,,p, ¢, l3) =,p ))(x, ,p, ,', 93). 17.2.53. If 4, is content-like and p is 4,-continuous2, then every Vitali derivate of p with respect to yG is *-measurable.

For if $13 is a Vitali system and D*(x) is a derivate of (p with respect to 4, on Q, then by 17.2.52 D*(x) = y D(x,.p, ¢, Y8); thus the contention follows from 17.2.2. 17.2.54. If 4' is content-like and ip is 4,-continuous', then for every M e 9 and every Vitali derivate D*(x) of rp with respect to ,y:

001) = (M) f D*(x) d4'. This follows from 17.2.51 and 12.1.52, since D*(x) =, D(x, ,p,

4', 113) by

17.2.52.

17.2.541. If 4, is content-like, if jp is 4,-continuous, and if M =, A, then for every Vitali derivate D*(x) of .p with respect to 4: D*(x) _ c 0 on M. This follows from 17.2.54 and 12.1.521.

17.2.542. If 4, is content-like, if Z is a Vitali system, and if Eo is the set of all a e E for which in Z. there is a sequence ((V,)) converging to a with 4'(V,) = 0, then Eo =,p A. We set p = 4'; then there is a derivate D*(x, 4', ¢, 113) which equals 0 on Eo and equals 1 on E - Eo . Thus by 17.2.53 Eo is 4,-measurable. But by 17.2.54 we have for every 4,-measurable subset M of Eo : 4,(M) = (1M1) f D*(x, 4, 4', Z) d4' = 0,

and hence Eo =.p A. and 4- are i-continuous; moreover, J is content-like According to 5.7.1 (§7, 4) with ,'. Thus there results from 17.2.52: 17.2.55. If 4, is content-like, then the set E* of all x e Eat which the three Vitali derivatives D(x, ¢, , 113), D(x, ¢+, tG,113), and D(x, J,-, ¢, 113) simultaneously exist =,P E.

17.2.551. If E* is the set of 17.2.55 and Eo is the set of 17.2.542, then on E* - Eo:' 'The condition that yc is 4-continuous is superfluous in 17.2.52 and 17.2.68; this is shown by 17.2.61 and 17.2.82. * Cf. footnote 1, p. 250.

4 If 4, is content-like, then by 17.2.55 and 17.2.542 E' - Bo -# E.

[CHAP. v

DIFFERENTIATION

252

D(x, 4+, , 93) + D(x, 0-, , Ql) = 1 and

D(x,

Q3) - D(x, V, , Q3) = D(x, 4G,

- = since and 0 5 ¢- S imply

and 4+

By 17.1.1 this follows from the fact that +y+ + 4- =

D(¢+, ) and D(4' , ) are not infinite; for 0 is 4,+ 5 0 9 D(4,+, ¢) 5 1 and 0 D(¢-, ) <= 1. According to 3.4.71, we form the decomposition: (2.4)

E = E+ + E" with E+E- = A, ¢-(E+) = 0, and 4+(E-) = 0.

Then we have: 17.2.552. If 4, is content-like and Q3 is a Vitali system, then

D(x,

4, 1 on E+,

D(x,

D(x,

# 0 on E+,

D(x,

D(x, 4', t4, Q3)

= t i on E+,

Q;) = y 0 on Lam;

-,

D(x,'G,

Q3) _,.1 on E-; ,

3) _; -ion E".

By 17.2.55 and 17.2.54 we have for M e E-112:

4,+(hfl = (fit) f D(x, 4+, , $3) dd.

But for every such M we have 4+(M) = 0. Thus it follows from 12.1.521 that D(x, 4'+, j., Z) = i 0 on E-; by 9.4.6 we can replace =,10 by = # 0. One shows in the same way that D(x, 0-, , B) = 0 0 on E+. The other contentions now follow from 17.2.551 and footnote 4, p. 251. Now we show that in the theorems 17.2.52 and 17.2.53 we can eliminate the assumption that p is 0-continuous. 17.2.6. If 4, is content-like and Q3 is a Vitali system, then in order that ,p be purely 4'-discontinuous Q5, 7), it is necessary and sufficient that D(z, gyp, 4', $i) = # 0 on E.

NECESsITY: Because of 5.7.25 and 17.1.1 rp can forthwith be assumed to be monotone increasing. First we assume also ¢ to be monotone increasing. It suffices to show that 15(x) = ,, 0. By 17.2.213(x) is 4'-measurable. since p and ¢ are monotone increasing, .D(x) ii 0. Suppose D(x) 0 were not true; then we would have [D(1) > 01 *,p A.

Hence if we set E. = Cls(t) >

I],Letit would

follow from [15(z) > 0] - SE,,, that E. $ # A for almost all m. m be chosen such that E. $,, A. Since ip is purely 4'-discontinuous, by 5.1.541, 5.1.55, and 5.2.12 there is a decomposition E. = E* + E**, such that ROE** = At, E* = 0 A, and E** = + A. Since E. $,# A and E** = i A, we have E* * # A. Thus since¢ is content-like, by 7.4.22 there is a closed F C E*, such that F $# A,

and hence ¢(F) > 0. Now let it > 0 be an integer. Since f(x) > m on F, to every a e F there are sets V e!8, such that a e V, V Q; S.IUKI , and `AV) > (V)

m

253

DERIVATION OF SET FUNCTIONS

§171

We designate the system of all these V (for all a e F) by 0. ; it satisfies the conditions imposed on Q3' in 30. Thus there are countably many disjoint sets V., , in Q3. , such that setting SV., = B, we have F - B. = A. Va , , V.. ,

Hence, if we set B. -f- F = B9 , then B; _ ,, B, and F C B9' . On the other hand, B '.c U,(1/.)(§2, 3). Since F is closed, we have F = DU,(11.) = Lim U,cu.> ; thus . from F Q B; C U, -p(F).

>

Furthermore, since

and

is monotone increasing,

we have cp(V.,) > m (V.,), and hence by summation over v also p(B.) > 1 , 1 m (B) = --4(B.). Since B..2 B. , B; a F, and (p and ¢ are monotone increas-

ing, we have cp(B.)

cp(B,) and 4'(B.) k 4'(F), and hence also ,p(B,) >

Thus,p(B.') -- w(F) implies also cp(F) z

(F).

(F).

m

But since Cl?) > 0 and F C E*,

So m the contention is proved for a monotone

this is contradictory to E* =, A. increasing 4'.-Now let 4, be arbitrary. Then is monotone increasing, and hence, as just proved, D(x, v, , Q3) - i 0; that is, D(x,,p, j, Z) = * 0. Thus by

17.2.552 we obtain for the set E* of all a e E at which simultaneously D(a, rp, , Q3) = 0 and either D(a, ¢, J, Qi) = 1 or D(a, 14, ¢, Q3) = -1, that E* _ ` E. Now let a e E* ; then for all sequences ((V,)) of Q30 converging to a 1t(V.) --+I or we have: V(V') PV.) -- 0 and either

RV.)

41(V .)

y (V-)

--+ -1. Therefore

41W.)

0

also; that is, for all a e E* we have D(a, sp, 0, 18) = 0, and hence D(x, q', 4', 23) = ,y 0

on E, as contended. SUFFICIENCY: By 5.7.3 p = p* + N** where rp* is ,'-continuous and {p*" is purely ¢-discontinuous. As we just have proved, D(x, **, P, Q3) = , 0. Thus from D(x, .p, 4', Q3) =,p 0 it follows by 17.1.1 that. 4', Q3) = ,c 0 also. Therefore it results from 17.2.54 and 12.1.52 that D(x,

9*(M) - (M) I D(x, o*, 4', Q3) d4' = 0 for all M e 0; hencep(M) = yo**(M) ; that is, 9, is purely ,'-discontinuous. 17.2.61. If 4, is content-like, then the set of all x e E at which the Vitali derivative D(x, gyp, ,y, 18) exists = + B.

According to 5.7.3 we form the decomposition 9 = rp* + v** where P* is 4'-continuous and 0** is purely 4y-discontinuous. By 17.2.52 the set of all x e E at which D(x, rp*, 4', Q3) exists = 0 E; by 17.2.6 the set of all x e E at which D(x, #**, 4G, Q3) exists and vanishes = # E. Thus the contention follows from 17.1.1. From 17.2.61 and 17.2.2 there results: 17.2.62. If 4, is content-like, thus every Vitali derivate of rp with respect to ' is 4,-oneasurable.

254

[CHAP. V

DIFFERENTIATION

Again according to 5.7.3 we form the decomposition tp - e + qo** where rp* is k-continuous and v** is purely 44-discontinuous. Then we have: 17.2.63. If tD is content-like and M e X71, then for even/ Vitali derivate D*(x) of go with respect to ¢:

V*(M) = (M) f D*(x) d4'.

For by 17.2.52 and 17.2.54 p*(M) = (M)

f D(z, e, ¢) d4. By 17.2.6

D(x, rp**, 4,) = # 0; hence by 17.1.1: D(x, p*, 4') = ,r D(x, g, p). According to 17.2.61 D*(x) = i. D(x, go, 4,).

Thus by 12.1.52 also `p*(M) = (M)

fi'(x) d4,.

From 17.2.63 there results according to 12.1.521: 17.2.64. If 4, is content-like, then for any two Vitali derivates D*(x) and D**(x) of p with respect to 4,: D*(x) =,. D**(x).`

3. Paving derivates. following properties:

We call a system i3 S* 0 a paving system if it has the

1,) Every set P e 13 is non-empty.

2,) To every a e E and every p > 0 there is a set P e $, such that a e P

andPcS,,.

3,) If A is any subset of E and i' is a subsystem of $ which to every a e A

and to every p > 0 contains a set P with a e P and P C S., then there are countably many disjoint sets P1 , P2, , P, , . in J,3', such that A Q SP'. A paving system 93 is also a Vitali system (for every 4'), but in general the converse is not true. If q3 is a paving system, then we call every Vitali derivate of /p with respect to 4, on $ a paving derivate of gp with respect to 4, on 13. In particular, D(a) cp, P, $), D(a, go, tp, %3), and D(a, 1p, 4,, ¶) are called the upper, the lower paving derivate, and the paving derivative, respectively, of 0 with respect to 4' on 3 (at the point a). 17.3.1. If 1J contains the open sets of the space E, then in order that there exist a paving system, it is necessary and sufficient that E be separable. NECESSITY: Let K be a positive integer. According to 2,), to every a e E

there are sets P e $, such that a f P and P C S.(11,,). The system of all these sets P satisfies the conditions imposed on 'Y in 3,); thus there is a countable subsystem V(*) of 3 covering E and consisting of the disjoint sets Pi`?, , each of which lies entirely in a sphere P24), - .. , P;`), Obviously - -

5 Here the Vitali systems employed in forming DO(x) and D*'(z) can be diferast.-The corresponding result follows directly from 17.2.61 only if D*(x) and D**(x) are formed by the help of the same Vitali system 18.

DERIVATION OF SET FUNCTIONS

§171

255

the countable set of the points a;`) (K, r = 1, 2, - ) is dense in E, and hence6 E is separable. SUFFICIENCY: If E is separable, then by 2.2.2 among the spheres S.,(a a E) there are countably many, S.,, (i = 1, 2, - ), which cover E.

+ Q.-,)

We set 5.,, = Q, and Sari - (Qt +

Q, (i > 1) and designate

Again by 2.2.2 among the spheres ), which cover Sal (with a t P.) there are eountably many, 5..;t(i = 1, 2,

the non-empty Qi by P,, Ps,

, PM

,

.

P.. We set P.8.., l = Q., and P.Sa.,II - (Q., + ... + Q.,-i) = Q.;(i > 1)

Continuing in and designate the non-empty Q., by P., , P.2, , P.. , . this way one obtains the sets P.,.,.. ; these sets with the same number of

indices are disjoint and P.,.,....,,.,,+, C P.t.,....; ; moreover,.1S P., - is and S P-1....K,..+1 = t'.,....,, . We show that the system 0 of all the -K+,

(eountably many) sets

P.1.,.....(K = It 2, ... , 9nt : It 2, ... , tnt = 1, 2, ... .... ) form a paving system. Since 9 contains the open sete, all spheres S., and hence also all sets P.,.,...., belong to P. It is evident that the properties 1,,) and 2,) of a paving system are satisfied. In order to demonstrate the property 3p), let A be any subset of E and let 3' be a subsystem of 13 which to every a e A and to every p > 0 contains a set P with a e P and P Q S., . Then to every a a A there is exactly one set in '3' which contains a and has as few indices as possible; we designate it by P(a). Each two of these sets P* are identical or disjoint. Thus the different sets among the P*(a) (with a e A) form a countable system of disjoint sets of 3' which covers A. From 17.3.1 and 17.2.51 (or 17.2.54), together with 12.1.31 and 12.1.51, we obtain again the theorem 12.4.11, but here only under the particular assumption

that E is a separable metric space whose open sets belong to T7 and that the determining function of the integral is content-like. 17.3.2. If tp and >G are content-like', if A e Tl, and if for a paving derivate D*(a) of cp with respect to 4, we have I D* (a) - c i < S at all points a e A, then

I1a(A) - $(A) ( 5 S#(A). Since tp and $ are content-like, by 7.4.2 there are two F,-sets C' and C", both contained in A, such that A - C' = A and A - C" _+ A. If we set (3)

C'

C" = C, then C is also an F.-set C A and A - C = A, A - C =,k A.

Thus by 4.2.44 p(A) = cp(C), ,ji(A) = y'(C), and (A) = J(C); hence it suffices to prove (3) for F,-sets A. If A is an F then by 2.1.41 A is the sum of a monotone increasing sequence of closed sets A.. Since then by 3.2.2 p(A.) --> P(A), ¢(A.) -- FG(A), and : (A.) --* ?(A), it suffices to prove (3) for closed sets. Thus let A beclosed. Let K > 0 bean integer. T o everya f A thereare sets P e 13, such that a e P, P C S.(i!.i, and I,p(P)

- c <6. We designate the system of all 1

S Cf. footnote 5, p. 7. 1 If p is content-like and n is #-continuous (cf. 17.2.5), then by 7.4.4 v is also content-like.

256

[CHAP. V

DIFFERENTIATION

these P (for all a e A) by ¶3. ; it satisfies the conditions imposed on '' in 3,). in '3. , P.. , Hence there are countably many disjoint sets P., , Pd , which cover A. Thus if we set SP., = B., we have A B.. On the other

hand, D. Q U.(11.); therefore, since A is closed, A = DB. - Lim B., and .

hence by 3.5.2: p(B.) -+ p(A), 4,(B.) --+ 4.(A), and (B.) -+ J'(A).

I¢(P..)

Now from

c < S one obtains (3) by the method employed at the end of the

proof of 17.2.5. 17.3.21. If p and 4, are content-like, if M e V, and if there is a paving derivate D*(x) of p with respect to ¢ which is finite on M, then

p(M) _ (M) f D*(x) d,'. According to 3.4.71, we form the decomposition M = M+ + M" where M+M- = A, 4-(M+) = 0, and 4,(M-) = 0. Then it suffices to prove the two formulas:

p(M+) = (M+) f D*(x) d¢ and p(M-) _ (M-) f D*(x) d4'. We prove, say, the first. Let b > 0 and M; = M+[(i - 1)s S D*(I) < is] ++o f 1, ::t:2, ) . Since D*(x) is finite on M, we have M+ = S Mi. From 17.3.2 we obtain again (2.33) and from here one continues as in the proof of 17.2.51 (whereby only M* has to be replaced by M+ and 17.2.62 has to be employed instead of 17.2.2). From 17.3.21 and 12.1.51 there results immediately: 17.3.22. If p and ¢ are content-like and if there is a paving derivate of p with respect to 4, which is finite for all x e E, then p is J,-continuous.

Now, according to 5.7.3, we form again the decomposition p = p* p** where Then we have: 17.3.23. If p and ¢ are content-like, if 3 is a paving system, and if E** is the set of all a e E at which there is no finite paving derivate of p with respect to ¢ on p* is ¢-continuous and ip** is purely 4y-discontinuous.

t h e n f o r e v e r y M e B2: p**(M) = p(ME**).

We have M = (M - E**) + ME**. By assumption there is a paving deri-

vate D*(x) finite on M - E**. By 17.2.62 D*(x) is 4'-measurable, and hence by 9.1.2 E** a 01. Thus, according to 17.3.21, p(M - E**) _ (M - E**)f D*(x) d4'. Since by 17.2.41 E** =+ A, we can write instead,

according to 12.1.53: p(M - E**) = (M) f D*(x) d¢. Thus

p(M) = p(M - E**) + P(ME**). _ (M) f D*(x) d4 + p(ME**). By 17.2.63 (M) finite, we obtain

f D*(x) d¢ = p*(M); therefore, since p = p* + p** and p is p(ME**).

DERIVATION OF SET FUNCTIONS

§17]

257

Thus by M = (M - E*O) + ME** a "singular decomposition" (§5, 1) of M into a 4,-regular and a +,-singular subset is given.

If ¢ is monotone, this can be somewhat strengthened. First we prove the lemma: 17.3.231. If p is content-like and 4, is monotone increasing, if $ is a paving system, if M e fit, and if for all a e M: D(a, p, 4', 3) 4 0 and Q(a, (p, d+, $) 5 0,

then M =, A. Let F be a closed subset of M. Let e > 0 and let K be a positive integer. Since

D(a) k 0 on M, to every a e F there are sets P e3, such that a e P, P Q Sat,,.), and (P(P) > -e. The set of all these P (for all a e F) forma a system V, which satisfies the conditions imposed on 3' in 3,). Thus there are countably many

in $., such that setting SP., = Bk we , P., , disjoint sets P., , Pd , have F Q B.. Since F is closed, F B, a U,(u.) implies Lira B. = F, and hence by 3.5.2 p(B.) -* p(F) and 4,(B -4 4,(F). But from >F z 0 we obtain, -s4'(B.), according to the definition of J)., f(P.,) is - e,'(P.,) ; hence 9,(B.) Therefore, since e > 0 was arbitrary, cp(F) 0. and thus r(F) s As D(a) 5 0 on M, one obtains in the same way the result that o(F) 0. Thus ,p(F) = 0. Since this holds for every closed subset F of M and since .p is contentlike, it follows from 7.4.22 that (p(A) = 0 for every gyp-measurable subset A of M;

that is, M =, A. 17.3.232. If cp and 4, are content-like and ¢ is monotone, if $ is a paving system, ] + [D(x, cp, +G, 3) _ - 1, then for every

and if E** = [D(1, 0, t,, $) = + me sn2: 0**(M) = v(ME**). By 17.2.61, 17.2.62, and 9.1.2E

a P. Set

C = [D(x, , 4,,10) - + - ] [P(t,

') 3co 1

Since on B - (E** + C) either D(x) or V(z) is finite, we have E** Q E** Q E** + C. But by 17.3.231 C =, A, and hence and let E** be the set of 17.3.23.

E**=,, E**. Thus,p(ME**) = o(ME**), and the contention follows from 17.3.23. We shall see in No. 5 that the theorems 17.3.21, 17.3.22, 17.3.23, and 17.3.232 do not hold for arbitrary Vitali derivates. As the following example shows,

the assumption that 4, be monotone is essential in 17.3.232: Let a e E; let sp(M) = 1 if a e M, and c'(M) = 0 if a ' eM; let 4i({a}) = 0, and assume that in the paving system 'j there is a monotone decreasing sequence ((P,)) of sets converging to a with a e P. and 4'(P,) > 0 as well as a monotone decreasing sequence ((Q,)) of sets converging to a with a e Q, and 4(Q,) < 0. Then by 3.2.21 lim 4,(P,) = lim 4'(Q,) = *Q a }) = 0. Thus D(x, (o, 4,, J3) = 0 for z - a, D(a, p, 4,, $3) = + 00 , and .)(a, ,p, ¢, is empty, while c,**({a}) F-` 0.

3)

-o.

Hence the set E** of 17.3.232

4. The regular derivative. Let Cl (CO) be an indefinitely fine system (No. 1).

258

(CHAP. V

DIFFERENTIATION

Let ((M,)) be a sequence of sets of 0 which converges to a. Then we say ((M,)) converges regularly to a (for >G relative to C) if a number r > 0 exists, such that

there is a sequence of sets ((Q,)) in C., converging to a, with M, Q Q, and 1'(M,) I > r I #(Q,) I for almost all v. The number is called a parameter of regularity of the sequence ((M,)). Every sequence ((Q,)) of Q., converging to a, converges also regularly to a (with the parameter of regularity 1). If C' c 9)7 is a second indefinitely fine system and if to every a e E there is a parameter of regularity r (a) > 0, such that every sequence of sets of Ca , converging to a, converges regularly for >G relative to C with the parameter of regularity l'(a), then C' will be called a regular system (for') relative to C. If there exists the derivative D(a, gyp, bi, C), then for every sequence ((Q,)) of sets of 0. which converges to a we have: "(Q') - . D(a, , q,, ¢, Vii). If for every

sequence ((M,)) which converges regularly to a (relative to C) we have:

(M.)

-b D(a, gyp, ,x, C), then we say: the derivative D(a, rp, 4,, C) is regular at

the point a. Thus Ma, ,p, 4,, 0) is regular at a if and only if for every sequence ((M,)) convergent regularly to a there exists

lim'P(M,)

According to 17.1.1 we have obviously: 17.4.1. If the derivatives D(a, V1, ¢, 0) and D(a, cps,,k, C) are regular at a and are ,C) (or + not infinite of different (or equal) sign, then D(a, D(a, pi - , ¢, C)) is also regular at a.

17.4.2. If 4, is content-like, if Z is a Vitali system, and if E, is the set of all a e E for which there is a sequence ((M,)) converging regularly to a (for 4, relative to l3) with 4, (M,) = 0, then Eo = # A.

This is an immediate consequence of 17.2.542, since the sets designated by Bo in both theorems are identical. 17.4.3. If (p is monotone and if the derivative D(a, cp, ,y, C) = 0 at the point a, then this derivative is regular there. We assume V to be, say, monotone increasing. Let ((M,)) be a sequence of

sets converging regularly to a with the parameter of regularity . Thus there are sets Q, e Ca , such that for almost all j ,'we have M, Q Q, and

14,M) I ? r

I #(Q.) I.

From D(a, (p, J,, C) = 0 it follows that!G(Q,) -+ 0.

Since {p is monotone increas-

ing, we have for almost all;,: 0 S ip(M,) < w(Q,), and hence,

.`tp,(M,) I

,',"((Q,) S 1.1 p

Thus we obtain also M) -0. 17.4.31. If J, is content-like and (p is purely #-discontinuous, then for every Vitali system ll the set of all x e E at which the derivative D(x, gyp, ,I', 23) is regular and vanishes =,p E.

According to 5.2.15, gyp} and w- are also purely #-discontinuous. Thus, by

DERIVATION OF SET FUNCTIONS

§17]

17.2.6,

259

for the set Eoe of all x e E at which both D(x, p+, 4', Q3) = 0

and D(x, 'p-, ¢, Z) = 0 we have: Eeo =,p E. By 17.4.3 D(x, w+, 0, Q3) and D(x, (P , 0,!B) are regular on Eoo. Thus from p =,p+ - 'eit follows because of 17.1.1 and 17.4.1 that D(x, jp, 4', Q3) is regular and vanishes on Eoo (=,p E) also. 17.4.32. If 4, is content-like and Q3 is a Vitali system, then the set of all x e E at which the derivatives D(x, ¢+, ',Z), D(x, V, 1Z, Q3), and D(x,,', , Q3) are simultaneously regular = * E.

We form the decomposition (2.4) ; then if A is the set of all a e E- at which

D(a, ¢+, j, Q3) = 0 and D(a, V, y , Q3) = 1, we have by 17.2.552 A = 0 E. According to 17.4.3, D(a, 4,+, , Q3) is regular at all points of A; that is, if a e A,

then for all sequences ((M,)) converging regularly to a we have: P(M,) --> 0. But then, since ¢+ +

we obtain

M,)) --> 1, unless a is a point of the

set E0 of 17.4.2; that is, the derivative D(a, ¢-, 0, Q3) is regular at all points of A - En ; and by 17.4.2 A - Eo =,. E- also. One sees in the same manner that, if B is the set of all a e E+ at which D(a, 41, ., Q3) = 0 and D(a, ¢+, , Q3) = 1, both D(a, 4-, j, 93) and D(a, ¢+, , Q3) are regular at all points of B - Eo and that B - Eo - * E+. Hence (A + B) - ED =,s E. But at all points x e (A + B) - Eo the derivatives D(x, 4+, , Q3) and D(x, 0-, , Q3) are regular, and thus, since P = 4,+ - V, by 17.4.1 D(x, 4', ', Q3) is also regular. 17.4.33. If 4, is content-like, then for every Vitali system Q3 the set of all x e E at which D(x, gyp, 0, Q3) is regular = # E.

According to 5.7.3, we form the decomposition + ** where v" is 4'-continuous and p** is purely 4'-discontinuous. By 17.4.31 the set of all x e E at which D(x, rp**, 0, Q3) is regular and vanishes = 0 E; thus, according to 17.4.1, it suffices to prove the contention for the 4'-continuous function,0*. Therefore we can forthwith assume ro to be ¢-continuous. Moreover, first we assume 4' to be monotone increasing, a condition which we shall eliminate at the end of the proof. We set [D(f, ,p, 4', Q3) 01 = Co ; because of 17.2.61, 17.2.2, and 9.1.2 we have Co e y2. Now we set: g1 = D(x, gyp, 4', Q3) on Co and g, = 0 on E - Co , g2 = D(x, re, ¢, Q3) where D(x, gyp, ¢, Q3) > 0 and otherwise g2 = 0; then by 17.2.61 D(x,,p, 4', Q3) _ g, + g:. Furthermore we set for all M e 1112: jp,(M) _ p(MCo) and rp1(M) = rp(M - Ce); then 01 and GPs are totally additive and (like cp) 4'-continuous. By 17.2.54 and 17.2.52: (4)

,(M) = (M) f D(x, v t

, 4', 93) d4'.

Since 02(-41') = rp(M - Co), we have again by 17.2.54 and 17.2.52: (4.1)

= (M - Co)

f g2 d4 = (Al) f g2 d4';

and in the same manner one obtains: (4.2)

'PI(M) .= (M) f g1 d,'.

260

[CHAP. V

DIFFERENTIATION

From (4) and (4.1) it follows by 12.1.521 that g2(x) _, D(x, 92 , 4, *3); and thus, since g2 = 0 on Co , we see that there is an No. _ k A, such that D(x, t , ¢, Q3) = 0 on Co - No. But since gz 0 and 4, 0, by (4.1) y,$ is monotone increasing; hence there results from 17.4.3: if a e Co - No, then for every sequence ((M,)) converging regularly to a we have

(

) --+ 0. Since g, S 0 and 4'

lows from (4.2) that 4,(31) 5 0 and hence lim

Mr) 5 0.

0, it fol-

Thus rp = Sq + 92

implies lim (M) 5 0, and so we have obtained the result: If Co = [D(1, jP, 4', Q3) 5 01,

then there is an No =,. A, such that for every a e Co - No and every sequence ow') S 0. Now we replace ((M,)) converging regularly to a we have Iim o(Mr) Then V -- co is also ,,-continuous and by 17.1.1 D(x, co - c4', 4', Q3) = D(x,,p, ¢, Q3) - c if x ti E Eo (where Eo =#A is again the set of 17.2.542 and 17.4.2). Thus we obtain: If C. = [D(t, gyp, 4', 18) 5 c},

V by 4' -- eye (where c is a finite constant).

then there is an N. = # A, such that for every a E C,, -- N. and every sequence sv(M ((M,)) converging regularly to a we have f in 'E(Mr) 5 c. In the same manner r

one sees: If C' = [D(z, (p, 0,Z) =a c], then there is an NQ = # A, such that for every

a e C,' - No' and every sequence ((M,)) converging regularly to a we have HM

, (M 4,M)

i= c.

Now let ci , c2,

,c,

be all the rational numbers. For

each of them we form the sets N.,, and N',1 introduced above; let N* SN,_ + SN.'. ; then by 4.1.32 N* = # A also. Moreover, let N** be the set of

M

all x E E at which D(x, qp, 4i, Q3) either does not exist or is not finite; by 17.2.61 and 17.2.41 N** = * A. Thus if we set N = N* + N**, we have N = # A also. $3) and for every sequence But at every point a e E - N there exists D(a, ((M,)) converging regularly to a we have: P(M,) --- D(a, gyp, 4, ,Q3), since for

Cml)

every rational c 2 D(a,,p, 4', Q3) we had lira

r

c 5 D(a, cp, 4', 18) we had lim

w(M.)

5 c and for every rational

z c. Thus at every point of E - N the

0(m.) derivative D(a, (p, ,', Q3) is regular and, since N = . A, the contention is proved

for monotone increasing 4,.-Now we eliminate the assumption that 4' is monotone increasing. Since is monotone increasing, according to the above proof we have for the set A of all a e E at which D(a,,p, , Z3) is regular: A = i E; that is,

by 4.2.4 A = # E. Thus for each such a and for every sequence ((M,)) con(M,) But according to 17.4.32, the verging regularly to a there exists lira C ml) set B of all a E E at. which D(a, 4', , $3) is regular =,k E. Hence for every a E B

§171

DERIVATION OF SET FUNCTIONS

261

and every sequence ((M,)) converging regularly to a there exists lim k(M,) 51

-M

,

and by 17.2.552 this limit has either the value 1 or --1, except for a set N* = , A. Thus for every a e AB - N* and every sequence ((M,)) converging regularly to a there exists Jim I(M,) ; that is, D(a,,p, 4', $3) is regular at every point a s AB -- N*.

But A = # E, B := , E, and N* _ A imply AB -- N* = E, and so the contention is proved. We now call an indefinitely fine system 8 Q EW an orderly system (for 4) if there is a Vitali system Q3 (cf2 and for ¢), such that Q13 is a regular system relative to t (for 0). Every Vitali system is also an orderly system. If 9 is an orderly system and if the function 0°(a), for every a E E, equals a derivate of V with respect to ¢ on G. , then we call D°(a) an orderly derivate of ,p with respect to 4' (on 3). 17.4.4. If 4' is content-like, if D°(x) is an orderly derivate of rp with, respect to and if D*(x) is a Vitali derivate of p with respect to 4', then D°(x) = # D*(x).

By assumption there is a Vitali system Q3', such that D*(x) is a derivate of (p with respect to P on W. Moreover, let D°(x) be an orderly derivate of gp with respect to i, on l3 and let 3 be the Vitali system relative to which lB is regular. By 17.2.64 and 17.2.61 D*(x) =,p D(x, cp, 4', Q3).

According to 17.4.33,

the set A of all a e E at which D(a, rp, ,', Q3) is regular =,, B; but for all a e A we have D°(a) = D(a, q,, P, Q3). Thus D°(x) D(x, rp, ¢, Q3), and hence D°(x) =,c D*(x) also. 17.4.41. If 4, is content-like, then for any two orderly derivates D°(z) and D°p(x) of 0 with respect to 4, we have: D°(x) =,, D0O(x).

For if D* is a Vitali derivate of i' with respect to 4', then by 17.4.4 D°(z) = # D*(x) and D00(x) = + D*(z), and hence D°(x) = # D°°(x).

5. Covering systems. A subsystem 19 of 1't shall be called a covering system" of A(CE) (for 4') if it possesses the following properties: 1.) Every set C e (S is dosed and non-empty. 2) To every a e A a subsystem W. of 19 is attached in which to every p > 0

there is a set C Q S., . 3,) If A' a A and ( ' is a subsystem of ( which to every a e A' and to every p > 0 contains a set C e L. with C a S., , then there are countably many disjoint sets C, , C, , , C. , in W, such that A' - SC, = y A. Every Vitali system ( consisting of closed sets obviously is a covering system of E if C means the set of all C e (Y with a e C. Conversely a covering system 4£ of E is a Vitali system if C consists of all sets C e (S with a e C. The word "cover" is here employed in a wider sense than in §2, 2.

ICHAP. V

DIFFERENTIATION

262

Now we extend the definition of +' to all sets A a E by

(Ms92).

>4(A) = inf RM)

(5)

MBA

Then according to 6.5.2 and 6.5.22, ' is a regular measure function. 17.5.1. Let A C E; in order that a system (S C T? possessing the properties 1.) and 2,) be a covering system of A, it is necessary and sufficient that (9 have the follow-

ing property: If A' C A and (' is a subsystem of (,£ which to every a e A' and to every p > 0 contains a set C e ( with C C; S., , then to every e > 0 there are finitely many disjoint sets C,1 , C2, ..

,

C. in (', such that

(A' - 8 CY < E.

Nr cESSrry: If CS is a covering system of A, then there are countably many disSince the joint sets C1 , C2, , CY ,. - in (s', such that .(A' - SC,) = 0. -

(C,) _ .(Sc) ,c '(E) ; thus E >
C. are disjoint sets of 9)?, we have

,

I

Y

is finite, and hence E , (C.) < e for almost all m. Since A' - S C, c ,-1 '7 (A' - SC,) + S C. , we have according to §6, 1, condition 3.) for measure functions: (A' - ,C C,) ,.e1

1' - SC,) + , (S C,). Thus from (A' - SC,) = 0 ,

,>M,

a

,

and ( S C,) = F, (C,) < t it follows that (A' - S C,) < e for almost all m. -1 ,>m ,>m SUFFICIENCY: By assumption there is a sum B1 of finitely many disjoint sets

of C, such that J (A' - B1) < 1. We set A' - B1 = Al . Since B1 is closed, there is a subsystem (_F1 of (' whose sets are disjoint from B1 and which to every a e A, and to every p > 0 contains a C e (rya with C, C S., . Thus there is a sum

B2 of finitely many disjoint sets of, (which are disjoint from the terms of B,),

such that '(A1 - B2) < J; that is, ¢(A' - (B1 + B2)) < J. Now we set A' - (B, + 132) = A2, and continuing in this manner we obtain a sequence ((Be)) of disjoint sets, each of which is a sum of finitely many disjoint sets of (', and

(A' - (B1 + B2 +

+ B5)) < k . Thus (A' - SB{) = 0, and hence by

4.1.1 the property 3.) is satisfied. 17.5.11. Let A a E, let (i be a subsystem of T? which possesses the properties 1,) and 2.), and assume that there is a g > 0 with the following property: if A' Q A and (1' is a subsystem of ( which to every a e A' and to every p > 0 contains a C e (F. with C Q S.,, , then there are finitely many disjoint sets C, , C2, , C. in W, such that

( S C;1

gkA'). Then (,f is a covering system of A.

,.1

Let A'

A. By assumption there is a sum B, of finitely many disjoint sets

of (s', such that kB1) k

We set A' - B, = A, I. Since B1 is closed,

there is a subsystem (S, of (' whose sets are disjoint from B1 and which to every a e A l and to every p> 0 contains a C e (, with C c S . Thus there is a sum B2 of finitely many disjoint, sets of (S1 , such that .(B2) z gi (A,). that is,

DERIVATION OF SET FUNCTIONS

§17]

263

(B2) z 9(A' - B1). Now we set A' - (B, + B2) = A2, and continuing in this manner we obtain a sequence ((B,)) of disjoint sets, cacti of which is a sum of + Bk_,)). -:z,:! g '(A' - (B1 -I- B.. + finitely many disjoint sets of t`', and Moreover, E (Bk) = ;y(,SB;, i < ,ylE), and hence E'(Bk) is finite. Therefore k

k

It

0, and thus (A' - (B1 + B2 +

+ Bk_,)) -- 0 also. Hence

¢(A' - SBt) = 0, and so the property 3,) is satisfied.

Q E and if

17.5.2. If A =

is a covering system of Am for every m, then

m

(j is also a covering system of A.

It is obvious that the properties 1,) and 2,) are satisfied' Let A' C A and let (' have the same meaning as in 17.5.1. We set A'A, = A i ; since G is a covering system of A, , by 17.5.1 there is a sum B, of finitely many disjoint sets

in C', such that (A,' - B,) < 22. We act A'A2 -. B, = A2' ; since B, is closed, there is-as in the proofs of 17.5.1 and 17.5.11--a sum B2 of finitely many dis-

joint sets of (' which are disjoint from B, , such that (A2 - BB) < 2a .

We

set A'A8 - (B, + B2) = A; , and continuing in this manner we obtain a sequence ((Bm)) of disjoint sets, each of which is a. sum of finitely many disjoint sets of (s',

and, if we place A'A,.. - (Bt + B2 + (:1,

B.,,) < 2 +,

.

+ B._,) = A., , we have:

Since A' - S'B, C S (An' - Bm) and is a measure funcm

m

tion, we have i (A' -n, SB,,,) 5. E t (Aw. - Bm) < E2 +r = m

A' - (B, + B2 +

E

2'

From

+ Bk) C (A' - SB..) + S B. it follows that m

m>k

(A' - ( B , -F- B2 +- ....+ Bk))Is (-4' - SB,.) + >y(S B,.) < ( S B.) -[- e Since the B. are disjoint sets of 0, we have ( S Bm) _ E 1J(Bm). (v

m

(Bml = (SBm) S m

S Bm) <

E

2

m>k

na>k

(E), and hence E

,i'(B.)

is finite.

m

Moreover, Therefore

for almost all k. Thus since B, + B2 + ...+ - Bk is a sum of

finitely many disjoint sets of W, the contention follows from 17.5.1. Let s., be the closed sphere with center a and radius p (§2,3). Then we have: 17.5.3. If E is separable, if the closed sets belong to 1.1%, and if to every a e E there

is a finite number p(a) > 0, s itch that i(&'.. 2,) p(a) then the system C of the &, (for all a e E and all p > 0) is a Vitali system. We have to show: if (F. is the system of all S. p with a e S., , then (S is a covering system of E. Set E. = [p(l) 5 m)(m = 1, 2, ); then E = SE- and, ac-

s

cording to 17.5.2, it suffices to show that (S is a covering system of E. for every m. 9 We assume hereby that. U. is independent of rn.

264

[CHAP. V

DIFFERENTIATION

Thus we can assume from the first that p(a) is bounded: p(a) S p for all a e E. Let A Q E and let 1 ' be a subsystem of ( which to every a e A and every o > 0 contains an Sa, with a e S. and p < a. By 2.2.2 there are countably many in Cf', such that A a SS.,.2, ; we can hereby assume: spheres , ,,,, , ,3,,,! , . . k p, 9 ... We omit all those 5.,,,(v > 1) which are not dispi 9 ps joint from . The S.,. :,, corresponding to the omitted S.,,, are contained be the 11rst in S.,. ,, , as the triangle inequality 12 in §2, 31 shows. Let

of the remaining S.,,, (if there are any such). Now we omit also all those corresponding to The 34,9.(v > v,) which are not disjoint from be the first of the the S.,,, just omitted are contained in S.,,. ,,,, . Let (if there are any such). Continuing in this manner and setting remaining

S.,P, -

we obtain a (finite or infinite) sequence of disjoint S.,,,, , Thus, since j is a measure (i = 1, 2, ) and A (3'.,s,,,t) p' function, we have: y (A) S and hence

z p i(A). Therefore if g < -1, then there is a k, such that gRA) and hence ( S

z O (A). Thus it follows from

17.5.11 that 11 is a covering system of E and, since ( with a e C, also a Vitali system.

consisted of all C e iZ

17.5.4. If J, is content-like, then event/ orderly system Q2 for

which consists of

closed sets is a covering system of E for ,&.

By definition of 3 [No. 41 there is a Vitali system Q3 (9.0), such that Ql3 is a regular system for relative to Q3; for every a e E let 1'(a) > 0 be the corre-

sponding parameter of regularity. We set E.

r (I) z 1 ], (m = 1, 2, . ) ; in

then SE,. = E and by 17.5.2 it suffices to prove that 8 is a covering system of E. (for every m).

Thus we can assume from the first: there is a p with I z p > 0,

such that r(a) z p on E. Let A 9 E and (A) > 0. Since 4, is content-like and

A with j(G) < 3 3 (A). Let p FM' be a subsystem of Z which to every a e A and every p > 0 contains a set W e $33. with W C-. S., . Then to every a e A and every p > 0 there are also a set V e Q3 and a set W e J', such that a e V, W C V G S., , V C G, and finite, by 7.4.1 and 7.2.4 there is an open G

(W)

If Wis the system of all these V, then according to 3,) [No. 21 there are countably many disjoint sets V1 , V2, , such , V, ,

that X4(A - SV,) = 0. To every V, there is a W, C V, in 81 with RW,)

ptiG(V,).

Since

A - (W, + Ws + .....F WK)

(A - SV.) + S. (VI - W.) -i- S.>KVI , 1

265

DERIVATION OF SET FUNCTIONS

117]

and (A - SV,) = 0 and since is a measure function, we have:

(A - (W1 + W: -}- ... + We)) s

,r1

(V, - W,)) + tZ(S V,). >R

As the V, (e U) are disjoint, ,>R ( S V,) = ,>R E (V,).

Since ,

(E) and thus E (V,) is finite, we have lim E RV,) = 0; hence

j(SV,)

R

,>R R

SR V,)

E r.l

Y

(Y

69

? y4(G)

2

for almost all

r - II7r) ; and (W,)

(V,) implies ¢(V. - W,) 6 (1 -

(1 - p)E ¢(V,) S (1 -

(S (V, - W,))

(A - (W1 + W2 +

Therefore

+ WS)) S (1 - 2 JA(G), and thus from (G)

3 , (A) it follows that (A -- (W1 + W, +/. -p

But (A)

W,)) _

Moreover, ( 8 W.

R

R

thus

1

K.

(A(W, +

--P) + WR)) < 3(2 2(3-p)

+ TV )) + ,7(A - (1V, +

+ Wa); hence

+...+Wj)

t'(W1+...+W,) z (A(W,

- p) ?J(A) - (A - (W1 + ... + We)) > (A) - 3(2 2(3-p) and thus J(W1 +

+ WR) ?

L(3

p

(A).

This inequality derived under

the assumption that (A) > 0 holds also if j(A) = 0. Therefore by 17.5.11 8 is a covering system of E for 4,. Now we prove Vitali's Covering Theorem: 17.5.5. Let A Q R. and let (1 be a system of closed, non-empty sets of R. ; to every a e A let a number r(a) > 0 and a subsystem 6E. of W be attached in which to every/ p > 0 there is a C Q S. and to every C there is a s > 0, such that C Q S.,

r(a) Then there are countably many disjoint sets C, , C:, ,C,, -in(9, such that #,(A - SC.) =0. ,

and µ,(C)

R. can be represented many ways in the form: R = SE; + N where the E, , are disjoint open sets with finite and µ (N) = 0. By 17.5.3 the closed spheres contained in E; form a Vitali system in the space E, . We designate by (,E, the system of all sets of Qi which are contained in E; . Moreover, we designate by Q13; the sum of 4L: and the system of all closed spheres in Ei whose centers do not belong to A; then Si is an orderly system for A. in the space E; . Thus by 8.2.1 and 17.5.4 there are eountably many disjoint sets Ca, , Ce ,- - , C;, , . of C, , such that A.(E{A - SCs.) = 0. Then the system of the oountably many

disjoint sets C,, (i, v = 1, 2,

-) provides what we desired.

266

[CHAP. V

DIFFERENTIATION

The conditions of 17.5.5 are satisfied in particular if the sets of CT, are closed , c.] (§8, 1) which contain the point a and intervals [b, , b2 , . , b, ; c, , c ,

for which the quotient of the edges

c,

-

b;

z t(a) > 0 (for i, j = 1, 2,

,

n)

c:; - b; where (a) depends only on a.--Thus it suffices in R, for the sets of (Y, to be arbitrary closed intervals containing the point a. But for R. (with n ? 2) the contention 17.5.5 does not always hold if the sets of (9, are arbitrary closed intervals containing the point a. For we show: 17.5.51. In R2 there is aµ2-measurable set Pond to every a e P a sequence (, of intervals with center a and with diameters converging to 0, such that for every sequence C, , C2 , . of disjoint intervals chosen from S Gf, we have: , C, ) aeP

µ2(P - ,SC,) > 0Y

In R2 we consider the closed sets W. defined by the inequalities

0<x5 m1,

(5.1)

05y6 1, m

and

(c>0).

m2

Then we have µ:(Wm) = c2 e`(c + 1).

(5.11)

y

Let a e R2 and let W.,(a) be the set derived from W. by the translation which brings the point (0, 0) into the point a. Let Q be the open square (0, 0; 1, 1) and let mo be any positive integer. To every point a e Q we attach the system of sets W,,, (a) ( Q (with m z mo) ; then the conditions of 17.5.5 are satisfied for A = Q. Thus among the sets TV. (a) (m z mo) there are cotintably many disjoint sets bV m;(ati) (i = 1, 2, ), such that SWm;(a;) C Q and µ2(Q - STd'mt(a)) i

= 0, and hence µ2(SWm;(a;)) = 1.

Now let ((c,)) be a sequence of positive

numbers. If in the definition of W_(a) we replace the number c by c" , then we

designate the resulting set by Thus, if v is fixed, among the sets W,(.')(a) (m z v) there are countably many disjoint sets such that calling their open kernels 11'11. ;(a,,;) and setting P, we obtain:

P, Q Q and µ2(P,) = 1.

(5.12)

Now we set P = DP,. Then P c Q also; moreover, by §1 (1.21) Q - P = Q - DP, = S(Q - P,), and hence µ2(Q - P)

µ.,(Q - P,). Thus µ2(Q - P,) = 0 implies µ2(Q - P) = 0, and hence µ2(P) = 1. Let a e P; then to every v there is exactly one set Ti";.;';(a,,;) with ae 1'1'm,,f(a,,,); we designate this set simply by W(''(a), the corresponding point a,,; by p,(a), and the corresponding index m,,; by m,(a); then m,(a) z P. Let C,,(a) be the interval with center a and With p,(a) as vertex. If a = (x*, y*) and p,.(a) _ (x**, y**), then by (5.1) V

1

Y

267

DERIVATION OF SET FUNCTIONS

§171

v and0
(5.1): (x* - x**) (y* - Y*)

e

".

v

Moreover by

(m.(a))2'therefore p,(C,(a)) s 4e-".

(m.(a))2

and hence by (5.11) : K2(C.(a)) µ2(W,(a))

{5.2)

4

c, + 1

Now we designate by %. the system of intervals C,(a) (v = 1, 2, ) attached in the above manner to the point a e P. If a e P and a' e P and if the two intervals C,(a) and C,(a') are disjoint, then W(')(a) and W')W) are not the same

Thus from P, = SWT,' ,(a,,;) it follows by (5.2) that (if v is i fixed) for every system of disjoint intervals C,(a1), C,(a2), . . , C,(ai), - we set liw':,,).,(a,,i).

have E µ:(C,(ai)) < c,

c,

} I wr(P,.), and hence because of (5.12)

4 1 . Thus for every sequence C, (aft)), C,2(a(2)),

intervals C,(a) we obtain: E µ2(C.k(a«°)) k

the c, such that

c,

+1 1 <

4 E c, +1 1

, C., (a'),

µ2(C,(a1)) S

of disjoint

Therefore if we choose

v

1

then E µ2(C,,k(a'k))) < 1; and so, since µ2(P)

= 1, by 3.1.42 the contention is proved. Now finally we show by an example that the theorems 17.3.21, 17.3.22, 17.3.23, and 17.3.232 do not hold for arbitrary Vitali derivates. Let E be the square (0, 0; 1, 1) of R2 and let B be the piece of the straight line x -}- y - 1 = 0

which lies in this square. We designate by Q3 the system of all squares [x, y; x + h, y + h] (with h > 0) which are contained in E and, except for the point (x, y), are disjoint from B. It follows from 17.5.5 that Q3 is a Vitali system for k. Now let 9 be the v-field of all u2-measurable sets M C E for which M1MB is A,-measurable and set P(M) = µ1(MB). By 8.2.1 A, and ;ft are contentlike. But V is also content-like. For by 8.2.52 to every e > 0 there is a set A Q B, closed in B, with MB Q; B - A and u1(B - A) < µl(MB) + e. 'Then

E - A is a set open in E and containing M with p(E - A) = µ1(B - A) < µ1(11B) + e = (p(M) + e; and hence by 7.4.3 p is content-like.

Moreover, one sees immediately that for every a e E the Vitali derivative D(a,,p, A2, 93) = 0. But, in contradistinction to 17.3.21 and 17.3.22, 'p is not u2-continuous and, in

contradistinction to 17.3.23 and 17.3.232, cp** _ 'p and hence we do not have (P**(M) = 0 for all M e 9. BIBLIOGRAPHY to Nos. 1-5: Vitali's Covering Theorem 17.5.5 is essentially due to G.VITAL), Atti Accad. Torino 43 (1907/1908), p. 229; subsequently more general formulations

268

DIFFERENTIATION

[CRAP. V

in H. Lssssous, Ann. to. Norm. (3) 27 (1910), p. 391; C. CAaATutoDoRY [11, p. 299; B. Jnsa3N, J. MABCINSIEWIO%, A. ZYOMIIND, Fund. math. 25 (1935), p. 224; A. P. Moass, Trans.

Amer. Math. Soc. 55 (1944), p. 205; A. S. BESZCovrrca, Proc. Cambridge Philos. Soc. 41 (1945), p. 103; 42 (1946), p. 1; a simple proof in S. BADzAcs, Fund. math. 5 (1924), p. 131; (for the case in which in R. the "Carathdodory linear measure" [footnote 63, p. 105] is considered instead of t,,,: W. SIERPthSKI, Fund. math. 9 (1927), p. 177; J. F. RAN-DOLPH, Annals of Math. (2) 40 (1939), p. 299).-Theorem 17.5.61 is due to S. BANAOH, W. cit., p. 134, and to H. BOHR (in C. CARATH&ODORY [1], 2"d edit., p. 689); of. also: 0. NIKODYM, Fund. math. 10 (1927), p. 168 (note of A. ZYOMUND); H. BUSnMANN-W. FsLLEa, Fund. math. 22 (1934), p. 255.

The theory of the derivation of totally additive set functions (in R. and for y a µ.) has been founded by H. Lnnnsaus, loc. cit., p. 887 (who hereby essentially relied on Vitall's Covering Theorem 17.5.5); one owes to him also the introduction of the parameter of regularity (No. 4). Subsequent to H. LnnEsous: Ca.-J. Dn LA VALLEE Pousslx [1], vol. II, p. 109; Trans. Amer. Math. Soc. 16 (1915), p. 485; [2], p. 59, 90; C. CARA'rn onoRY [11, p. 480; S. BANACS, Fund. math. 6 (1924), p. 170; H. BusnuANN-W. FELLER, loc. cit., p. 226; A. S. BEszcovrrca, Fund. math. 25 (1935), p. 209; A. J. WARD, Fund. math. 26 (1936), p. 167; 28 (1937), p. 265; 30 (1988), p. 100; S. SASS, Fund. math. 27 (1936), p. 72; [21, p. 105; E. J. McSaANS [11, p. 366; of. also the bibliography to §18,2 and §18,3.-Derivates of totally additive set functions with respect to a general totally additive J, have been discussed : in R. by P. J. DANIEia., Bull. Amer. Math. Soc. 26 (1919/1920), p. 444; Proc. London Math. Soc. (2) 26 (1926), p. 95; J. RIDDEa, Nieuw Archief v. Wiskunde (2) 21 (1943), p. 212; A. S. BEBI-

covlTca, Proc. Cambridge Philos. Soc. 42 (1946), p.1; in abstract spaces by W. FELLSa, Bull. internat. Acad. Yougoslave 28 (1934), p. 40; R. DE PosszL, Paris C. R. 201 (1935), p. 579; Journ. de math. (9) 15 (1936), p. 891; E. ToaNzRR [1), p. 76; S. SASS [21, p. 152; B. JESSEx, Mat. Tideskrift B 1938, p. 23; 1939, p. 14; A. RosnNTHAL, Bull. Amer. Math. Soc. 48 (1942), p. 414; A. P. Monsa, Trans. Amer. Math. Soc. 55 (1944), p. 205; 61 (1947), p.418;

W. Nnr, Festschrift zum 60. Geburtitag von Andrew Speiser, Zurich 1945, p. 201.

6. Tile derivates. As we saw in 17.5.5 and 17.5.51, one can form Vitali systems by means of intervals of R. (for n a 2 and >' = µ$) only if a condition of regularity is satisfied. We shall here consider other systems to which (for 4' = µ.) the set of the intervals of R. (without any condition of regularity) also belongs.10 So far (for the Vitali systems and for Vitali's Covering Theorem) the sequence of the selected sets V. (v = 1, 2, . ) was to be disjoint. We shall abandon this condition for the systems to be defined here. Again let the assumptions of No. 1 be valid. According to 16, 5, we again extend the set function which in $)t is totally additive and monotone. increasing to a regular measure function in 12, which we designate also by J. Now we call a system Z Q T1 of sets a tile system" (for 4') if it possesses the following properties: lt) Every set T e is non-empty.

2,) To every a e E and every p> 0 there is a set T e X, such that a e T

andTaS..

3,) If A is any subset of Eu and ' is a subsystem of X which to every a e A IY Cf. 18.2.28.

A' We think here of (overlapping) roofing tiles.

If Equivalent to 3,) is the same property but required only for those A C E for which

§17]

DERIVATION OF SET FUNCTIONS

and to every p > 0 contains a set T with a e T and T G S , then to every in ', such e > 0 there are countably many" sets T, , Ts , , T, ,

that A - ST, = ,r A and E (T,) < (A) + E. r

Because of 4.1.1 and §6(1.1) it follows from 3 j that

J(A) s i(ST,) s r

r

Thus we have for the sets T, of V: (6)

j (A) .9 i(ST.)

E? (T,) < J(A) +

According to 17.2.31 we have: 17.6.1. If 4, is content-like, then every Vitali system for ¢ is also a tile system for 4'.

Conversely, a tile system need not be a Vitali system (not even for contentlike 4). This is shown for 4, = pz by the example of the set of all intervals of R$ . For these do not form a Vitali system according to 17.5.51, but by 18.2.23 they form a tile system. 17.6.11. Every tile system for 4, is also a tile system for ' and conversely. Proof as for 17.2.1. Let Z be a tile system. We designate the system of all sets T E T containing a by Z, . Then Z is an ordinary indefinitely fine system (No. 1) and, by No. 1, in every point a e E we can form the derivates of rp with respect to 4, on T. , in particular the upper and the lower derivate 1) (a, ,p, ¢, T,) and Q (a, gyp, 4', To); we denote them by 1) (a, gyp, 4', ` ) and L) (a, gyp, 4', Z), respectively, and call them

the upper and the lower tile derivate (on Z) of p with respect to 4' at a. A function which at every point a equals a derivate of rp with respect to 4' on T. shall

be called a tile derivate (on ;l;) of (p with respect to 4'. If at the point a

1) (a, gyp, 4', ¶) = p (a, p, ,', T), then this value shall be called the tile derivative D (a, cp, 4', Z) on T of (p with respect to ¢ at a. St) and (x, 9,, 4', St) are4-measurable 17.6.2. If is a tile system, then 1) functions of x on E. Proof as for 17.2.2. 17.6.3. If I jp(M) 1 S q j,(M) for every M e T1, if A e 0 and if for a tile derivate D* (x, cp, 4', St) we have I D* (a, gyp, 4i, T) - c I < 6 at all points a e A, then (6.1)

Let K > 0 be an integer.

I

(A) - ab(A) C 5 To every a e A there are sets T e to , such that

,I(A) > 0. This property is automatically satisfied for (A) - 0. For either one can allow that in ' the system of the countably many sets T, may be empty; or, if one will not allow this for A sd A, one can draw the following conclusion: Let iG(A) - 0 and a e A(O A); let p. -+ 0 and T. t Z' with a e T. and T. Q S.,.. Then Lim T, - {a l and hence (Lim T.) - 0; K

thus by 3.6.2 VT.) --* 0.. Hence there is a T., with (T,.) < e. u Not necessarily disjoint.

e

270

[CHAP. V

DIFFERENTIATION

T c So(1i%) and I `P(T)

- c < 6.

We designate the system of all such T (for all

a e A) by X, ; it satisfies the conditions imposed on Z' in 3,). Thus there are countably many sets T., , T.2 , , T., , of X. , such that setting ST., = B,

we have A - B. = y A and, according to (6), (A.) < (B.) S D(T.,) < Thus since '(B.) = (A) + (B. - A) (A - B.), we have'y(B. - A) -0, and hence '(B. - A) -p 0 also. Therefore 4,(B.). = (A) + 4'(B. -- A) - 4,(A - B.) implies ¢(B.) -> 1P(A). Since jG(A) + K

.

Therefore RB.)

F

v(M) 5 q .(1f) (for all M e TO), we have ,*(A - B.) = 0 and cp(B, - A) - 0 also; and hence c(B1) _ p(A) +V(B. - A) - (p(A - B.) implies (B1) -->,p(A). `'p" (T") - c < 8, and thus Now we had 1/(T.) (6.11)

(with { e..1 < 1).

(p(T.r) = cO(T.r) +

V-1

T., and for v > 1: T.', = T., - S T.,, and T..

We set

µ-1

then the TKr a EU2 are disjoint and B. = ST., = STk, r

r

.

-

By 3e) we have:

r

+ G(TKY)) _r (T.,) + E r(TK.)

r

= i'(B1) + E (7 'Xo) < (A) + 1x . Thus since (A) 5

we have E (7") < 1 K, and hence I E#(T,,) I < v

also.

K

Therefore

4,(B.) = #(ST:,,) _ P

r

Tay)

r

_ E (TRr) - G(Ti,) _ EO(T.,) + e. 1 (with 10, 1 < 1). K r

Analogously, from (6.11) and I -p(M) S gi(,11) it follows that we have also:

so(T.r - TRV) _ r(T=r) - Ego(TR,) = r +

c .y (Tsr) + 8;1)6. CA(A)

e: 5 1,

eR1)

r

\ + K) + 0K=)q

r

,

(where 0K) < 1,

{ < 1, and { 9;2) 1 < 1). If herein (6.2) is substituted, then

we obtain: So(B.) = c

((B1)_e.

(,i(A)+!:)+o2)q!:. ti)+e"o.

DERIVATION OF SET FUNCTIONS

§171

271

Thus since p(B.) - co(A) and ,l,(B,,) --> P(A ), it follows for K --' + ao that v(A) = c4G(A) + O (.4) with I 0 I <_ 1. 14, for every M e 9)Q then for every M C. D1 and 17.6.31. If I ,p(M) ! 5 every upper (or lower) tile derivate B(x) (or D(x)) of q with respect dtG\\)

(6.3)

so(M) = (M) f D(x) dj (or 'v(M) = (M) f D(x)

.

//every

Since I v(M) S qi(M) for every M e l)l, we have 115(x) I s q for x e B. Thus by 17.61 and 12.3.52 D (x) is -summable on M. Now let a > 0 and , ±k; kS > q); then Mc = M((i - 1)5 D(M) < iS1 (i = 0, f 1, f 2, +-+t M = S M, . Because of 17.6.2 and 9.1.2 Mi a M. By 17.6.3 p(M1) _ i-+k

(i - 1)8 (Mr) + Oja (M,), (I 0; I S 1). Therefore cp(M)

E (i

e ,(M), ((e ( 5 1), and hence by 13.1.1: cp(M) = (M) f 2e'5j (M), (I 0'

D(x)

d, +

S 1); since herein 5 > 0 was arbitrary, (6.3) is proved.

17.6.32. If I p(M) (< qy (M) for every M e T1, then the set of all x e E at which the tile derivative D(x, gyp, , Z) exists = , E. For from 17.6.31 it follows by 12.1.521 and 9.4.6 that

17.6.321. If jp(M) 15 q¢011) for every M e 91, then every tile derivate of p with respect to j' is 4'-measurable.

Proof as for 17.2.53. 17.6.322. If I ,p(M) I S qq(M) for every M e T1, then for every M e D1 and every tile derivate D*(x) of Sp with respect to

: ,p(M) = (M)

f D*(x)

Proof as for 17.2.54. 17.6.323. If X is a tile system (for 4,) and Bo is the set of all a e E for which in X. there is a sequence ((T,)) converging to a with (T,) = 0, then Eo =,c A. Proof as for 17.2.542 whereby 4, always has to be replaced by j. and 41 5 4 [§3 (4.21) and (4.2)1, there results from Since I ¢ 15 17.6.32:

17.6.33. The set E* of all x e E at which the three tile derivdtives D(x, ,y, , Z), D(x, 4,}, , 2r), and D(x, 4,-, +7',T) simultaneously exist = E. Then one proves in the same manner as 17.2.551:

17.6.331. If E* is the set of 17.6.33 and Eo is the set of 17.6.323, then

on E* - Eo(=# E): D(x, tG+, t7, :) + D(x, V, >

,

5) = 1

1' It follows from 17.2.63 that such a condition is essential.-As to the significance of the above condition, cf. 17.6.5.

[CHAP. V

DIFFERENTIATION

272

and

XT, e,1, X) - D(x, V, V r) = D(x,,P, , X) If E = E+ + E- is again a decomposition (2.4), then one proves in the same manner as 17.2.552: 17.6.332: If Z is a tile system, then D(x,

Z) =#.1onE+,

D(--,

Z)

0 on 9-;

D(x,

T) = # 0 on E+,

D(x,14

Z)

1 on E-;

D (x,

T) = 1 on E+,

X) = ,r -1 on E.

D (x,

17.6.4. If Z is a tile system and if I p(M) I S q &(M) for all M e FM, then

I D(x,,p,,P,T) 15i.gandI ?)(x, By definition I D(x, tp, ,fi, Z) 15 Tun

15,p q.

`for all sequences of sets T e Z,

I

which converge to x. According to 17.6.332, I D(x, ,y, that is, for all x e E except a zero-set for ¢,

(

I = 1 on E;

we have Jim - 1 for every T-+a

sequence of sets T e X. which converges to x. Therefore I D(x, ,e, Tun

I

14,(T) I

Tim

I

Thus from I ce(T) I 5 O (r it fol-

lows that I D(x, p, 4,, T) I ;5 # q. Now we generalize theorem 17.631, considering the tile derivates of 0 with respect to 4, (instead of ) : 17.6.41. If J 1o(M) 15 qy'(M) for every M e t', then for every M e 1R and every upper (or lower) tile derivate D(x) (or D(x)) of ,p with respect to it': (6.4)

'a(M) _ (M) f D(x) dip (or 'p(M) = (M) f D(x) 4)

.

If ft is the set of a l l x E M for which 11'5(x) 15 q, then by 17.6.4 M - i M. Now one proves 17.6.41 in the same manner as 17.6.31, replacing M by M there and taking into consideration 12.1.53 as well as the fact that cp(M - M) - 0 (as a consequence of i/i(M - M) = 0). The following theorems are now proved (by means of 17.6.2 and 17.6.41) as were the theorems 17.2.52, 17.2.53, 17.2.54, and 17.2.541: 17.6.42. If I p(M) 15 for every M e Itfl, then the set of all x e E at which the tile derivative D(x,,p, CZ) exists =,c E. 17.6.43. If I p(M) 15 O(M) for every M e J1, then every tile derivate of so with respect to ' is 4-measurable. 17.6.44. If I p(M) I S for every M E l , then for every M e R and every tile derivate D*(x) of co with respect to 4,:

,p(M) = (M) f D*(x) do. 1f Cf. footnote 14, p. 271.

§17]

DERIVATION OF SET FUNCTIONS

17.6.441. If I p(M) j 5 q (M) for every M e

273

and A =, A, then for every

tile derivate D*(x) of rp with respect to 4,: D*(x) =,p 0 on A.

From 17.6.44 there results according to 12.1.521: 17.6.442: If I V(M) 15 O(M) for every M e SJJI, then for any two tile derivates D*(x) and D**(x) of p with respect to ¢: D*(x) =,. D**(x)'o

One proves in the same manner as 17.2.542 (taking *3(4.21) into consideration): 17.6.45. If X is a the system (for P) and Eo is the set of all a e E for which in 7, there is a sequence ((T,)) converging to a with ib(T.) = 0, then E == # Al. 17.6.451. If 2r is a tile system (for ,y) and Eo is the set of all a e E for which there is a sequence ((M,)) converging regularly to a (for ¢ relative to Z) with 4'(M,)

=0,then Eo='A.

This is an immediate consequence of 17.6.45, since the sets designated by Eo in both theorems are identical. Analogously to 17.4.32 one proves (by means of 17.6.332 and 17.6.451): 17.6.452. If X is a tile system (for 4'), then the set of all x e E at which the derivatives D(x, 4+, lG, Z), D(x, 1F, ¢, Z), and D(x, ¢, yG, Z) are simultaneously regu-

lar = # E. 17.6.453. If I v(M) I S q'(M) for every M e 0 and if Z is a tile system (for 4L), then the set of all x e E at which D (x, .r, 4, Z) is regular = ,c E. Since I cp(M) 15 qr (M), v is 4i-continuous. Now the proof of 17.6.453 is

quite analogous to that part of the proof of 17.4.33 in which rp was assumed to be 4-continuous. Hereby take into consideration that I p(M) - c4'(M) 15 (q + jcl+'(M) and that one has to employ 17.6.4 (instead of 17.2.41). For V and 4, the previous assumptions (No. 1) are to hold. 17.6.5. In order that I V(M) 1 :5 qRM) for every M e 41Jt, it is necessary and sufficient that ,y(M) = (M)

f f d4' for every M e SR where f is a 4.-bounded Q9,1)

function on E. NECESSrrY17: Sp is totally additive and, as a result of I p 15 q, ', also }-con-

tinuous in 0; hence by 12.4.1 ce(M) = (M) f f d¢. If q = 0, then by 12.1.521

f = # 0. Thus let q > 0. Again let E = E+ + E- be a decomposition (2.4). If f were not f-bounded on E, then f would not be 4.-bounded on at least one of the two sets E+ and E-. Let f be non-'-bounded on, say, E+. Then thereis a

set Mo C E+ with Mo 0 A, such that on Mo either f z 2q or -f 2q; we assume, say, the first to be the case. Then by 12.1.32 2go(Mo) = 2g4(Mo) 5 (Mo) f f d# = p(M.), while by assumption j p(Mo) 15 qti (Mo).

SUFFICIENCY:

1' Here the tile systems employed in forming D*(z) and D**(x) can be different. IT If in B there is a tile system Z for 4,, then it is simpler to reason in the following way:

By 17.8.41,(M) - (M) f D(z) d¢ and by 17.8.4 1 D(z) 1

;5,c q.

CHAP. V

DIFFERENTIATION

274

If p(3f) = (111)

hp and I f 1 <,F q, then by 12.1.53 and 12.2.311 1 V(M) 1

qi (il) for all M e S)?2.

17.6.51. In theorem 17.6.5 the inequality I (p(M) 1 5 by ,x(11)

can be replaced

<_

NEe1 ssrrY: This follows from 17.6.5, since I (p(M) 1 S rp(M). SUFFICIENCY:

if p(M) = (.11) J f dv', then by 12.2.3 p(M) = (M) f I f j and 12.2.311 it results from I f 1 5 y q that FP(M)

Thus by 12.1.53

qkM).

Analogously to the orderly system .i] (§17,4) we formulate the definition: :in indefinitely fine system 1 c 9W shall be called a normal system (for 4,) if there is a the system Z (c 9t and for ¢), such that B is a regular system relative to (fur y). Every tile system is also a normal system. If P is content-like, then by 17.6.1 every orderly system (for ¢) is also a normal system (for 4,), but in general not converse] y.

If lB is a normal system and if the function D°(a), for every a e E, equals a derivate of e with respect to 4, on V a , then we call D°(a) a normal derivate of cp with respect to 4, (on i3). 17.6.6. 1 f '; 9(11) { < q¢ (M) for every M e I)t and if D°(x) is a normal derivate of

with respect to ,r and D*(x) is a tile derivate of rp with respect to ¢, then

1)°(x)

_

D*(x).

Proof is for 17.4.4. 17.6.61. If I p(1L1) ; 5 4(11) for every 31 a 97t, then for any two normal derirates D°(ax) and D`p(x) of ,p with respect to 4' we have: D°(x) _ D00(x).

Proof as for 17.4.41. Boo I" ,hu ov: R. De PossEL, Paris C. R.. 201 (1935), p. 579; Journ. de math. (9) 15 (1936), p. 391; B. lou,oviTCn, C. R. Acad. URSS.(N.S.) 30 (1941), p. 112.

§18. Applications 1. Density of a set. Again let E be a metric space, let T1 be a a-field consisting of subsets of E, and iet E e R)1. Moreover, let 1G be a totally additive set function in 9)1 and let 9)1 be complete for ¢. Furthermore we assume 4' to be finite. But the results can then immediately be generalized for the case in which E = SE, where the Ei are open sets with finite 4'(6)18; (1)

in this case one has to perform the analysis for the single E; and then subsequently one has to pass on to E (taking 4.1.32 into consideration). ' Then every sphere Sap , for sufficiently small p, is contained in an E; and for every 'k-measurable suI,s,r; 11 of &, the value ¢0I) is finite.

275

APPLICATIONS

§18]

According to 6.5.2 and 6.5.22, the absolute-function of f can be extended The corresponding inner ,i-measure (§6,2) is designated by 'fix . Let C. be an indefinitely fine system (§17,1) with C c fit and let O, be the subsystem of C. which is attached to the point a e E. Let A be any subset of E. For every sequence ((Q,)) of sets of C. which converges to a we form

to a regular measure function.

the sequence of

(Q,)

(in case i(AQ,) = (Q.) = 0 this quotient has

to be replaced by 0). Among all values of accumulation of all these sequences of numbers there is a greatest one and a smallest one; we call them the upper and the lower outer density of A at a for ¢ on fD and designate them by

d(a, A, , C) and respectively.

d(a, A, 1G, e),

Obviously we have:

0 5 d(a, A,'', 0) S d(a, A,

1) 5 1. If herein d = d, then this common value is called the outer density of A at a for on C and is designated by d(a, A, , C). We have: (1.11) 0 S d(a, A, , C.) 5 1. If in these definitions one replaces the -measure J by the inner -measure (1.1)

,

'x , then one obtains the upper and the lower inner density of A at a for y' on C, which shall be designated by dx (a, A, i , C) and dx (a, A, ', £1), respectively,

as well as the inner density dx (a, A, , C). Then we have the inequalities: 0 5 dx (a, A, C) 5 dx (a, A, ,7, 1) 5 1; (1.12)

(1.13) dx (a, A,', C) 5 d(a, A, yG, C) and dx (a, A,

5 d(a, A, , Q);

0 5 dx (a, A, C) S d(a, A,', Z1) 1. If dx = d (or dx = d)-which by 6.2.3 certainly is the case if A is J,-meas-

(1.14)

urable-, then we simply speak of upper and lower density, respectively. Then

if d(a, A, 'L, 0) = d(a, A, 17, 0) = dx (a, A, ', C) = dx (a, A, ', C), this common value is called the density d(a, A, of A at a for ' on Q. 18.1.1. If A Q B, then d(a, A,

dx(a, A,

C.) 5 d(a, B, y , £1.),

e) 5 dx (a, B, 'G, Q,

a(a, A,'., £) 5 d(a, B, ' , 0), dx (a, A,

C) 5 dx (a, B, t7, Q.

For from A C B it follows that F(AQ) 5 ; (BQ) and 'X (AQ) 5 'fix (BQ). 18.1.11. If Ax is a measure-cover of A, then for every a e E:

d(a, A, ', C) = d(a, A X, ,G, C),

d(a, A, ', Q) = d(a, Ax,

If Ax is a meamire-kernel of A, then for every a e E:

d x(a, A, ', 0) = d(a, Ax

,

t7, Ci),

dx (a, A, t7, C) = d(a, Ax

, +G,

0)

276

DIFFERENTIATION

[CHAP. V

For since all Q e £Z are *-measurable, we have F(AQ) _ &(AxQ) and it'x (AQ) = #(Ax Q) We desigflate by the set of all a e E for which in l;.la there is a sequence ((Q,)) converging to a with (Q,) - 0. Then .

0forallaeL.

(1.2)

18.1.2. If C is a Vitali system (§17,2) or a tile system (§17,6) (for ¢), then =j, A. Let C = Q3 be a Vitali system. To every a e . let a sequence ((V.,)) of Q3a19 be attached which converges to a and for which (Va.) = 0. Then, according to the property 3e) of the Vitali systems, among the Va, (a a L+s, v = 1, 2, ) there are countably many disjoint sets V1, V2, , V' , such that

R-SV'=,`A. But from y (V) = 0 it follows that (SV) = 0 also, and hence (R) = 0; that is, 2 =,c A.-If C is a tile system, the discussion is analogous. 18,1.21. For every aeE - R: (1.21)

dx (a, A, , 0) + d(a, E - A,

1 and

+d(a,E -

1.

If C is a measure-kernel of A, then by 6.3.31 E - C is a measure-cover of

B - A. Since for every Q e Z: ¢(CQ) + y ((E - C)Q) = (Q), we have also T(CQ) + C)Q) = 1, if 4,(Q) 0 0. Thus the contention follows from 18.1.11.

18.1.22. In order that d(a, A + B, ¢, £l) == 0, it is necessary and sufficient

that d(a, A, , Cl) = 0 and d(a, B, , 1;1) = 0. NECESSITY: This follows from 18.1.1 and (1.1). Stpplcmxcy: If ((Q,)) is a sequence of C, converging to a and if S > 0 is arbitrary, then (AQ,) 5

$ ,(Q,) and (BQ.) J(BQ,)

SJ(Q,) for almost all v.

Thus ¢((A + B)Q,)

24(Q,) for almost all v, and hence ((A + B)Q')

d(a,A+B,

(AQ,) +

0; that is,

C) = 0.

18.1.221. In order that dx (a, AB, yZ, £Z) = 1, it is necessary and sufficient that

dx (a, A, , £1) = 1 and dx (a, B, ', £1) = 1. NEcBssrrY: This follows from 18.1.1 and (1.12). Surrxcmxcr: Certainly a e E - 2, since otherwise by (1.2) we would have dx (a, A, , C) = 0. Thus

it results from 18.1.21 that d(a, E - A, , 0) = 0 and d(a, E - B,, 0) = 0. Therefore by 18.1.22 d(a, E - AB, J, £Z) = 0, and hence again by 18.1.21:

dx (a, AB, , 0) = 1.

10 According to 117,2, FAa designates the system of all V E S which contain a.

§181

277

APPLICATIONS

18.1.23. If p > a(a, A, , C), q < d(a, A, &, C), px > ax (a, A, , V, qx < dx(a, A, J, C), then there is a p > 0, such that for all Q e Q. contained in S,,:

F(AQ) 5

F(AQ) ' lax (AQ) 5 p%(Q),

?x (AQ) ? q% (Q).

in Q. with

But

Otherwise there would be a Q.

p, and hence

then ((Q.)) is a sequence of Q. which converges to a with

we would have d(a, A, , C)

p.

Now Z is to designate a tile system for 4, (and hence by 17.6.11 also for r'): 18.1.3. For every A a E the set of all a e A at which d(a, A, , X) = 1 does not hold = . A.

Let B be a measure-cover of A. We set for all M e 92: p(M) = (BM). If we designate by f the function which equals 1 on B and equals 0 on E - B, then by 12.1.61 sp(M) = (M) f f day. Simply writing D(x,

D(x)

and D(x, rp, y ,.T) = D(x), we obtain from 17.6.31:

,P(M) = (M)

f f d _ (M)

f D(x) d;7, = (M)

f D(x) di.

Thus by 12.1.521 .(x) =; D(x) = f on E, and hence 15(x) =; D(x) = i 1 on B. But f(a,,p, , T) = d(a, B, X) and D(a, p, , Z) = d(a, B, yTi, Z), and thus by 18.1.11 also D(a, (p, ,b, X) = a(a, A, , T) and D(a, cp, , Z) = 4(a, A, , Z). Therefore d(x, A, j', Z) _; 1 on B. Thus the set of all a e B (and hence also the set of all a e A) at which d(a, A, I does not hold A, which by 4.1.12 is equivalent to =* A. 18.1.31. For every A E the set of all a e E --- A at which dx(a, A,, 0 does not hold =,p A. Let C be a measure-kernel of A. We set for all Mefil: p(M) A(CM). If we designate by f the function which equals 1 on C and equals 0 on E - C, then

f

by 12.1.61,p(M) _ (M) f 4. Then we obtain again (as in the proof of 1&14).,

15(x) =; D(x) =; f on E, and hence D(x) =; D(x) _; 0 on E -- C. But

.D(a, j o, y;, Z) = d(a, C, ,7', Z) and D(a, cp, , Z) = d(a, C, , X), and thus by 18.1.11 also 15(a, c, tG, X) = dx(a, A, , Z) and D(a, gyp, 1 , ) = dx(a, A, ¢, Z). Therefore dx(x, A, vy,) _; 0 on E -- C. Thus the set of all a e E - C (and hence also the set of all a e E - A) at which dx(a, A, ,11r) = 0 does not hold = y A. The following theorem is contained in 18.1.3 and 18.1.31: 18.1.32. If A is 4,-,measurable, then the set of all a e A at which d (a, A, Z)

1 holds =# A and the set of all a e E - A at which d(a, A,

==,E-A.

S) = 0 holds

18.1.33. In order that the set A Q E be ,y-measurable, it is necessary and sufficient that the set of all a e A at which dx(a, A,', 1) = 0 holds = 0 A.

278

[CHAP. V

DIFFERENTL4TION

NECESSrry: If A is 4,-measurable, then by 18.1.32 the set of all a e A at which

dx(a) < 1 holds =, A. SUFFICIENCY: Let C be a measure-kernel of A; then it

suffices to show that A - C =,. A. We set: A - C = A' + A" where A' designates the set of all points of A - C at which dx(a, A, , T) = 0 and A" designates the set of all points of A - C at which dx(a, A, ,y, Z) > 0. By assumption A' = v A. Because of 18.1.31 the set of all points of A '- C at which dx(a, C,

T) > 0 holds = , A; hence by 18.1.11 A"

A also.

There-

fore A - C = A' + A" implies that A - C =,p A also. If 4, is content-like, then by 17.6.1 every Vitali system 3 for 4' is also a tile system for 4,. Thus we have: 18.1.34. If ¢ is content-like and $3 is a Vitali system for 4', then in the theorems 18.1.3, 18.1.31, 18.1.32, and 18.1.33 T can everywhere be replaced by Z. We say that for an indefinitely fine system e the density theorem with regard

to 4, holds if for every A C E the set of all a e A at which d(a, A, 4, C) = 1 does not hold =,, A. The statements that the density theorem holds with regard to 4, or with regard to are equivalent, because of 4.1.12. By 18.1.3 and 18.1.34 the density theorem with regard to ¢ holds for every tile system $ associated with ¢ and, if 0 is content-like, for every Vitali system associated with ,'. 18.1.35. In order that for an ordinary indefinitely fine system C the density theorem (with regard to ¢) hold, it is necessary and sufficient that Z. be a tile system (for 4').

NEcEssrry: Let A c E with (A) > 0 [cf. footnote 12, p. 2681 and let C' he a subsystem of C which to every a e A and to every p > 0 contains a set Q e C. with Q c S.p. We have to show that for every given e > 0 the property 3,) (§17,6) is satisfied (if there Z and T is everywhere replaced by C and Q). We choose y with 0 < y < 1 such that

y (A) 5 '(A) + e; moreover, let X

be a fixed number with 0 < A < 1. We designate the subsystem C. C' by 0.a. If B C A, then let CB be the system of all those sets Q e Ca with b e B for which (1.3)

+G(BQ) > y.(Q).

Furthermore, set as = sup j(Q) for all Q e CB. If (B) > 0, then because of the density theorem there is a point bo e B with d(bo , B, case;, * A and ae > 0. Therefore: (1.31)

1; thus in this

as = 0 implies ¢(B) = 0.

Now we define the Q, (of 3,)) by induction. First let Q, a 0; with (Q,) > Aa4 ; we set A, = A and A2 = A -- Q, . If (A2) = 0, then 3,) is already satis-

fied for our given e, since by (1.3) (A) z j(AQ,) > y'(Q,), and hence (Q,) < I y4(A) 5 y (A) + e. But if >ji(A2) > 0, then we repeat the same procedure for A2 (instead of A) and obtain Q2

.

Let Q. (x = 1, 2,

, v) be already defined

279

APPLICATIONS

§18)

and let A.+1 = A - S Q; with (AK+1) > O (K = 1, 2,

, v).

Then let

_

Q,+1 E OA.+1 with y'(Q,+1) > XQA,}, ; we set A,+2'= A,+1 - Q,+1= A - S Qi i_i

;!K

then by 6.3.34 the set Ax - S Q; _

Let Ax be a measure-cover of A (for

;..1

;_K

A x 1 is a measure-cover of A K., _ A - S Q; . By 6.3.23 A; QK is a measurei_1 , v + 1) are disjoint. Now if cover of A.Q. , and the sets A xQ.(1; = 1) 2, (A,+2) = 0, then 3e) is again satisfied for our e; for by 6.3.32 and (1.3) we have

K_,+1

FG(A)

(A

S QK)

=G

K_1

(1.32)

T

K..1

K_,+1

and hence

Kr,+1

K_Y-f-1

S AKQK) K_1

= i(

K..,+

K_V +1

E

(

S A XQK

[_3

K_,+.1

i(AK QK) > ! (A QK) = E K_3

E K_1

T

(QK),

1

(QK) < ¢(A) 5 (A) + e. But if (A,+2) > 0, then continue

-

the procedure. If no ¢(A,) = 0, and hence the sequence ((Q,)) is infinite, then 3,) is also satisfied for our e. First we obtain, just as in (1.32), that <

' (A) + e. Herein, since,y(A) is finite, E (Q,) is also finite, and hence (Q,) > XoA, implies that E QA, is also finite; thus oA, -+ 0. Since A*

.9 ,.(v = 1, 2,

), we have;,

Set A* = A - SQ,

A, and

.

5 QA, ; therefore

QA, = 0, and hence by (1.31) z(i(A *) = 0; that is, A - SQ, =,& A. Thus in every case 30 holds for our arbitrarily given e > 0. SuFFICIF,Noy: This is theorem 18.1.3. 18.1.36.

If 2 is a tile system (for ¢), then the density theorem (with regard to ,l.) holds also for every system Z which is regular (for ¢) relative to 2 (§17,4). Let A C E and let B be a measure-cover of A for jfi. Moreover, let E be the set of all a e E for which in T. there is a sequence ((T,)) converging to a with

y (T.) = 0; by 18.1.2 E = A. According to 17.6.332, 1 D(x, 4,, E; that is, for all x e E, except a zero-set B for ', we have lim

O(T)J

on

=1

for every sequence of sets T e Z. which converges to x. By 18.1.32 the set of all a e B at which d(a, E - B, , T) = 0 holds =,p B. Let oo be such a point of

B - (R + P) and let ((Q,)) and ((T,)) be sequences of sets of 0% and .,, respectively, which converge to ao, such that for almost all v: Q, Q T, and 14,(Q,) 1 z r(ao) 14,(T,) I (with r(ao) > 0). Then we have for almost all v:

[cHAP. V

DIFFERENTIATION

280

((E - B) TO j(T,) I i(T.)1

(T.)

4W.) I 14,(Q.) 1

(Q,)

((E - B) TO

= FR

-41 (T

(T,) I,b(T.) I

Thus from B) T,) RT,)

0 and

+G (T,)

-' 1

14, (T.) I

- B) - Q`) -- 0; that is, the set of all a e (B - (E + X)) at PQ,) B. Hence for every such a we have by which d(a, E - B, , 0) = 0 holds 18.1.21: d(a, B, y , 0) = 1, and thus by 18.1.11 d(a, A, , C) = 1 also. Thereit follows thatj((E

fore, since B Q A, the set of all a e A at which d(a, A, , C) = 1 does not hold =,p A; that is, we obtain the density theorem for £1 with regard to 0. From 18.1.35 and 18.1.36 the following two theorems result immediately: 18.1.361. If Z is a tile system (for 4'), then every ordinary indefinitely fine system C which is regular (for 4') relative to is also a tile system (for 4'). 18.1.362. If for an ordinary indefinitely fine system £1 the density theorem (with, regard to 1G) holds, then it holds also for every regular system (for ,') relative

to Z. Now we assume temporarily: for every a e E we have {a} a V. (1.4) Then let £1 be an indefinitely fine system. If Q e £1, , we set Q + (a) = Q*; the system of all these sets Q* (for all a e E) shall be called the ordinary system

0* (C. SD?) associated with 0, provided that 0; means the subsystem of all Q* a £1* which contain x. 18.1.4. Let (1.4) be satisfied. Let 0 be an indefinitely fine system and assume the density theorem (with regard to 4') to hold for the associated ordinary system £Z*30. Then in order that the density theorem (with regard to 4') hold also for £1, it is necessary and sufficient that every discontinuity-point a of 4, be contained in

all sufficiently small Q e W. Nucassrry: Suppose there is a discontinuity-point a of 4,(%5,3) and a sequence

p, , 0 with Q. C S, and Q, a >0a , but a - e Q,. Then ,((a } Q,) = j(i(A) =0, and hence

({ a) Q,) = 0 also; thus we have d(a, Jai, j, 0) = 0. But since

by 5.3.3 J ((a }) > 0, the density theorem does not hold for the set A = jai. SUFFICIENCY: Let ((Q,)) be a sequence of Q.,, converging to a, and let Q; _20 Or what by 18.1.36 amounts to the same thing: assume' to be a tile system (for PP). 4l That is, to every discontinuity-point a of 4, there shall be a p > 0, such that for every Q e Q. with Q C; Sap we have also a e Q.-By 6.8.51 this condition is satisfied trivially if 4, is continuous.

281

APPLICATIONS

1181

Q, + {a} (e

*). If a is a discontinuity-point of J,, then by assumption for almost all n: Q.* = Q, . But if a is a continuity-point of 4,, and hence .({ a)) = 0,

1 ({ a 1) _ (Q,.); thus for all v: (Q:) = (Q,). then t (Q.) 5 S (Q,) Therefore C is a regular system for relative to C. Hence by 18.1.362 the density theorem holds also for C (with regard to j' and ¢). The converse of theorem 18.1.4 is not true; that is, even if (1.4) is satisfied

and every discontinuity-point a of 4, is contained in all sufficiently small Q e ;tla , the density theorem can hold for C without holding for 0*. Example: Let E be an open interval of R2 and let 4' =A2 (which is continuous by 8.2.22). The closed

squares in E which contain a shall be designated by I.. Let the system £0a consist of all sets Ia + {x} where x denotes any arbitrary point of E, and let

0 be the sum of all Ca (for all a e E). According to 17.5.5, 8.2.22, 18.1.3, 18.1.34, and 8.2.1, the density theorem with regard to A2 holds for Q. We have

always Q* = Q, and hence £1* = C. But C. 0a ; for 0; consists of all sets Q e 1;1 which contain a, that is, of all sets Ia + ( x) and all sets 1, +- { a). Now let A be a closed, nowhere dense set in E with p2(A) > 0 (cf. 8.2.8) and let a e A. We designate by 1; the I. which are disjoint from A and set Q/* I9+ {a} (e0Q). Then p2(A.Q'*) =p2({a)) =0,and henced(a,A,p2,£.1*)=0 for every a e A, while A2(A) > 0; that is, the density theorem (with regard to p2) does not hold for W. 2. Density of a set in R. . Now in particular let E = R. and 4' = A. ; then (1) is satisfied. If A is an arbitrary set of R. and Sa, is the open sphere with center a and radius p, then we set: (2)

d(a, A) - lim

,-o+ n(Sap)

dx(a, A)

dim

P-O+

px(ASa,)

d(a, A) = lim `"(ASa,) pu(Sa,)

dx(a, A) = lim

/-.o+

Aftx(ASa,) Aw(Sa,)

and we call these four numbers the upper and the lower outer density of A at a and the upper and the lower inner density of A at a, respectively. If d(a, A) =

d(a, A) (or dx(a, A) = dx(a, A)), then we call this common value the outer density d(a, A) (or the inner density dx(a, A)) of A at a. If d = dx, d = dx, d - dx , then one simply speaks of the (upper, lower) density of A at a; this is certainly the case for a µ,-measurable A. 18.2.1. For every A C & the set of all a e A at which d(a, A) = 1 does not hold = Px A. Since (1) is satisfied for E = R. and 4, = it suffices to prove the contention for the set AE in the space Er E. By 17.5.3 the system of the spheres S,, C Ei forms a Vitali system Si of the space E: for A. . Since at every point at which d(a, AE; , A. , (a;) = 1 we have also d(a, AE{) = 1, the contention follows from 8.2.1, 18.1.34, and 18.1.3.

DIFFERENTIATION

282

[CHAP. V

Now from 18.1.21 there results: R. the set of all a e R. - A at which dx(a, A) = 0 18.2.11. For every .4

does not hold =a, A. The following theorem is contained in 18.2.1 and 18.2.11: 18.2.12.

If A C R. is }In-measurable, then the set of all a e A at which d(a, A) = 1

holds = µn A and the set of all a e R. - A at which d(a, A) = 0 holds =,,,, R. - A. 18.2.13. In order that the set A C Rn be g.-measurable, it is necessary and sufcient that the set of all a e A at which dx (a, A) = 0 holds =,,,, A. Proof as for 18.1.33.

By 17.5.5 or directly by 18.1.362 one can replace the spheres used in the discussion above by n-dimensial cubes (parallel to the axes) or by intervals regular relative to the spheres; that is, for every a e R. the system of the spheres with center a can be replaced by the system of the intervals which contain a and for

which the quotients of the edges are not smaller than >:(a)(> 0), where (a) depends only on a. Then one obtains also the theorems corresponding to 18.2.1, 18.2.11, and 18.2.12. Thus for n = 1 the system of all intervals can hereby be employed. But, over and above this, one can obtain the corresponding theorems also for the system of arbitrary intervals of Rn , as we now will show, although accord-

ing to 17.5.51 this system (for n > 1) does not form a Vitali system for µ . Let 3n be the system of all open intervals Jr. of R. ; to every a e R. let the subsystem n,a of all those J. be attached which contain art The (upper and lower, outer and inner) densities of A(C; Rn)(,, formed in this way on d(a, A, A.

, J),

4(a, A, An

,

Jdx(a, 4, An

dx(a, A, An , Jr.),

Jn), r

3n)2

d(a, A, An

,

3.,

Jn);

dx(a, A, IA., an)

shall simply be designated by S(a, A), E(a, A ), S(a, A) ; Sx(a, A), bx(a, A), Sx(a, A), respectively. 18.2.2. For every A C R, the set of all a e A at which S(a, A) = 1 does not hold Mn

A.

For n = 1 our proposition is true, according to the above remark. Now we prove the theorem for it = 2; the proof is analogous for n = 3, 4, .-If A>' is a measure-cover of A (which by 6.3.41 exists), then it suffices, according

to 18.1.11, to prove our theorem for Ax (instead of A). By 7.2.251 Ax =

SI', + N where N is a zero-set and the F, are closed, bounded sets; thus, according

to 18.1.1, it suffices to prove our theorem for these F,

.

Hence from the first

n Thus.Jn is defined as an ordinary indefinitely fine system. This is essential for the validity of the subsequent density theorem 18.2.2. For let 3' be the system of all open intervals J. of R. where to every a e R. the set 3n',a of all those intervals J. is attached which are contained in Sai ; and let A be a nowhere dense, closed set of R with is. (A) > 0 (cf. 8.2.8). If we now employ the intervals Jn e &. which are disjoint from A, we we that for every a e R. we obtain: 4(a, A,,,,,, 3,',, - 0.

283

APPLICATIONS

§181

A can be assumed to be closed and bounded.-As in §16,1, we designate by M the

set of all x for which (x, y) e Al. Now let e > 0. For every integer K > 0 let B. be the set of all points a* = (x* , y*) e A for which in all linear open intervals z (1 - e)µ,(J). The seJ containing x* with length < 1 we have: quence ((B.)) is monotone increasing. Moreover, every B. is closed. For if the points a, _ (x, , y,) a B. converge to a' = (x', y') and if J' is a linear open

interval containing x' with length < 1 , then x, e J' for almost all v and hence K

Ml(AY,J') z (1 - e)µ,(J'). Since A is closed, we have A,,.J' Q Lin (A,,,J), and thus by 3.5.11 µ,(Aq.J') z Ml (Lim (A,,,J')) z lun µ1(A,,,J') z (1 - e)p1(J'). , Therefore a'e B. , and B. is closed.--Thus the set T = A - SB. is µs-measurable.

If (x, y) a T, then for every x there is a linear open interval J containing x with

length < 1, such that µ,(TYJ) 5 M,(A,,J) < (1 - e)µ1(J). Since this holds for K every x a T, and our theorem 18.2.2 is already proved for n = 1, it follows that 0. Therefore by 16.1.4 and 16.1.24 µ2(T) = 0 also. Thus since the sequence ((B.)) is monotone increasing, we have for a sufficiently great Ko by 3.2.21: i.12(A - B.,) < e. Now Ave simply write B instead of B., .-If one replaces A by B and interchanges x with y, then one finds in the same way a closed set C., Q B (with K, > Ko), such that p4(B - C.,) < E and C.1 is the set of all points a* = (x*, y*) e B for which in all linear open intervals J containing y*

with length < I one has: µ,(B, J) z (1 -

µ,(J).-Now let a* _ (x*, y*)

be any point of C., and J* = (xi, yl; x2 , y2) an open interval containing a* with diameter < 1 ; we set: (xi , xx) = J', (yl, y2) = J". By 16.1.4 and 16.1.24: Kl

Y

(2.1)

f µl((AJ*).,) dy.

µ:(4J*)

Yl

Since x* a J' and x: - xl < 1 < KI

we have re

M1((-4J*)Y) = µl(A0J') ? (1 - e)(xs - xl),

provided that (x* , y) a B(= B.,) and y e J", that is, provided that y e B=,J". Moreover, since a* e C,., , y* e

J", and y2 - yl < 1,K1 we have

µ1(B.. J") z (1 - e)(y2 - yi) Thus by (2.1), 12.1.7, and 12.1.6 142(AJ*) it (1 - e)2(x2 - x,)(y2 - y,) (1 - e)2µ2(J*). Hence b(a*, A) z (1 - e)2 for every point a* a C., , and

µ2(A - C.,) = p 2(A - B) + M2(B - C.,) < 2e.

_

CHAP. V

DIFFERENTIATION

284

Therefore if D. is the set of all a e A with $(a, A) < (1 - e)=, then p,(D,) < 2e. But e > 0 was arbitrary; with e, -+ 0 the sequence ((D.,)) is monotone increasIf D;, is a measure-cover of D., , then by 6.3.32 ing; we set SD., = Do SD ; = D(X) is also a measure-cover of D., and SD;, - SD2 = Do is a measure-01

cover of Do. Thus from WD.°) < 2*, it follows by 3.2.2 that ,(D,) = 0, and hence p,(Do) = 0 also; that is, the set of all a e A at which 3(a, A) = 1 does

not hold =,A. 18.2.21.

not hold =

For every A Q R. the set of all a e R - A at which S x (a, A) = 0 does A.

0. This follows from 18.2.2 according to 18.1.21, since by 8.2.3 always p The following theorem is contained in 18.2.2 and 18.2.21: then the set of all a e A at which 6(a, A) = 1 18.2.22. If A Q, R. is holds =,,, A and the set of all a e R - A at which 3(a, A) = 0 holds =,, R. --- A. The following theorem results from 18.2.2 and 18.1.35, if we take (1) and 6.1.32

into consideration:

1*23

of all open intervals of R. forms a tile system for p . From 18.2.23 and 18.1.33, in consideration of (1), there results: it is necessary and suf18.2.24. In order that the set A C R. be cient that the set of all a e A at which $X(a, A) = 0 holds =R, A. 18.2.23.

The system

Now we prove a theorem which contains 8.2.7 as a particular case. To each , b.), we attach , a,,) and b = (b, , b2 , pair of points of R. , a = (a, , a2 ,

a point a - b = (a, - b, , a2 - b2 ,

, an -

(which can also be considered

as a representative of the vector ab). Let A and B be two point sets of R. ; to every point a e A and every point b e B we form the point a - b and designate the set of all these points a - b by V(A, B). We employ again the densities defined by (2). sets A and B of R have the density 1 in each of 18.2.3. If the their points, then the set V (A, B) is open.

Let c e V (A, B) ; it has to be shown that there is a a > 0, such that c'c < o implies c' e V (A, B).

Since c e V (A, B), there is au a e A and a b e B, such that c =

a - b. Since d(a, A) = 1 and d(b, B) = 1, to every S > 0 there is a p > 0, such that designating by

the n-dimensional measure of a sphere with

radius p we have:

S.,) > (1 -

and

(1 - 3)8.(p), and hence

p,(S., - A) < Ss (p) and p (Sb, -- B) < Ss (p). Moreover, there is a a > 0, such that pa < u implies: (2.2)

(2.21)

/1.(Sovsn) > (1 - 5)8.,(P)-

- One should notice here that a was defined as an ordinary indefinitely fine system; of. also footnote 22, p. 282.

APPLICATIONS

§181

285

, c;,). Let b' be the point which from b Now let c'c < a and c' = (ci , c2', , n) and let B' be the set is derived by the translation xi = x; + c{(i = 1, 2, which from B is obtained by the same translation. Then from b' - b = c' and and a - b = c it follows that b' - a = c' - c; thus b'a = c'e, and hence b'a < c. Furthermore, because of §1(1.2) and 3.1.42 we have: AB' a

[(S.,, - A) + (S,', - B')) and .(AB') A.(S,PSe'P) - >a.(S.P - A) - pn(Sb'P - B'). and since Since b'a < o, we have herein by (2.21): p.(S41Sb-P) > (1 B') = p.(&, - B), it follows from (2.2) that p.(AB') > by 8.2.6

(1 - 38)s.(p). Therefore as soon as we choose 6 < }, we have AB' $ A.

Now let a* a AB'; then by definition of B' there is a b* a B, such that a* - b* = c', and thus, since a*e A, we have c'e V(A, B), as contended. 0, then the 18.2.31. If A and B are sets of R. with px(A) > 0 and open kernel (§2,1) of V(A, B) is not empty.

By 6.3.411 there are measure-kernels A' and B' of A and B, respectively. By 18.2.12 the set A" of all a e A' at which d(a, A') = 1 holds =,,,, A'; analogously B'. Thus we have the set B" of all b e B' at which d(b, B') = 1 holds

d(a, A') = d(a, A")

and d(b, B') = d(b, B"), and hence d(a, A") = I for all a e A" and d(b, B") = 1 for all b e B". Therefore by 18.2.3 V(A", B") is open. Since p.(A") = u.(A.') =

p.x(A), we have p.(A") > 0, and hence A" * A; analogously B" * A; thus V (A", B") * A. From A" c A and B" Q B it follows that V (A", B") Q; V (A, B), and so 18.2.31 is proved. 18.2.32. If A C R. and p.x(A) > 0, then V(A, A) contains a neighborhood of the origin.

Let A" have the same meaning as in the proof of 18.2.31; then A" $ A and by 18.2.3 V(A", A") is open. Thus since the origin is contained in V(A", A")

and since V(A", A") C V(A, A), 182.32 is proved. In 18.2.32 theorem 8.2.7 is contained as a particular case. BIBLtoozu rr to Nos. 1 and 2: The definition of the density of a set and theorem 18.2.12 (for n - 1) are due to H. LEBESOUE [1], p. 123; Math. Annalen 61 (1905), p. 266; Rendic. Accad. Lincei (5) 15, (1906), p. 8; Ann. ]to. Norm. (3) 27 (1910), p. 406; subsequent to him: CH,-J. nE LA VALUE POUSSIN [1], vol. II, p. 114; C. BURBTIN, Sitzungsberichte Akad. d. Wiss. Wien 123 IIa (1914), p. 1534; A. DENJOY, Journ. de math. (7) 1 (1915), p. 132; H. RADEMACHER, Monatshefte f. Math. u. Phys. 27 (1916), p. 192; W. SIERPI1 SKI, Paris C.R. 164 (1917), p. 993; Fund. math. 4 (1923), p. 167; N. LuSIN-W. SIERPI sRI, Rendic. Circ. mat. Palermo 42 (1917), p. 167; H. BLUMBERG, Bull. Amer. Math. Soo. 25 (1919), p. 350; Trans. Amer. Math. Soc. 24 (1922), p. 122; J. RIDDER, Nieuw Archief v. Wiskunde (2) 16, (1930), p. 72; F. Riasz, Acta Univ. Szeged 5 (1932), p. 218. As to 18.2.13: E. KAMxa, Fund. math. 10 (1927), 433; S. KAMETANI, Proc. Imp. Acad. Tokyo 16 (1940), p. 350:-As to 18.2.3-

18.2.32: H. STEINHAus, Fund. math. 1 (1920), p. 93; "Annexe" to it (tome 1, nouv. td., 1937), p. 232; 1i. RADEMACHER, Jahresber. d. Deutsch. Math: Verein. 30 (1921), p. 130.As to 18.2.2-18.2.22: S. SAxs [1), p. 231; [2], p. 129 (the above proof for 18.2.2 is taken after this); F. RIEsz, Fund. math. 22 (1934), p. 221; H. BUSEMANN-W. FELLER, Fund. math. 22

286

DIFFERENTIATION

[CHAP. V

(1934), p. 226; L. CssARI, Ann. Scuola norm. P a (2) 8 (1939), p. 301. While according to 18.2.2 the density theorem with regard to /s holds for the system of all intervals of R,,, it is remarkable that it does not hold for the system of all rectangular (arbitrarily oriented) parallelepipeds of R. ; this was shown by 0. NIKODYM-A. ZYGHUND, Fund. math. 10 (1927), p. 167, and H. BUBEMANN-W. FILLER, loc. Cit., p. 243.-As to 18.1.3-18.1.32, 18.1.35, and 18.2.23: R. DE PosssL, Paris C. R. 201 (1935), p. 579; Journ. de math. (9) 15 (1936), p. 391; B. YouxovJTcrf, C. R. Acad. URSS. (N.S.) 30 (1941), p. 112; cf. also H. BusEAfANN-W. FELLER. IOC. cit.

As to 18.1.362: H. BUSEMANN-W. FELLER, 1ee,. cit., p. 237.

It may be mentioned that for the "Caratbdodory linear measure" in IRA (§8,5; footnote 63,p. 105) the analogue to the second part of 18.2.12 holds, but not the analogue to the first part of 18.2.12 (or to 18.2.1); cf. as to it: A. S. BESICOVITCH, Paris C. R. 183 (1926), p. 553; Math. Annalen 98 (1927), p. 422; 101 (1929), p. 161; 110 (1934), n. 331; 115 (1938), p. 296; 116 (1939), p. 349; W. SIERPIfSKI, Fund. math. 9 (1927), p. 172; G. WALKER, Fund.math. 16 (1930), p. 108; Proc. London Math. Soc. (2) 30 (1930), p. 481; A. S. BESlcovrTcx and G. WALKER, ibid. (2) 32 (1931), p. 142; R. L. JEFFERY, Trans. Amer. Math. Soc. 35 (1933), p. 633 (for µk in J. GILLIS, Fund. math. 22 (1934), p. 57; Journ. London Math. Soc. 10 (1935), p. 234; G. W. MORGAN, Proc. London Math. Soc. (2) 38 (1935), p. 481; J. F. RANDOLPH, Annals of math. (2) 37 (1936), p. 336; D. R. DICKINSON, Math. Annalen 116 (1939), p. 358; A. P. MORSE and J. F. RANDOLPH, Trans. Amer. Math. Soc. 55 (1944), p. 236 (linear

measures in R, in general).

3. Approximately continuous functions. The general assumptions of the begin-

ning of §18,1 are to be satisfied. Again let C(c 9) be an indefinitely fine system (§17,1). Let the function f be +'-defined on E and let a e E be a point at which f is defined. Then f shall be called somewhat continuous above (or below )24 for P at a relative to `. if for every y > f (a) the set [f(!) < yj (or if for every y < f (a) the set [f(1') > yj) has a positive upper inner density for t at a on C. 18.3.1. Let k be a finite constant. If f is somewhat continuous above (below) for 0 at a relative to C and if f >,# k If 0 k), then f (a) ? k (f(a) S k). Suppose f(a) < k. We set [f(i) < k] = A; then f >_p Ik implies that A =,p A. On the other hand, since f is somewhat continuous above at a, the set A has a positive upper inner density at a and, since by 6.1.5 A is IL-measurable, also a positive upper density at a (for on Q. Thus there is a Q e C, with ijy(A.Q) > 0,

and hence '(A) > 0, contrary to A =

>.

A.

be a tile system for 4, (§17,6). 18.3.11. Let A be the set of all a e E at which the function f, 0,-defined on E, is somewhat continuous above (below) for 4, relative to Z. If A = ,c F, 0ten f is 0-measurable on E. Now let T.

Since A = 0 E, we have to show that for every y the set Ajf(1) < y] = B is ¢-measurable. Let a e B, and hence f(a) < y. By assumption dx(a, B, i-, Z) > 0. Thus it follows from 18.1.33 that B is 4,-measurable. A converse of 18.3.11 is contained in 18.3.4.

Again let the function f be -defined on E. 24 Cf. §11,2 and in particular footnote 48, p. 144.

If for every y > f(a) the set

287

APPLICATIONS

§181

Lf(&) < y] (or if for every y < f (a) the set U(i) > yj) has the density 1 for at a on 0sb, then f is called approximately continuous above (or below) for ¢ at a

relative to C.

As in No. 1, we again designate by R the set of all a e E for which in Ca there is a sequence ((Q,)) converging to a with (Q,) = 0. By (1.2) we have: 18.3.12. If f is approximately continuous above (below) for ¢ at a relative to 0,

thenasE - ..

18.3.13. In order that f be not approximately contit umus above (below) for ¢

at the point a e E - . relative too, it is necessary and sufficient that there be a y > f (a) (a y < f (a)), such that the set [f() >_ y] (th.e set [ f (:r) S y]) has a positive Upper outer density for ' at a on C.

NECESSITY : There is a y > f (a), such that the set [f() < y] = A has a Hence by 18.1.21 d(a, E - A, , C) > 0. Let N be the zero-set for ' on which f is not defined; then E - (A + N) _

lower inner density at a smaller than 1.

[f(i) ? y]. Thus, since by 6.1.512 i((I: -- (A + N))Q) = ((E - A)Q), we have also d(a, E - (A + N), C) > 0. SUFFICIENCY'. Employing the same notation we have a(a, E

(A + N), Vii, 0) > 0, and hence

d(a, E - A, , 0) > 0 also. Thus by 18.1.21 dx(a, A, y, 0) < 1; that is, f is not approximately continuous above at a. The following theorem is contained in 18.3.11: 18.3.14. Let A be the set of all a s E at which the function f, 4'-defined on E, is E, then f is approximately continuous above (below) for 4, relative to T, If A ,y-measurable on E. A converse of 18.3.14 is contained in 18.3.4.

If f is both approximately continuous above and approximately continuous below at a (for 4' relative to 0), then f is called approximately continuous at a (for ¢ relative to C,). If f is approximately continuous at every point a e E, then f is called approximately continuous on E (for 4, relative to 0). 18.3.2. In order that f be approximately continuous at a for 4, relative to 0, it is necessary and sufficient that for every3 > 0 the set of all x e E at which

II f(x) - f(a) II < have the density 1 at a for

on

sze

77

NEcEssrTY: For every y > f(a) and for every y' < f(a) the sets [f(l) < y] and {f(1) > y'] have the density 1 at a. Thus the same is the case by 18.1.221 also for [f(i) < y] .[f(t) > y'] and hence also for [11f(t) - f (a) 11 < a]. SumcroscY: If y > f(a) and y' < f(a), then for a sufficiently small a >0 we have:

1I1 f(x) -- f(a) II < 61 C [f(:) < yl and [II f(i) - f(a) II < al Q [f(x) > y'].

Thus it follows from 18.1.1, (1.1), and (1.12) that f is approximately continuous

at a. 9a Or, what by (1.14) amounts to the same thing, the inner density I for ; at a on 0. 2 If f(a) is finite, then E1 f(x) - f(a) 11 < S can here be replaced by I f(x) - f(a) I < S. 21 Because of (1.14), "density 1" can here be replaced by "inner density 1".

[CHAP. V

DIFFERENTIATION

288

Now we consider an indefinitely fine system C which at the point a e E satisfies also the following condition: (3) There is a monotone decreasing sequence ((U;'))) of neighborhoods of the

point a which converges to a, such that with every Q e Z. we have also

Q, U.", E e.18.3.3. If f is ,&-defined on E, then in order that f be approximately continuous at a e E for 4, relative to Cl, it is sufficient and, if (3) is satisfied, also necessary that there be a set A a E with a e A and dx(a, A, 128 on which f is continu-

ous at a. then we have by 18.3.2 Nscnssrrr: We set Ck = [II f() - f(a) 11 < dX(a, Ch) , £.1) = 1, and hence by (1.2) and 18.1.21 d(a, E - Ck, , C) = 0. Thus by 18.1.23 there is a pk > 0, such that for all Q e Q. contained in Sa,k: (3.1)

(Q - k) 5 k2

(Q);

hereby we can assume: pk+, < pit . Now let Uak) a S.,k (k = 1, 2, ) be a subsequence of the sequence of neighborhoods which exists according to (3); Uak+1)) thus Uak) Q Uak+'). We set A = S(Uak) Ck + (aj. From Cki.1 Ch

-

it follows that A Uak) C Ck , and hence for all x e A Uak) we have I I f (x) - f (a) I <

Therefore f is continuous on A at a.-The neighborhood U.() of a cbntains a

sphere Sa,{. Now let Q e Q. and Q C Sa,.{; then Q - A = Q. S (Uak) - U?*" ) (Uak) - Uak+o) Ck = S (Uak) - UU.+i)) (Q - Ck) c S (Uak)Q - Ck),

-S kgi

kzt i

and hence by (3) and (3.1) : +'(Q - A) 5 kE y(Uak)Q - Ck)

kyi

kE k2

y(Uak)Q) S

$(Q) kE k2' Since this holds for all Q e Z. contained in Sao,, we obtain d(a, E - A, +&, C) = 0, and hence by 18.3.12 and 18.1.21 dx(a, A, Vii, >ia) = 1. SUFFICIENCY: Let S > 0 and set C = E[ 11 f(2) - f(a) II < S]. If ((Q,)) is a sequence of Q. converging to a, then since f is continuous on A at a, we have

Q,A Q C for almost all v; thus by 6.2.21 x(Q"A) 5 X(Q.C) 6 x(Q,). But x(Q'A)--* 1 and 17 -(Q sinced (a,A,>Ji,C) = 1, we have also; therehencJX(Q,C) fore dx(a, C, ¢, Cl) = 1. Thus by 18.3.2 f is approximately continuous at a. 18.3.4. Let ¢ be content-like, let Z be a tile system for 4', let the function f be 4G-defined on E, and designate by A the set of all a e E at which f is approximately continuous for ¢ relative to Z. Then in order that f be &.-measurable, it is necessary

and sufcient that A = j. E. NECEBBITY: First we assume 4'(E) to be finite. If we have furnished the proof under this assumption, then the result can immediately be generalized 28 Because of (1.14), "inner density 1" can here be replaced by "density 1".

§18)

APPLICATIONS

for the case in which (1) is satisfied.

such that >,G(E - Bk) <

289

By 11.3.1 there is a4'-measurable Bk a E,

and f is continuous on Bk . Let B,, be the set of all

x e BA, at which d(x, Bk , , Z) = 1; by 18.1.32 Bk =,p Bt, and hence (E - Bk) <

also. We set B = SBk ; then according to §1 (1.21) E - B = D(E - BI),

and thus y (E - B) 5

(E - BA';) < k for every k. Hence B =, E and it

suffices to show that f is approximately continuous at every point a e B. Thus let a e B. Then there is a k, such that a e Bk , and hence a e Bk also; therefore d(a, Bk , , T) = 1. Thus since f is continuous on BA, , by 18.3.3 f is approxi-

mately continuous at a, as contended. SuFmc mcy: This is contained in 18.3.14.

is continuous the point (a,, a2, 18.3.5. If F(yi, y2, , a.) of &, if , , n) are approximately continuous at a e E for ¢ relative to the f;(x) (i = 1, 2, O, i f f;(a) = a; , a n d i j (3) i s s a t i s fi e d at a f o r C, t h e n F(f1 , f2, , is approximately continuous at a for 1G relative to C.

By 18.3.3 there is a set A; Q E, such that a e A; , f; is continuous at a on A; , and dx(a, A; , y , 1 . ) = 1 . If we set A = AI A2 . A , then by 18.1.221 dx(a, A, , 0) = 1 also. Thus, since (by a well known theorem on composite continuous functions) F(f, , f2 , , fn) is continuous on A at a, the contention follows from 18.3.3. BIBLIOGRAPHY: The notion of approximate continuity (without this notation) is already to be found in H. LEBESGUE, Paris C. R. 137 (1903), p. 1228; the notation and most theorems of this No. are essentially due to A. DENJOY, Paris C. R. 158 (1914), p. 1003; Bull. Soc. math. France 43 (1915), p. 165; cf. also: W. SIERPIksm, Fund. math. 3 (1922), p. 320; H. LooMAx, Fund. math. 5 (1924), p. 105; J. RIDDER, Fund. math. 13 (1929), p. 201; A. ROSENTHAL, Bull. Amer. Math. Soc. 48 (1942), p. 414. As to 18.8.14, and 18.3.4 ("sufficiency"): M. H. A. NEwMAN, Trans. Cambridge Philos. Soc. 23 (1923), p. 10; W. STEPANOFF, Mat. Sbornik Moscow 31 (1924), p. 487; E. KAxan, Fund. math. 10 (1927), p. 431; S. KAMETANI, Proc. Imp. Acad. Tokyo 16 (1940), p. 350; (moreover, cf.: H. BLUMBERG, Proc. Nat. Acad. U. S. A. 8 (1922), p. 283; Trans. Amer. Math. Soc. 24 (1922), p. 126; Fund. math. 32 (1939), p. 29; W. SIERPIksHI, Fund. math. 4 (1923), p. 124).

4. Integration and differentiation. We again use the terminology and the assumptions introduced at the beginning of No. 1. Moreover, let f now be a

4,-slunmable function on E. We set p(M) = (M) f d4' for every M e D1; then f ,p is totally additive in the o-field V. We again designate by C an indefinitely fine system, by t a Vitali system, and by T a tile system for ¢.

18.4.1. If p(M) = (M) f d¢ and if ,P is content-like, them the set of all a e E f at which D(a, rp, ¢, 83) = f(a) holds =,p E. For by. 12.1.51, 17.2.52, 17.2.51, and 12.1.52 we have

CHAP. V

DIFFERENTIATION

290

,P(1f) _ (M) f D(x, v, ,G,18) di', and hence the contention follows from 12.1.521.

From 18.4.1 and 17.4.33 there results: 18.4.11. If p(M) = (M) f f d¢ and if .1G is content-like, thin the set of all a e B at which D(a, v, 4, Q3) is regular and D(a, (o, }', X23) = f(a) holds =, E.

18.4.12. If p(M) = (M) f f d¢ and if f is 4'-bounded on E, then the set of alt a e E at which D(a, sp, ¢, T) = f (a) holds = j E. For by 17.6.5, 17.6.42, 17.6.41, and 12.1.52 we have

V(M) = (M) f D

gyp,,',

') d4',

and hence the contention follows again from 12.1.521. The density theorem 18.1.3 is a particular case of 18.4.12 if one replaces 4, by ' and assumes f to equal 1 on a measure-cover B of A and to equal 0 on E - B

(cf. the proof of 18.1.3).-We now obtain also the theorem corresponding to 18.1.35:

18.4.121. Let O be an ordinary indefinitely fine system and tet,p(M) = (M)f f d¢.

Then in order that for every 4'-measurable and +'-bounded function f on E the set of all a e Eat which D(a,,p, ,y, 0) = f(a) holds =,p E, it is necessary and sufficient that .0 be a tile system (for 4').

NEcEssrry: According to 3.4.71 there is a decomposition E = E+ + E" E+E_= A, tL+(E-) = 0, and f-(E+) = 0. Let A Q E and let AX be a measure-cover of A for . We set f = I for x e AxR+, f = -1 for x e AXE-', and f = 0 for x e E - Ax. Then by 12.1.61, 3.4.7, and §3 (4.2): p(M) = '(=l)`E+M) - 4,(AxK M) = P+.(AXM) + 4' (AX,U) = , (AxMI). Thus by with

assumption - ,c 1 D(a, v, 4', 0) = l em _(qXM) = Q.

ti-a

,G(M)

on AXE', =# -1 on AxE-, and =j. 0 on E - Ax. In particular for Ax = E this gives: Urn PM) x. n . ID (M)

=y 1

M -a

on E+ and

-1 on E-. Hence lim. -(AxM)

Ma

O

M)

lim 4(M))

N
11

A.r f

L(AxM) +G(.tf)- - 1

291

APPLICATIONS

§181

on Ax and =,c. 0 on E - Ax. Thus (by §6(3) and because of the regularity of the measure function ) we have also: lim

(AM) = i 1

McCa F(W M-.a

on Ax; that is, the density theorem holds for C with regard to 0. Therefore by 18.1.35 Q is a tile system (for 4'). SUFFICIENCY: This is theorem 18.4.12 if one takes 12.3.52 into consideration.

From 18.4.12, 17.6.5, and 17.6.453 there results:

18.4.122. If p(M) _ (M) ffd4' and if f is 4-bounded on E, then the. set of all a e E at which D(a, gyp, 4', Z) is regular and D(a, gyp, 4', `l:) = f(a) holds = E. Now let R be again the set defined on the occasion of (1.2).

18.4.2. If w(M) = (M) ffd4' and if at the point a e E - P the derivative D(a, gyp, 0, Q) is regular and equals f (a), then f is approximately continuous at a

for ¢ relative to C.

Suppose f not to be approximately continuous at a (for 4, relative to C). Then by 18.3.13 there is either a y > f(a), such that the set [f(t) yj has a positive upper density at a, or a y < f (a), such that the set [f() S y] has a positive upper density at a. We suppose, say, the first case and we set V(14) (z) yj B. According to 3.4.71, we form the decomposition B = B+ + B- where B+B- = A, V(B+) = 0, and ,P+(B-) = 0. Then by 18.1.22 we have eitherd(a, B+, J', C) > 0 or d(a, B-, ¢, 0) > 0. We suppose, say, the first case. Then there is a se-

quence ((Q,)) of Q. converging to a, such that

Q') -- a(a, B+, ;, C) (>0).

4G(Q.)

for almost all v. But Thus there is a > 0, such that (B+Q,) > (B+) = 0 implies , (B+Q,) = ¢(B+Q,), and thus, since (Q,) z 14,(Q.) 1, we have k(B+Q,) > r- `,'(Q,) I for almost all P. Hence the sequence ((B +Q,)) converges regularly to a.

But since 4' (B+) = 0 and f *e y on B+, we have by

12.1.32: ,p(B+Q,) = (B+Q,) ffd4' k y4(B+Q,). Thus since y > f (a), the sequence

p(BQ,) +Q,) cannot converge to f(a). 'G(B

We obtain a converse of 18.4.2 in the following form: 18.4.21. Let 4' be monotone.," let f be ¢-measurable and 0-bounded,80 and let _ This condition is essential. Example in Itl : Let f - 1; let J7t be the or-field of the µi-measurable sets in (-1, +1) and let. t he that totally additive set function in XlYl which equals µ, for the sets M. SDl in (0, +1), equals -µi for the sets Ma fit in (-1, 0), and equals 0 for (0(; moreover, let Co be the system of all intervals in (-1, +1) which are symmetrical to x. - 0. Then ,p(M) _ (M) ff dIp - ¢(M), and hence D(0, to, ,p, Oo) D(O) is not regular).

0 * f(0) (moreover,

30 This condition is essential. Example: Let E n (-1, +1), let

- /al, and let a - 0;

[CHAP. V

DLJTERENTIATION

292

ip(M) _ (M) J f 4. If now f is approximately continuous at a for ¢ relative to 0, then at a p has the regular derivative D(a, p, ', 1:2) = f(a). We assume ' to be, say, monotone increasing. Since f is approximately continuous at a, by 18.3.12 a e E - E. Moreover, since f is f-bounded, by 18.3.1 f(a) is finite. Let ((M,)) be regularly convergent to a; then there is a sequence ((Q.)) of fan convergent to a, such that M, Q; Q, and #(M,) z 34,(Q,) (with t > 0) for almost all Y. Since a e E - 2, we can assume that 4'(Q,) 0 0. Let a > 0 be arbitrarily given and set C = 11 f (I) - f(a) I < 8J. By 18.3.2 d(a, C, 4,, C1) = 1, and hence by 18.1.21 d(a, E -- C, 4,, £1) = 0. Thus '1(Q' - C) 4,(Q,)

Q. and ¢(M.) Z l¢(Q,) that

--+ 0 and then it follows from M.

also; therefore t4(M,? --* 1. *(MI)

4,(M' ¢(M.1

-* 0

Moreover, p(M,) _ (M,C) f f d# + (M, - Off d#

Herein, since I f (x) - f (a) 1 < a on C, we have by 12.1.32: (M,C) f f d# f(a)¢(M,C) + e*(M,C), (with 101 5 1). Since f is 4,-bounded, there is a p, such that 1 f 1 S ; p; and hence by 12.1.53 and 12.1.32 (M, -- C) f f d# = O'pO(M, - C), (with 19' 1

#(M' -

Therefore it follows from

1).

Mr

- + 1 and

f(a) < 26; that is, V(M,) --> f(a).

-+ 0, that for almost all r:

I

0(M,)

I

The theorems 18.4.2 and 18.4.21 together give the following result: 18.4.22. Let 4, be monotone, let f be 0-measurable and 4-bounded, and let c(M) _

(M) f f d*.

Then in order that,, have the regular derivative D(a, co, y4, 0) = f (a)

at the point a e E - 2, it is necessary and sufficient that f be approximately continuous at a for 4, relative to C.

If we drop the assumption that y' is monotone and f is #-bounded, then we can show: 18.4.3. Let f be *-integrable and let p(M) = (M) ff d#. If f is approximately continuous at a for * relative to h4, then there is a sequence ((Me)) regularly convergent to a, such that `p(M;)

IPM)

- f(a).

moreover, let On be the system of the intervals in (-1, +1) which are symmetrical to 0; i 1 1 and set f - 2" in A. 1 (for m - 1, 2, ) and f 0 otherwise. Since 2"`

2t 2

d., +

0, f is approximately continuous at z = 0. On the other hand, i

tilt

KRh f dx -

and hence D(0) - i and D(0) _ }, while f(0)

0.

§181

298

APPLICATIONS

Let ((Q,)) be any sequence of A, converging to a; since by 18.3.12 a s E - 1, we can assume that y4(Q,) 54 0.

If we set C: _ [Jt f(t) - f (a) II <

i], then by

18.3.2 d(a, C; ,, Q) = 1, and hence lim 71. Thus there is a v: , and we can forthwith assume that v:+l > v: According to 3.4.7 and 3.4.71 we form the decomposition C,Q,, - At' + AT with A;Ai = A, *-(At) = 0, ¢+(A;) = 0, and (C.Q,,) _ #(A{) + I #(A{) 1. Thus at least one of the two following inequalities holds: 4 (A t) > 1 (Q.) or Assume that, say, the first holds and we set At = M. I> that is, ((M;)) converges regularly to a relative to £l. Then O(M;) >

such that j(C,Q,,) >

If now f (a) is finite, then ) f (x) - f (a) I I < _ , for sufficiently great i, amounts to the same thing as ( f (x) - f (a) I < b; where the b, are certain positive numbers with b; -- 0. Thus on M: we have f (a) - S < f (x) < f (a) + it, and hence by 12.1.32: (4)

V(1t:) = (M:) f f d,. = f(a)4'(M:) + e.4.'(M:),

Q 01 I

1).

But if f (a) = + cc (or - co), then I J f (z) - f (a) 4I < amounts to the same thing

as f (x) > N, (or f (x) < -M;) where the N; are certain non-negative numbers with N1 - + co. Thus again by 12.1.32 we have: (4.1)

'KM:) = (M{) fi d4 > N,¢(M:) (or < -N(M:)).

Since #(111:) > } (Q,,) > 0, it follows now from (4) and (4.1) that -P(M:) >G(M:)

as contended. If f is 4'-integrable and approximately continuous on E (for i6 relative to 0)

and if again e(M) _ (M) f f d*, then by 18.4.3 to every point a e 1 a sequence ((M;)) regularly convergent to a with'n(M) --+ f(a) (and with #L(M:) 0 0) can be attached. If now C is a Vitali system Ql (for ¢), then all the sets M: appearing in these sequences ((M1)) form an orderly system QB for 0 (¢17,4), and hence we have! : 18.4.31. If f is 4'-integrable and approximately continuous on E for +y relative to a Vitali system Q3 and if p(M) = (M) f f d+¢, then f is an orderly derivate of p with respect to 4..

If above C is a tile system Z (for 4'), then all the sets M: appearing in those it In ¢17,4 $* is assumed to be finite; correspondingly f should be assumed here to be But since the definition of the orderly derivate can be generalised immediately to the case of a non-finite v, it suffices here to assume f to be i-integrable. y,-summable.

294

[CHAP. V

DIFFERENTIATION

sequences ((M,)) form a normal system ff for 4, (§17,6), and hence we haves': 18.4.311. If f is ¢-integrable and approximately continuous on E for # relative to a tile system T_ and if p(M) = (31) ffd#, then f is a normal derivate of V with

respect to ¢. BIBLIOGRAPHY: For R. and 4, - A. (in particular as to the beginning of this No.): H. LEBESOI?E [1], p. 124; Ann. Ec. Norm. (3) 27 (1910), p. 395; G. VITALI, Atti Accad. Torino 43 (1907-1908), p. 237; CH.-J. DE LA VALUE POUSSIN [1], vol. II, p. 115; Trans. Amer. Math. Soc. 16 (1915), p. 457; [2], p. 72; A. DENJOY, Bull. Soc. math. France 43 (1915), p. 172, 204; C. CARATHI;ODORY [1], p. 496; J. WOLFF, Bull. Soc. math. France 52 (1924), p. 581; J. RIDDHR,

C. R. Soc. sc. Varsovie 23 (1930), p. 10; S. SAxs [1], p. 232; Fund. math. 22 (1934), p. 267; 25 (1935), p. 235; [2], p. 132, 147; H. BUSEMANN-W. FEI.LER, Fund. math. 22 (1934), p. 226; A. ZYGMUND, Fund. math. 23 (1934), p. 143; B. JESSEN, J. MARCINSIEWICZ, A. ZYGMUND, Fund. math. 25 (1935), p. 217; L. CESARI, Ann. Scuola Norm. Pisa (2) 8 (1939), p. 301. For abstract spaces and general .: It. DE PossEr., S. SAKS, B. JESSEN, A. Ros>9NTSAL, loc. cit.

(117,5).-Cf. also the other bibliography to §17,5 (second paragraph).

5. The limit functions of a derivate. We retain the assumptions made in No. 1, designate by

tion that 4, is monotone and content-like. Moreover, we assume that for all sets V of the employed Vitali system Q3 we have: V 00 A, that is, J,(V) 0 0. Then for every non-empty, open set G we have: G 9d ,c A. For if G =,. A and a e G, then in every sequence ((V,)) of Q3 converging to a we would have V, G for almost all v, and hence V, = ,c A for almost all v, contrary to the assumption. 18.5.1. Let 4, be monotone and content-like, let rp be 4-continuous, and let V $,. A

for all V e Z. If D*(x) is a Vitali derivate of 9 with respect to 4, on Q3 and if there

is an a e E, such that D*(a) > z (or D*(a) < z), then [D*(i) > z] 04, A (or [D*(z) < z] ;d,, A). We assume that 4, is, say, monotone increasing.

Suppose, [D*(:9) > z] _ A.

Then since by 17.2.54 v(M) = '(M) f D*(x) d4,, we would have (p(M) 5 zO(M)

by 12.1.53 and 12.1.32, and hence D*(x) 5 z for all x e E, contrary to the assumption. Now we modify the notion of the upper and lower limit functions of a given function f (cf. footnote 48, p. 187) in the following way, neglecting the zero-sets for ¢: Let f be defined on A and let 91 be the system of the zero-sets for 4,. For a e A consider the set of all those numbers y e RI to which for every N e T there is a sequence ((a,)) in A - N with a, -' a and f(a,) --> y. One sees immediately 32 With regard to §17, 6 (normal derivate) the same remark holds as in footnote 31.

295

APPLICATIONS

X18]

that the set of all those numbers y is closed. Now we call the greatest and the smallest value of those y the upper and the lower limit function off at a neglect-

ing the zero-sets for sk. We apply this to the derivate D*(x), setting A = E. 18.5.11. Let ¢ be monotone and content-like, let p be ,P-continuous, and let V 3y A

for all V e Z. Then for every Vitali derivate D*(x) of a with respect to ,y on Q3 the upper (lower) limit function and the upper (lower) limit function neglecting the zero-sets for y coincide.

Let d(x) be the tipper limit function of D*(x) and let d°(x) be the upper limit function of D*(x) neglecting the zero-sets for ¢. At the point a let d(a) > z. Then in every neighborhood U. there is an a', such that D*(a') > z. By 18.5.1 (employed for E = U.) Ua[D*(I) > z] 0,k A. From this it follows that d°(a) z z

and, since this holds for every z < d(a), we obtain d°(a) z d(a); hence 18.5.12. Let be monotone and content-like, let p be 4,-continuous, and let Z and 0 be Vitali systems, such that V # 0 A for all V e Q3 and W # ,c A for all

W e G. If now D*(x) is a Vitali derivate of p with respect to ¢ on Q3 and D**(x) is a Vitali derivate of p with respect to 0 on ll33, then D*(x) and D**(x) have the same upper (lower) limit function. By 17.2.64 D*(x) =,p D**(x), and hence by 9.4.6 D*(x) and D**(x) have the same upper limit function neglecting the zero-sets for 4'. Now the contention follows from 18.5.11.

18.5.13. Let 0 be monotone and content-like, let p be ¢-continuous, and let ,c A for all V E Q3. If now D*(x) is a Vitali derivate of cp with respect to ¢ on 93 and f =,p D*(x), then for the upper and the lower limit function d(x) and d(x) of D*(x) and for the upper and the lower limit function f(x) and /(x) of f(x) we have the inequality: f(x) 5 4(x) 5 d(x) s f(x). If d°(x) and f°(x) are the upper limit functions of D*(x) and f(x) neglecting the zero-sets for ¢, then by 9.4.6 d°(x) = 1°(x). But by 18.5.11 d°(x) = d(x), and hence the contention follows from low S f(x). V

18.5.2. Let 4, be monotone and content-like, let ip be ¢-continuous, and let V Op A for all V e Q3. If D*(x) is a Vitali derivate of So with respect to 4' on Q3 and if G

is open, then sup D*(x) (or inf D*(x)) coincides with the supremum s (or the inzia :.o

fimum i) of (Al) for all 4,-measurable M F_ G which # t, A. IVAI)

We assume that 4, is, say, monotone increasing. Since D*(x) (x a G) is the

limit of certain quotients Q(M) (where M Q G is 4,-measurable and M #,p A),

we have sup D*(z) 5 s. On the other hand, to every z < s there is a 4-measze 5

urable M c G with M # j. A, such that

z; thus p(M) > O(M), and 1P (IV)

hence by 17.2.54 (M) fD*(x) 4 > z4'(M). According to 12.1.32, this is

CHAP. V

DIFFERENTIATION

296

possible only if there is an a e M with D*(a) > e. But then sup D*(x) > z sW

and, since this holds for every z < s, sup D*(x)

s also.

¢eo

If in the above proofs 17.6.44 and 17.6.442 (instead of 17.2.54 and 17.2.64) are employed, then we obtain: 18.5.3. AU above theorems of this No. hold for tile systems (instead of Vitali systems A3) if the assumption that

by the assumption that { q(M) 19

is content-like and tp is 4,-continuous is replaced

i(M) for every M a V.

BmuoaaApur As to 16.5.2: U. DINT 111, p. 192; of. also P. Do Rots-RZYMOND, Math. Annalea 16 (1880), p. 119, 128; L. Scnnarrna, Acta math. 5 (1884), p. 196. As to 18.5.11: H. L5BnsGU f11, p. 80. As to 18.5.13: C. CAB. ngonoar 111, p. 501.

6. Trems[ormatlon 1 integrals. We retain the assumptions made in No. 1 for'; and let rp be a finite, totally additive, and ,y-continuous function in U. Moreover, P is assumed to be content-like; then by 7.4.4 gp is also content-like. 18.6.1. Let ¢ be content-like, let rp be y-continuous, and let f be p-summable on E. If 111 is a Vitali system for t' and D*(x) is a Vitali derivate of so with respect to on 113, then for every M e T'l: (6)

(M) ff(x) dw = (M) ff(x) D*(x) d#,

where we have to set f (x)D* (x) = 0 wherever D*(z) = 0."

We set X(M) = (M) fAx) dgo; then X is finite, totally additive, and sp-continuous, and hence by 5.7.13 also 4k-continuous in $t. By 17.2.11 113 is a Vitali system also for V. Let Dk(x) and D,,(x) be the Vitali derivatives (on 113) of X with respect toand rp and let Al', be the set of all z e M at which D#(x) exists; by 17.2.52 M, + M. We designate by M* the set of all x e M at which D*(x) is finite; according to 17.2.4 M* = 0 M, and hence, because of the 0-continuity of q,, M* _, M also. Moreover, let M2 be the set of all x e M* at which D,(x)

f (z) holds and D,(x) is finite; by 17.2.4 and 18.4.1 M2 =, M. Let a e M,M2 and let ((V.)} be a sequence of Z. which converges to a and for which n(y.) -4G(v.)

D*(a). Then we have also X(V) - Dc(a) and X(V) --- f(a).

But from this it follows that Dp(a) - f(a)D*(a) for all a e M,M:, and hence by 17.2.54: (M,M2) f f dsp= a(MIM,) - (M,M2) f f(x)D*(x) 4. Since M2 =,111, we have k v-)

go(V,.)

u Even if Ax) is infinite or is not defined.-For the ease f (x) e 0 and D*(x) o f - a particular stipulation is superfluous, since this case, according to 17.2.4, can take place only on a zero-set for 4,.

INTEBYAL FUNCTIONS

§191

297

MI - M, _, A, and hence by 17.2.541: D*(x) =,. 0 on MI - M2. Thus, since by 12.1.5 (M1 - M2) f f dip = 0, we have (MI - M2) ff dip

(MI - M,) f f(x)D*(x) d#. But M1

M implies MI =, M, and hence we

have because of 12.1.53: (M) ffdrp = (MO ffdcf = (M1M2) ffdco

+

(MI - M2) f f drp = (M1M2) f f(x)D*(x) dip + (M1 - M2) ff(x)D*(z) d# _ ff(x)D*(x) d¢.

ff(x)D*(x) d¢ _ (M)

(M1)

BI3LIo4RAPaY: J. VON NEVMANN [11, p. 232; S. SARK [2], p. 86.

§19. Interval functions 1. Interval functions and associated set functions. Let H be an open set of R.. As in §8,1, a finite. set function X(I) defined for the system of all closed intervals I Q H shall be called an interval function. A finite set of closed intervals of H, no two of which have inner points in common, shall again be called a simple finite system of intervals. If e5 is the simple finite system consisting of the closed intervals I1 , It , - , , then we define: -

(1)

I.

x(S) = x(II) + x(I2) + ... + x(Im) and x(S) = I x(II) I + I x(I2) I + ... + I X(Im) I

.

For the empty system 8 of intervals we set X(V) = 0 and X(2) = 0. If C and C' are two simple finite systems of intervals and if the intervals of

6 have no inner points in common with the intervals of e', then we designate by C5 + Co' the simple finite system which consists of the intervals of (B and Let 0 be an indefinitely fine system (§17,1) consisting of closed intervals of H. Let e, (for n - 1, 2, ) be a simple finite system of intervals of Ci and designate by d. the maximum of the diameters of the intervals of S,. If d,--+ 0, then (((s,)) shall be called a distinguished sequence of interval systems. Now let 911 be a a-field consisting of subsets of H and containing the closed intervals in H. Moreover, let the set function ¢ be totally additive, finite, and content-like (§7,4) in 9 and let TZ be complete for 4," If ( C C is a simple

finite system of intervals and S is the sum of the intervals of e5, then we set

Re) = ks)

The interval function x shall be called ,'-continuous on C if for every distin-

guished sequence ((5,)) of interval systems of C with (C5,) -a 0 we have

x(c,) - 0.

at These assumptions on Dl and ¢ will not be used before 19.1.2.

[CHAP. V

DIFFERENTIATION

298

19.1.1. In order that the interval function x be 4-continuous on 0, it is necessary and sufficient that for every distinguished sequence ((C,)) of interval systems of J. with >y(5,) --+ 0 we have X(5,.) --+ 0.

NECESSITY: Let 5, = 5: + 50" where C5,' consists of the intervals I' e 5, consists of the intervals I" e C, with x(I") < 0. If with X(F) >_ 0 and 0 and X(5,") --- 0, and hence from X(C,) X is ,'-continuous, then

x(Z/) - x(5/") it follows that x(5,) --+ 0 also.

SUFFICIENCY: This results

from I x(5.) I <

19.1.11. In order that the interval function x be'-continuous on 0, it is necessary and sufficient that to every f > 0 there be an n > 0 and a d > 0, such that for every simple finite system 5 C !a of intervals with diameters 9 d and with < n we have X(C5) < e. NEcF,ssITY: If the condition is not satisfied, then there is an e > 0 and a simple

finite system e5, c C of intervals with diameters S 1 (for v = 1, 2, v

), such

that (C5,) < 1v and 5(5,) > e. But then by 19.1.1 x is not 4-continuous on

,.. SUFFICIENCY: If ((5,)) is a distinguished sequence of interval systems of 0 with >y(C5,) --+ 0, then for almost all v (5,) < ,t and for almost all v the intervals of C5, have diameters 5 d. Thus X(C5,) < e for almost all v; that is, X(" ,)

--+ 0.

Let M e ST1, let ((5,)) be a distinguished sequence of interval systems of C, and let S, be the sum of the intervals of C, . Then we shall say: ((e,)) converges to M for 'y, written symbolically: Cam, --+,o M, if

(S, - M) -i 0 and (M - S,) -+ 0." Obviously C5, -->* M is equivalent to C5, --+; M. Now let $3 be a Vitali system (for ') consisting of closed intervals in H (cf.

§17,2 and 17.5.5). 19.1.2. If M e E, then there is a distinguished sequence ((C5,)) of interval systems of 93, such that S. -,. M. This follows from 17.5.1 and 17.2.3, if one sets, say, e = 1 and chooses the v

subsystems lt3' in 3,) such that their intervals have a diameter smaller than (Then every C -.(C Q3) consists even of disjoint intervals.) Now let p be a finite set function in 9Y1 and let x be an interval function.

If

for every set M e 9U1 and for every distinguished sequence ((5,)) of interval systems of Z with 5, Al we have x(5,) --+ tp(M), then we call p associated with x (with respect to 4, and S3). 19.1.3. In order that there be a set function cp associated with the interval function x with respect to J- and 23, it is necessary that x be &-continuous on Z. Let (( *)) be a distinguished sequence of interval systems of % with 0

J3 From this it follows by 3.1.42 that then ;(fi,) - .(M) also.

INTERVAL FUNCTIONS

§](.)]

and let S* be the sum of the intervals of e*. By 7.4.31, to M e TJ there is a sequence of open sets G, D M, such that 4.(G, - M) --> 0. Since % is a Vitali

system and S* is closed, there are in 23 countably many disjoint intervals I,; Ss G, - ';*,(j = 1, 2, ) with diameters smaller than 1 , such that v

(ry

S 1'j.

,S*

i

Then by 4.2.42 and 4.2.31 we have G, Cc S* + S I ,j i

that ¢( S I,i) < v and set I,, +

=f

We choose m, so great

I,m, = S, ; then

M - S.cG,-S.C S*+i>m. S I.i Thus, since (S*) --a 0 and ;y( S I,i) --- 0, we have also (M- S,) .- 0; moref>.n, over, S. - M a (}, - M, and hence (S, - M) --+ 0. Thus if Cam, is the system , 1,., , then we have e, -+,, M. Since the intervals of of the intervals I ,, , CS, are disjoint from S*, the interval system C, + C* is also simple and finite;

moreover, from C, -,c M and (S*) - 0 it follows that C, + C* - .'. M also. Thus we obtain x(S,) -,p(M) and x(e, /+ C*) But since

x(C, + C.) = x(S,) + X(`' *)7 it results that x((S, - 0. 19.1.31. If the set function ip is associated w th an interval function x with respect to 4., then j(M,) , 0 (for M, a PI) implies p(M,) --* 0. Otherwise there would be a sequence ((M,)) in Tt and a p > 0, such that ¢(M,) -* 0 and I ,p(M,) I > p for all v. To M, there is, according to 19.1.2, a

simple finite system C, of intervals of IS with diameters smaller than 1, such

that (S,) < ' (M,) +

and I

x((e,) - p(M.) I < y.

Thus for the distin-

guished sequence ((s,)) of interval systems we have ((B,) - 0 and I x((S,) I > 2 for almost all v. Then x would not be 4'-continuous, contrary to 19.1.3. 19.1.32. If the set function cv is associated with an interval function x with respect to ,., then V is totally additive.

According to 19.1.31 and 3.2.43, it suffices to show that p is additive. Let M, and 31s be two disjoint sets of ',Tl and let 05;' ---j M: ; we designate by Sy' the sum of the intervals of C5;. By 7.4.31 there is in T1 a sequence ((G,)) of open sets containing M, , such that (O, - MI) -i 0. Since $3 is a Vitali

system and S' is closed, there are in ?3 countably many disjoint intervals I,i a G, -- S;' (j = 1, 2, - - -) with diameters smaller than 1, such that v

G, - S;'

S I,i . i

300

[CHAP. V

DIFFERENTIATION

Thus by 4.2.42 we have also G. _ , G,S' + S I,s . We choose in, so great that

¢(S I,;) <

1

and set I.1 +

t

+ I,.., = S'; then

M1 -- S. Z G, -- S, =, G,S; + S I,i . i> m.

Herein ( S 1q) -> 0. Moreover, since G,S;' c (G, - M1) + j>m,

S;'MI and

M1M2= A, we have: G,S'C (G, - M1) + (S; - M,), and hence also CG,S;) --' 0.

Therefore we obtain: (M1 - S;) -- 0. Furthermore, S' - M1 C G, - Ml and hence also (S' , - M1) -k 0. Thus if C; is the system of the intervals Since , I,.,, , then we have C; --', M1 , and hence x(C:) -- p(M1). 1.1 ) C;' -*,. M2, we have also x((S;') -+p(M2). Since the intervals of e, are disjoint from the intervals of C;.' , the sequence ((C5; + S, )) of interval systems

is also distinguished and e' + C." --> M1 + M, ; hence X(C: + tio(M1 + M2). But since x (S,' + C;) = x(C;) + x(e;'), there results that p(M1 + M2) = p(M1) + p(M?); that is, p is additive. The following theorem follows immediately from 19.1.31, 19.1.32, 5.7.2, and 5.7.21: 19.1.33. If the set function p is associated with an interval function x with respect to ¢, then p is ,J,-continuous.

Now we assume somewhat more for the I e il3 than 19.1.2 already contains; to be specific, we assume: (1.1) To every closed interval I e there shall be a distinguished sequence

((C,)) of interval systems of Q , such that C, C I and C, -+i I. In other words: To every I e !B there shall be a distinguished sequence ((C,)) of interval systems of Q3, such that C, Q I and i(I - S,) --+ 0. then (1.1) is satisfied. 19.1.34. If 4, is According to 19.1.2 there is a distinguished sequence ((C,)) of interval systems of Q3, such that Cs, -'I 1. Certainly we can assume that C, does not con-

tain any interval disjoint from I (for if one eliminates such intervals, then

(1 - S,) and (S, - I) do not increase). We now designate by C. the simple finite system of intervals which results if one eliminates from C, all intervals containing border points of I. Then C; C I and, since d, -+ 0, we have sc. (S, - S;) --> 0. Thus since ¢ is µ,-continuous, by 5.7.2 J (S, - S;) --+0 also. Moreover, since S. Q S' , , we have

(I - S.) _ i(I - S.) + R(S, - u;) 1), and hence it follows from 7(I -- S.) -- 0 that (I - S,) -+ 0 also. The intersection of finitely many closed intervals It, I2 , , I..(m 2) of no two of which have inner points in common, shall be called a boandarysegment T of each of these intervals (with respect to Q3). 19.1.35. If the set function p is associated with an interval function X with J8,se

36 Here one can replace 18 by any system of closed intervals.

301

INTERVAL FUNCTIONS

§191

reaped to lp and Q3 and if (1.1) is satisfied, then for every boundary-segment T of every interval I e Q3 we have: 1p(T) = 0. 2) where the I,(i = 1, 2, , m) designate the Let T. = I, I2 . 0 closed intervals of a simple finite system CD(Q; Q3), and set S = S I; . By i_1

of interval systems

(1.1) to every I; there is a distinguished sequence

of 18 with 6,,a -i, I; and 6;' c Ii ; hence x(e;'') -, rp(Ij). Let e, = c1> + e;_) +

- + (o;'); then ((C5,)) is a distinguished sequence of interval

systems of 13 with (S. c S and 1G(Ii -

Sri))

S;") -+ 0. But by 3.1.41 S (1, - SYi))) z i (S -- S,),

thus (S - S.) -- 0 also. Therefore, since S. C S, we have e, -,y S and hence X(e,) - cp(S). This and X(C5;`)) --i p(Ii) imply because of (1) that E p(1,). Now we designate by Ck the set of all points which belong i..1 to exactly k intervals Ii ; in particular C. = Tm . Then by 19.1.32 and 3.1.3 a E ,p(Ii) = P(S) + E (k - 1)ip(Ck). Thus there results: AI

i_i

k_,

(k - 1)0(C,,) = 0.

(1.2)

If in particular y, z 0, then by (1.2) .p(CA) = 0 for k = 2, , m. For a general (non-monotone) rp we prove this by induction with respect to m. If first m = 2,

then (1.2) means: c(C,) = p(Ti) = 0. Now assume it has already been proved that for every simple finite system of 2, , m - 1 intervals of Q3 we have: cp(T2) = 0, , 0, (m > 2); then we consider our system ( of m intervals. Let ii < is < < ii be l different of the numbers 1, 2, ... , m and let j, < j, < . . . < ja_i be the other m - l numbers. If we set A,1;, ... j1 = li,.1..... li, - (I;, + I;, + .....F I,.-), then these AiI+, ,e are disjoint and Ck is the sum of those Ai,i=... ik which have exactly k indices; thus ,p(Ck) is the sum of the corresponding ,(A,,i, ... ,k). Hence, if we first take

k = m - 1, then by 3.1.11 0 P(Cw-1) - E cp(Ii1 Ii, ...Ii.-t _. C.) (1.21)

!I-1

a

_

-P(Ii, I,, ...I,..-1) - VW(C.),

and thus by assumption: (1.22)

iv(C. -i) = -mcp(Cm).

, m -- 2 (provided m ? 4), then we have But if k = 2, Ai14 ...ik = Ii,Iy ... Ii, Ii,I;:... Iit.[It, + (I;. - IiIIn)

+ U h - I ill i. i.) + ... + (1a,-k - II,I ... I,-k)1

(CHAP. V

DIFFERENTIATION

302

and hence again by 3.1.11 ,n-k

cc(Ck) = E 'W,, I,s ... (k)

E EX-t tp(I(, I(, ... I,k Iia) (k)

+ z ,P(I;t I;= ... I(k lit Ii:) + .. .

(1.23)

(k)

+ E ,P(I;4 I;, ... f$y in Ii:...

+ (k )'P(C,n),

(k)

where E designates the summation over all k-tuples of indices (it i2 (k)

Thus by assumption we have for m

4, k = 2,

(1.24)

(k)'P(Cr)

IP(Ck)

,m-

. ik).

2:

For m = 3 it results from (1.2) and (1.22) that p(C3) _ jp(Ts) = 0. For m z 4 it follows from (1.2), (1.22), and (1.24) that c(Cm).[(s) + 2(i) + ... + (m - 3)(n;'s) (1.25)

-(m-2)m+(m-1)]=O.

The coefficient of tp(C,.) in (1.25) is always positive for m ? 4, since the sum of

the last three terms in the brackets in (1.25) already equals 2 (m - 3)2 - 1 and hence is positive. Thus V(C,,) _ -r(Tm) = 0, and now for every m >_ 2. 19.1.351. Let 932 be afield in R. and let (( be a simple finite system of intervals contained in 97t; moreover, let Ck(k 9 2) designate the set of those points which belong to exactly k intervals of ( and let p be an additive, finite set function in V. If now AP(T) = 0 for every boundary-segment T of every interval I e e (with respect to Cam),

then p(Ck) = 0 also for every k z 2. Let C`S consist of the intervals It , Is ,

, I,,,(m z 2). Taking into consideration that C. = 1112 ... I. , one obtains the contention as an immediate result

of the formulas (1.21) and (1.23). in order to find necessary and sufficient conditions that there be a set function associated with a given interval function, we now have to consider the differentiation of interval functions. 2. Derivatives of interval functions. For D't and the assumptions given at the beginning of No. I are to be satisfied; let X(I) again designate a finite interval function and let $1 be again a Vitali system (for 4,) consisting of closed intervals (c N). Then in the some manner as in §17,2 we can form the notion of the Vitali derivate, the upper and the lower Vitali derivate, and the Vitali derivative of the interval function X with respect to 4,." 37 If D is an indefinitely fine system consisting of closed intervals, then correspondingly

(according to }17,1) one can form the derivates, the upper and the lower derivate, and the derivative of the Interval function X with respect to V, on 0.

303

INTERVAL FUNCTIONS

§19]

In the same manner as 17.2.2 one proves: 19.2.1. The upper and the lower Vitali derivale D(x, x, ¢, Q3) and D(x, x, 0, Q3) of the interval function x with respect to ¢ are. ,'-measurable. 19.2.2. If the set function 9 is associated with the interval function x with respect to ¢ and 93, then

D(x,

X, 4 , , = ; )

D(x, p, 4,, Q 3 )

and

D(x,

Q3)

D

where D(x, tie, 4', V. designates the Vitati derivative of v with respect to 4i.

Let H* be the set of all a e H at which D(a, ,p, 4', Q3) exists and, moreover, D(a, ¢, , Q3) exists and equals +1 or -1; according to 19.1.32, 17.2.61, and 17.2.552, H* =,, H. For all a e H* we have: D(a, so, 0, Z) D(a, 0, , f8) = D(a, cv, , Q3) and

.(a, x, 4', !6) D(a, 4', t;, Q3) = D*(a, x, , Q3)

where D*(a, x, , Q3) designates a Vitali derivate of x with respect toy , which is 4'-measurable by 19.2.1, 17.2.62, and 9.2.3. Thus it suffices to show that D*(x, x, a[,, Q3) = i D(x, cp, ', Q3) on H*. Hence if we set D*(x, x, , Q3) = f and D(x, p, , 18) = g, we have to show that f =,. g on H*, and for this purpose it suffices to prove: for every c > 0 we have

H*[f(:t) > g(x) + cj =,a A and H*[g(l) > f(t) + cl = . A. We show, say, the first and set H*[f(t) > g(y) + c] = A; by 9.2.7 A e T1. Ac-

cording to 7.4.31, there is an open G, Q A, such that (G, - A) < 1 . Let a e A; since f(a) > g(a) + c, to every p > 0 there is an interval I e Q3, such that

a e I , I Q8 S, CG ,d(I) < 1,+G(I) > 0, and v

X(I)>W)

w)

;-(I)

The system of all such I (for all a e A and all p > 0) shall be called 13, . Then, according to property 3,) (§17,2), there are countably many disjoint

I,1,...)I,is...inQ, such that A - S1 =,p A. If we choose i, sufficiently great and if we designate by CO, the interval system 1.1 , I,2, .

, 1,,, and by S. the sum of these intervals, then according to 3.2.21 (A - S.) < 1v ; and since S, G, and .

(G, - A) < 1

,

DIFFERENTIATION

304

we have also (S. - A) <

v

Thus 6,

[CRAP V.

A. But from

x(I,.) > sp(I,c) + CAI.,) it follows that x(C,) > cp(S.) + 4(,S,). Herein, since C, --+,. A, we have x(e,) -> p(A) and (S,) -+ +'(A); and, since by 19.1.33 rp is P-continuous, we have also o(S,) -- p(A) (according to 5.7.21 and 3.1.11). But this is possible

only if ;y(A) = 0. 19.2.21. If the set function (p is associated with the interval function x with respect to ¢ and Q3, then the set of all x e H at which the derivative D(x, x, ¢, $1) exists = ,, H and D(x, x, 16, Q3) =,r D(x, p, %, Q3). For it follows from 19.2.2 that the set of all x e H at which D(x, X, J,, $3) _ D(x, x, 4,, Z3) = D(x, w, 44, Q3) holds = * H. From 19.2.21 there results by 19.1.32, 19.1.33, 17.2.54, and 12.1.52: 19.2.22. If the set function (o is associated with the interval function x with

respect to 4, and Q3, then for every M e $t we have: c(M) = (M)

fD(x, x,,+, Q3) d4.

From 19.1.35 and 19.2.22 there results: 19.2.23. If a set function ,p is associated with the interval function x with respect to ' and Qi and if (1.1) is satisfied, then for every boundary-segment T of

every internal I e Q3 we have: (T)

f D(x, X, P, 18) d# = 0.

From 19.2.21, 19.2.22, and 19.2.23 we obtain: 19.2.24. In order that there be a set function (p associated with the interval function x with respect to 4, and Q3, it is necessary that the set of all x e H at which the derivative D(x, x, 4,, Q3) exists =,c. H and that D(x, X, P, 48) be 4-summable on H; if (1.1) is satisfied, then it is also necessary that for every boundary-segment T of every interval I e Q; we have: (T)

fD(x, x, 1', B) 4 = 0.

Now we prove the converse of the theorems 19.1.3 and 19.2.24. For this ... , a(.i); bl", bai), ... , b,'3] purpose we first put the intervals Ic,' = [ai:1 , as) c' , , m) of the simple finite system S in a well-defined lexicographical (9 = 1, 2,

order according to the value of the numbers aid, ai , , a( V); that is, if vo in the first index for which a,, 4 a;,1 (while a," = a;' for v ro - 1), then J(.) is to precede or succeed P°, according as a;o) < a;) or a,(.) > a("). Let Im be the order of the intervals of e obtained in this way. From , m) of Ch. we procure disjoint, in general the closed intervals I, (j = 1, 2, 1, 2, , m) with the same sum S by retaining non-closed intervals I;` ( each point x e S only in the first of the intervals If which contain x. The intervals I c and Ii possess the same open kernel. If we have a sequence ((Cs,)) of simple finite systems of intervals, then we designate the closed intervals of

e, by I,,, , the intervals associated with them (as immediately above-that is, in general, non-closed intervals) by I' and the common sum of the Ii., and

1,, by S..

INTERVAL FUNCTIONS

1191

Now to the interval function X and the simple finite system (5(C $3) of intervals we attach a point function f(x, x, Co) according to the following definition:

f(x, x, 5) = (l) for x e I` (? - 1, 2, ... , m) (2)

andf(x,x,C) =0forxsH-S.

Obviously this function f(x, x, () is #-measurable. , k*) be the set of those points which belong to Again let Ck(k = 1, 2, exactly k intervals of ( and let k* be the greatest of the numbers k possible in 6; we have 1 S k* S 2". We now define the functions f°)(x, x, e) for 1 A k* in the following way: (2.1) for x e Ck with k < A let f °) (x, x, ') = f(x, x, e) ; for x a Ck with k ;--> A let f 00 (x, x, Cam) =

1G(I)

intervals of e which contain x; for x e H - S let f (1)(x, x, r) = 0.

where Iix is the at" of those

't'hus f(')(x, x, e) = f(x, x, e) [cf. (2)]. All these functions fc')(x, x, C) are also 4,-measurable.

.Let us formulate the definition (2.1) in another manner. From the If e C5 , k*) [which in m; A = 1, 2, we obtain the intervals I,())* (j = 1, 2, general are not closed] by retaining each point x e Ck for k < A only in the first of the intervals Ij which contain x and for k z A only in the ath of these intervals" We have I;')* = I°i`; the intervals If and I(x)* possess the same open kernel; for fixed A the I;x)* are disjoint; for every A we have SIix)* - SIi = S. i

i

Now we can formulate the definition (2.1) of the functions fx)(x, x, C5) for , k* as follows: A = 1, 2, (2.11)

f("')(x, x, (S) =

19.2.3. Let M E

x(I') for x e

I}")*

(3 = 1, 2, ..., m)

>G(li)

and f cx) (x, x, C5) = 0 for z e H - S. and e, -i M; if the derivative D(x, X,

3) is #-defined on

H, then the sequence of the functions f. = f(x, x, (s,) defined according to (2) is asymptotically convergent for ¢ on H to the function (2.2)

f (x) = D(x, x, P, Q3) for x e M,

f (x) = 0 for x e H - M.

By 10.3.41 it suffices to prove the asymptotic convergence both on M and on H - M; that is, it has to be shown for every q > 0: if we set

M[II f,(1) - f(t) I I ? q] = A. and (H - M) [II f. (x) -. f(f) II ? q] = B. , then (A,) - 0 and (B,) --> 0. In order to prove (A.) -a 0, it suffices to prove (A,S,) - 0, since (?l! -- S,) --i 0. First no point a eM at which Is If we have a sequence ((0,)) of simple finite systems of intervals, then we designate those intervals belonging to 6, by 1;3.

306

DIFFERENTIATION

ICHAP. V

D(a, X, 4,, Qi) exists can belong to infinitely many sets A.S.. For if ((v{)) were a sequence of indices, such that a e A,,S,, , then from a e S,,M it would result, according to the definition (2) of the f,,, that f, (a) --- D(a, X, 0, Q3) = f(a), contrary to a e A ,, . Thus Lim A,S,. =,p A, and hence it follows by 3.5.11 that r

'(:1,S,) ---> 0, which we had to prove.-Since f, = 0 on H - S,, we have B, C S. - M; thus'(S, - M) -* 0 implies also '(B,) --+ 0. Again let ((C,)) he a distinguished sequence of interval systems and let k; (5 2") be the maximal multiplicity of points of C, Besides the functions f, = f(x, x, (e,) defined according to (2) we consider also the functions fYr'

(with f :" = f.), = f `"' (x, x, 6.), defined according to (2.1). We arrange these functions f ;a1 (2.3)

1 S s k* 5 2") as a simple sequence and f+; = f,+1 the functions f,12), fya>

(v = 1, 2,

;

by inserting between f;') = f, Moreover, let ((CSC,, )) be that distinguished sequence of interval systems which results from ((C5,)) by repeating each e, k* times. If CSr -,p M, then -->,p M also, and for these functions we obtain the following theorem analogous to 19.2.3 by the same proof as used above for 19.2.3: 19.2.31. Let M e )2 and CS(,) -+,, M; if the derivative D(x, x, f, Q3) is #-defined on H, then the sequence ((f(,,))) is asymptotically convergent for' on H to the function f defined by (2.2). We now consider a simple finite system CS of intervals I f(j = 1, 2, , m), such that 4,(I;) * 0 (for j = 1, 2, , m). Then the corresponding functions f(x, x, CS) and f ('') (x, x, O [cf. (2) and (2.11)] are bounded and thus, since they are ¢-measurable, by 12.3.52 they are also ¢-summable. -

Again we employ the above notations (p. 304--305).

Then (Ii - I*) Ck

(2 < k 5 k*) consists of the exactly k-fold points of I; - I.

Let CIA,,), be

the set of those points x of (I i - 1;`) - Ck for which I; is the Aah interval of CS (2 5 X 5 k) containing x. Since C,,k,a consists of those A,,,, ... ,, (used in

the proof of 19.1.35) for which i = j, we have Cik,a e fil and, for a fixed j, the C;k,,, are disjoint. If a e I* and c;,k,a e C; k,,, , then by (2) and (2.1) f tA'(ci.x.a , x, 5} = f(a, X, 25). But according to (2), X(I i) = #(I3) f (a, x, e) for a e I;`; henceeo: x(I,) = 4,(I4') f(a, x, S) + E E 1G(C;.k,a) f k-2 a..2 Thus we obtain by 12.1.6: (2.4)

X(Ii) _ (Ii) ff(x, x, e) d¢

k-2 A..2

(Ci.k,)

J

(c,.k.a , X

c).

ft'''(x, x, ") d#.

The C,,k.a belonging to a fixed X are also disjoint. Since C;k,A Q Ck and since every point xo a Ck belongs exactly once to the Xth interval of CS containing

xo (a = 1, 2,

, k), we have for a fixed x: 8C;,k,,, =- Ck (for 2 5 k 5 k* and

i sa If herein a C;,k.x = A (such that c;,k.a does not exist), then we replace the corre-

spouding term of the double sum by 0.

INTERVAL FUNCTIONS

§191

307

2 5 A 5 k). Thus we obtain by summation over j, according to (2.4) and (1) : R

(2.41)

k

EE (Gk) ff°(x, x, Cam) d x(G) _ (S) ff(x, x, e5) d¢ + keg X_2

Now temporarily we assume ¢ to be monotone increasing. According to (2.11) we have for a e I;')*: x(I;) = #(I;) f ()(a, x, S). Thus since #(I;) we obtain by 12.1.6: i x(I j)

>-_

f

4, (J

*) I f

ca)(a,

x, S) I =

)

f If (",(X, x, S) 14,

and hence by (1) : (2.42)

}el

fI

f(a)(x,

x, e) I d¢ = (S)

fI

fa)(x,

x, e) I djF,

for 1 < X S k*. 19.2.4 Let ((5,)) be a distinguished sequence of interval systems of Q310, every interval of L, shall $,, A, and let ¢ be monotone increasing. If the interval function x is VI-continuous on 3, if the set of all x e H at which the derivative D(x, X, J, $3)

exists -_ , H. and if D(x, x, ', $3) is #-finite" on H, then there results for the functions f ° -- f '") (x, X, e,): to every e > 0 there is an ,i > 0, such that for every A e a)l with #(A) < il and for all v and all ). (1 5 X S k*) we have the inequality

(A) f f;)) do < e.42 I

By 19.1.11 there is an n > 0 and a d > 0, such that for every simple finite system Cam. C Z3 of intervals with diameters S it and with i4((e) < +i we have

X(S) < 2 . For every natural number p we designate by I°; ; those intervals P)')* [cf. footnote 38, p. 305] for which I f a)) I > p(> 0). We set S;'p* We designate by the simple finite system consisting of the closures of these intervals I; (for fixed v, X, p) and by St,) the sum of the intervals of Ca,,'D . Our first contention is the following: there is a p and a r,, such that ¢(S°'p) < 7' for v ? r, and 1 S X S k,,. For otherwise to every p there would be a vp > p and a X,a (with 1 <)`,y k!,), such that n ; then by 3.5.11 we would g)) 9 ii'. But since for no point a s Lien also have 4,(Lim f) the derivative P

P

D(a, x, +', l03) (if it exists) can be finite, we have a contradiction to the assumption that D(x, x, 4,, 'Z3) is #-finite; thus the first contention is proved.-Now

we choosey so great that for v z P the intervals of 5, have a diameter

d.

In 19.2.4 IS can be replaced by 0; cf. footnote 37, p. 302.

a According to 12.3.11, the derivative D(x, x, ', 2t) is certainly #-finite if it is 4-sum-

mable.

' Herein is contained, for a - 1, the corresponding proposition for the functions f, J(x, x, 60 .

DI TERENTIATION

308

fore

Then from #(S;"p) < +i' it follows that k(C') <

f j f, d# < for v Z s and 1 9 X S k*. We now set

(2.42) also that (S°y

SO)" - 88,

- S;''.

2 point of S,(")** we have If,(" I < q3, it Since in every

2 , in order to obtain that, according to 12.2.311,

suffices to assume n ' < %&.(A) < »" implies

is, and hence by

`(ASS''f f c'' d4,1

< 2 for all v and all A (1

X

k*).

We set I = min (v', ri"). Let. A e 1t and ¢(A) < L. We have (A) fi;"' d4

ff "' d4

(A - S.) f f.( dP +

r f1a)

+ (AS,("'")

d4,.

Herein, since

f;'? = 0 on A --- S, , we have (A - S,) f f!") df = 0. Moreover, as we just have shown,

(AS;"*) f

(AS;,'p s) fi' Ids

;5.

(A

day,

Sop'f

< 2. f,''>

j

Finally, by 12.2.31 we have:

j d¢ S (S ;) /' If,(") j d4,, and hence,

I

as we have shown above, (AS;'','') f f,''' dP, < 2 for v

is and l S 1

k*.

I

v < fi and 1

h

< e for v z i; and 1 S X 5 k,*. But by 12.3.7 to every k* there is an V;'' > 0, such that '(A) < :1;" implies

I (A) f f;') d# I

< E.

Thus if we set n = min (1, -9 ) for v S i, - 1 and

Thus I (A) f f,()) d4, I

X 6 k*, then #(A) < n implies the inequality v and all X (1 6 h S k*), as contended. 1

I

(A) fi;') dit' < e for all

192.41. We assume (1.1) to be satisfied." In order that there be a set function w associated with the interval function x with respect to 4, and Q3, it is necessary and sufficient that x be $i-continuous on Q3, that the set of all x e H at which thederivvative

D(x, x, 4', Q) exists -,p H, that D(x, x, 4', Q3) be P-summabte on H, and that for every boundary-segment T of every interval I e Q3 we have. (T')

fD(x,x,4',$l)d4' = 0.

NwEserry: This is contained in 19.1.3 and 19.2.24. SUFlclawcy: -We furnish the proof first for a monotone increasing 4'. Let M e TZ and let O (1.1) is used only for that part of this theorem which refers to necessity. then by 19.1.34 the assumption (1.1) is satisfied and, moreover " If 0 is (if D(x, x, o, T) is p-integrable in 9R), we always have: (T) by 12.1.51 and 6.7.13 (M)

f D(x, x, p, 8) d4. a 0, since

f D(x, x, p, $3) day is µ-continuous.

f19]

INTRRVAL FUNCTIONS

309

((c,)) be a distinguished sequence of interval systems of 83 with ei, --+# M. According to 19.2.3, the sequence of the functions f, = f(x, X, (B,) defined by (2) is asymptotically convergent for 4, to the function f defined by (2.2). Thus if we first assume that in C, none of the intervals - . A, then because of 19.2.4

(for X - 1), 15.2.22, and 15.2.221 we have: (A) fi. di, - (A) ffd4' for every A E M.

(2.5)

This holds in particular for A = M. But since by 19.2.4 (for X = 1)

¢(S, - M) --+ 0 and 4'(M - S.) -- 0 imply (S, - M) ff.d4 - 0 and (M - S,) f f, d4' --+ 0, it follows that (S,) f f. d¢ --, (M) f f d¢, and hence by (2.2) :

(S,) fi. 4 - (M) fD(x, x, 0, $3) d¢.

(2.51)

Since 4(S. - M) - 0, by 12.1.51, 5.7.2, and 12.2.3

(S, - M) f I D(x, x, #, Z) I d4. -, 0. Thus if e > 0 and we is sufficiently great, then (S, - M) f I D(x, x, 4, Q3) I d,/'

< E for P Z v . Let Ck'' be the set of those points which belong to exactly k intervals of e, (k = 2, 3, , 2*); then since Ca') - M Q S, - M, we have

by 12.2.31: E (k - 1) (Ck'' - M) f D(x, x, P, Q3) d., ( 5 Ift

E (k - 1) (C;'' - M) f I D(x, x, p, 93 j d4' < e(2" - 1)2"-' for e

ve

Y*

On the other hand by 19.1.351 E (k - 1) (C;')) fD(x, x, ,P, Q3) d¢ = 0. Thus 26

we have also I k (k - 1) (C.(,, M) r 2 ro , and hence by (2.2) also

fD(x, x, 4, *3) do

I

< o(2" - 1)2"-' for

R,

(2.52)

I E (k - 1) (C'') f f d4,1 <E(2" - 1)2"-' for v z vo.

Now we consider the functions f,(" defined by (2.3) and-as there--again arrange them as a simple sequence moreover, as there, we designate by ((tea(,)))

that distinguished sequence of interval systems which results from ((S,)) by repeating eache5, k, times. By 19.2.31 the sequence is also asymptotically convergent for iF to f. Therefore, just as above, we obtain the analogue of (2.5): (A) ff.) d4. (A) ffd# for every A E D2. There results by 15.2.2

[CHAP. V

DIFFERENTIATION

310

that (A) f I frµ) - f I d,& -* 0 for every A e U. Thia holds in particular also

for S* = S(S.); thus we can choose Ao so great that for µ have: (S*)

f

I I day < e. Therefore, since C(') C S*, we can choose

a v;, -v o) so great that for v _ vo we have: (C"))

I < k 5 k* 5

No we

f

f I d¢ < e (for

and 1 < X < k) ; hence k-,

r.-+

E (c ') Jf I

f'(a)

k.21-a2

_ f I dyi <

E(2"-.

1)2'

for v Z VOL Thus by 12.2.31 and 12.2.11 we have also e(2" - 1)2"-' >

E E (C:')) f(f a) - f) d4,1 ? k-2 X..2

I

for v z Yo'. Therefore, since E

k-2 X-2

k;

E (C;')) ff(A)

k-2 k.t

4 - k,2) E E.2 (C(')) ffd#1

(C;')) f f a# = E (k - 1).(C') f f d,/, k-2

k

it follows by (2.52) that I E E (C;')) f f? k-2 ).`2

d I < e(2" - 1)2" for v j.

VOL

f f, d4, < e(2" - 1)2" for v ?. vo; and hence

Thus by (2.41)

by (2.51) we obtain: (2.53)

(M) A (z, x, ¢, 18) dk.

This has now been proved under the assumption that in S, all intervals $ . A. If this assumption is not satisfied, then we form the decomposition C2, = C25; + C;' where S,' consists of those intervals of C, which * A. Now

, -i M, and hence by (2.53) : x(C5;) -3 (M) fD(z, x, 4G, 93) d+,.

C5, -->,. M implies S'

But in Vv' all intervals

A, thus ¢(S,') = 0, and hence the ,y-continuity of x

implies that x(251') - 0. Therefore, since x(SY) = x((S:) + x(S,'), (2.53) holds again.-Thus if we set o(M) = (M) JD(x, X, P, 13) dye, it results that Z, --+d M implies x(5,) -- 4p(M); that is, the set function p is associated with the interval function x with respect to ' and $; so the contention is proved

for a monotone increasing ¢.-But from this result the proposition follows generally by applying it to J instead of '. For if the conditions of 19.2.41 are satisfied for ,,, then they are satisfied also for : The interval function x is also y -continuous. By 17.2.552 the set of all x e H at which the derivative D(x, x, , Q8) exists =,p H also and, if we employ the decomposition H = H++ H-

according to §17 (2.4), we have D(x, x, , 13) =,c D(x, x, y', 93) on H+ and

INTERVAL FUNCTIONS

§ 19]

311

D(x, x, , Q3) = 0 -D(x, X, 1G, Q3) on W. Thus by 12.3.1, 12.3.3, and 12.1.53 D(x, x, , 98) is also'-sunnmable on H, and hence by 12.3.2 and 12.2.24 also iy-summable. Finally, (T) fD(x, x, ¢, Q3) dO = (TH+)

(TH-) JD(x, x, , Q3) d4e; thus since ¢,-(TH+)

fD(x, x, , Q3) d¢ -

0 and P+(TIr) - 0, it

follows by 12.2.23 and 12.1.5 that

(T) f D(x, x, 4,, 1) dpi = (TH+) f D(x,

Q3) d4*

+ (TH-) f D(x, x, , Q3) dti (T) f D(x, x, , Q3) d4,+ + (T) fD(x, x, $, $3) d>/+

,

and by 12.2.24 this equals (T) f D(x, x, 4, %) dlr.

Hence (T) fD(z, x, ¢, $3) d4, = 0 implies (T) fD(; x, Z, Q3) d4 = 0 for every boundary-segment T of every interval I E FS. Thus all conditions of 19.2.41

are satisfied also for J, as contended. BiRmooRAPHY to 519: H. LEBESOUD, Ann. Ec. Norm. (3) 27 (1910), p. 385, 408; J. RADON,

Sitzungsberichte Akad. Wiss. Wien 122 (1913), p. 1306; C. DE LA VALL*B POUssIN, Trans. Amer. Math. Soc. 16 (1915), p. 458, 478, 493; [21, p. 76, 98; C. CARATUtODORY [1], p. 502; J. C. BURKILL, Proc. London Math. Soc. (2) 22 (1924), p. 275; J. RIDDER, Nieuw Archief v. Wiskunde (2) 161 (1929), p. 55; (2) 16, (1930), p. 50; S. SAKS [11, p. 5, 46; [21, p. 59, 93, 105; P. REICEIELDERFER and L. RINOENBERO, Duke Math. Journ. 8 (1941), p. 231; L. A. RINaENBERO, Trans. Amer. Math. Soc. 61 (1947), p. 134. As to the above theory: A. ROSENTHAL,

Publicaciones Instit. de Mat., Universidad Nac. del Litoral, 51 (1945), p. 153.

BIBLIOGRAPHY

LIST OF BOOKS QUOTED E. BOREL [1] Lecons sur la ih6orie des fonctions, Paris 1898. Paris 1905. [2] Legons sur lee fonetions de variables C. CARATSiODORY [1] Vorlesungen Ilber recite Funktionen, Leipzig-Berlin 1918 (2. Auflage 1927). A. L. CAUcaY [1] Cours d'analyse de l'Ecole Polytechnique 1, Analyse algebrique, Paris 1821. U. DINT [1] Fondamenti per la teorica delle funxioni di variablli reali, Pisa 1878.

L. M. GRAvEs [1] The theory of functions of real variables, New York-London 1946.

H. HAHN [1 ] Theorie der reellen Funklionen, I. Band, Berlin 1921. [2] Reelle Funktionen, 1. Teil: Punktfunktionen, Leipzig 1932. G. H. HARDY, J. E. LIYTLEwOOD, G. P6LYA [11 Inequalities, Cambridge 1934. 0. HAUPT [11 Differential- and Integralrechnung (three volumes), Berlin 1938. F. H.AueDoRis [1] Grundzuge der Mengenlehre, Leipzig 1914. [21 Mengenlehre, 2. Auflage, Berlin-Leipzig 1927.

E. W. HossoN III The theory of functions of a real variable and the theory of Fourier's series, vol. I, 3rd ed., Cambridge 1927; vol.II, 2nd ed., Cambridge 1926. C. JORDAN [11 Cours d'analyae (2. ed.), vol. I, Paris 1893.

H. KrszsLiAN [I] Modern Theories of Integration, Oxford 1937. H. LEBESGUE [1] Legons sur l'integration, Paris 1904. [21 Legons sur l'integration, 2. ed., Paris 1928. [3] Legons sur les series trigonometriques, Paris 1906. E. J.. McSn&wz [1] Integration, Princeton 1944. H. MINKOWSKI [1] Geometrie der Zahlen, Leipzig-Berlin 1896 and 1910. J. voN NEUMANN [1] Functional Operators, Princeton 1933-1935. G. PEANO [11 Applicazioni geometriche del calcolo inftnitesimale, Torino 1887. J. PIEmoNT [1] Lectures on the theory of functions of real variables, vol. 1, Boston 1905; vol. II, Boston 1912. 8. SASS [1] Theorie de l'integral, Warszawa 1933. [2] Theory of the integral, Warszawa-Lw6w 1937. E. TORNIER [l ] Wahrscheinlichkeitarechnung and allgemeine Integrationstheorie, Leipzig-Berlin 1936. CR.-J. DE LA VALL* POUSSIN [1] Cours d'Analyse infcnitesimale (2. 6d.), vol. I,

Louvain-Paris 1909; vol. II, Louvain-Paris 1912.

[21 Int4grales de Lebesgue, Fonctions d'enaemble, Classes de Baire, Paris 1916. G. VrrALI [11 Sul problems delta misura dei gruppi di punti di una recta, Bologna 1905.

NV. H. YouNO and G. CHISHOLM YouNG (11 The theory of sets of points, Cambridge 1906. A. Z'eoMuND [1] Trigonmietrical series, Warszawa-Lw6w 1935. 312

LIST OF SYMBOLS AND SIGNS

Pap a ti e M (a is not an element of the set M) .......................................... I (a) (the set consisting of the single element a) ...................................... I

a M (a is an element of the set M) ................................................

, ak) ................

1

A (the empty set) .................................................................. A C B, B 2 A (A is a subset of B) .................................................

I

jai , at ,

, ak} (the set consisting of the elements a1 , a: ,

-

A C B, B D A (A is a proper subset of B) ......................................... A+ B (sum of the sets A and B) ................................................... A-B, AB (intersection of the sets A and B) .........................................

A - B (difference of the sets A and B)............ , ................................ E - A, -A (complement of the set A) ............................................ S Ak , SAk , S A,, (sum of sets) .................................................. k-I

k

k

k.x

(intersection of sets) ...........................................

((as)) (sequence of the elements a, , at , , ak , ) ............................... ((A,)) (sequence of the sets A, , A2, ... , Ak , ... ) ................................. Lim AS , _k Lim A,, , Lim A, (Limes superior, Limes inferior, Limes of a sequence k

1

1

1

4

k*K

M

D AS, , D Ak , D Ak

k-1

I

1

2 2

k

of sets) ..................................................... ............ .

.

.

.. .

.

.

2

3T4 , 101, (smallest o-system or a-system over WI) ....................................

3

lle,, MI,, ......................... ................................................ Ms, (Borel system over $R) ......................................................... U. (neighborhood of point a) ....................................................... A() (open kernel of the set A) ..................................................... A(,) (border of the set A) ...... ...................................................

3

AO (closure of the set A). - .... .................................................... A' (derivative of the net A) ........................................................

Aa (nucleus of the set A) .......................................................... As (scattered part of the set A) .................................................... P 131 .............................................................................

lim ak - a, ak -* a (a is limit point of the sequence ((a,))) .......... .

...............

3

4 4 4

6

6 6

6

J;

inf x, sup x (greatest number to which x e A is inferior, smallest number to which no zed

.*A x e A is superior) ..............................................................

8

max(a, b), min(a, b) (greater and smaller of the numbers a and b) ................... 118 sgi (signum) ....................................................................... 199 ab (distance of the points a and b) .... ............................................ 8 aB (distance of the point a from the Bet B)... ................ .................... AB (distance of the sets A and B) .................................................. d(A) (diameter of the set A).. ......................... S., (sphere with center a and radius p) ............................................. (closed sphere with center a and radius p) ...................................... Ua, (neighborhood p of the set A) .................................................. R. (n-dimensional Euclidean space) ................................................. .

8 8

8

8 9 8

R, (Euclidean straight line) ........................................................ 8 $l ............. .......................... ........................................ 23 S(a) (bounding transformation) .................................................... 23

318

LIST OF SYMBOLS AND SIGNS

314

Page

II a, -- at II (distance of two points a, and at of Ri) ..................................

(a, b) (open interval in R,) ......................................................... [a, b] (closed interval in R,) ........................................................

]a, b) and (a, b] (half-open intervals in R,) .........................................

X A. (combinatory product of A, , At , , A, X As X , a b, , b3 , , b,) (open interval in R,) ............................. (a, , a: , (a, , a2 , ... , a ; b, , b: , ... , b,.] (closed interval in R.) ............................

23 91 91

91 91 91 91

(half-open

and

intervals in

91

I, J, I (closed, open, half-open interval) .......................................91,93,94 '' X C" tproduct of subdivisions of a system of intervals). ........................ 91

K (power of the continuum) ........................................................ 100 f, f (upper and lower limit function of f) ........................................... 187 LEI, f;:, ti42, (Zx (systems of functions of first and second order) ....................... 146 (, (- C= (S o .......................... .... ........................ 146 (g s" (,

A (5) (nucleus of the Suslin scheme ;) . .. ........................................ 71 BI f (partial function of f, restricted to the subset B) .. ........................... 144

01,p (partial function of p, restricted to the partial system Z) ...................... Al 3l (system of all subsets of A belonging to `.1R) ........................ ......... ositive-function, ne ative-function absolute-function of A = , A (zero-set for,p)

..........................................................

A $ , A_ ............. ......

..............

.

.

.

.

.

.

.

.

.

.

.

. .......................

Ace B (A is a subset of B, neglecting a zero-set for ,) ............................. A = B (.4 and B differ only as to zero-sets for ,o) .

................................

17 17

18 27 27

29 30

A *, B ...... ..................................................................... Limo A,,,................... .......................................................

30

.4(= ,)A (zero set for p in the wider sense) ..........................................

..................................................................... A (-,)B ............................................................................

33 83 33

T2° (smallest complete field containing 912) ..........................................

33

,p° (extension of p front D2 to 912°) .................................................

84

A(F_,)B ..

..

81

(system of singular sets) .........................................................

36

92(,c) or 9? (system of the sets regular forjp) .........................................

36

...................................................................

37

us ,,30 , an

o`,jr** (regular and singular part of p) ................. .......................... 38,43 rx, px (outer and inner p-measure) ..................... ............................ 65 A', Ax (measure-cover and measure-kernel of A) ................................... 68 )t (complete cover of a field $) ..................................................... ,u (n-dimensional [outer] measure) ..................................................

76 93

k.x (n-dimensional inner measure) .................................................. 93 Mk (k-dimensional ]outer] measure in R,,) ............................................ 106 107 axx (k-dimensional inner measure in

Uf(1) > y], AIf(x) > y] (the set of all x e A with f > y) ............................. 110 (, < s f"

,

f^ _ I f1 ; f, <,, f2 , f2 >, f,....... ......................................... 118

f, _ J>(f, and fs are p-equivalent) ................................................. 118

f,

f (i.e., lion ff =,, f) ........................................................... 121

(A)J f d4 (c-integral off on A) ...................................................... 150

(A) f f d', (A) f f (hp (upper and lower c-integral of f on A) ......................... 171

LIST OF SYMBOLS AND SIGNS

315

Page L'(f,o, [z.]), L'(f,cc, ]z.l ), L'(f,q,, ]z:]), L"(f,o,, ]z:)), L(f,,v, ]z,)) (Lebesgue sums)... 180

S(f, V, Z) (Riemann sum).... ...................................................... 183 S(f, v, T)), (f, ,p,

) (Darboux sums, [upper and lower sum]). . ...................... 190

+ +(system of the sums of finitely many sets of

i) ................................... 225

ff' X ( and a' X ( X a"' (product of fields). . .................. ............. 226,232 v' X and ,p' X o' X m"' (product of totally-additive set functions) ......... 228, 233, 235

M. (set of all y for which (x, y) a M) ................................................ 229 M (set of all x for which (x, y) eM) ................................................ 229 Z, (ordinate set of the function.f) ................................................... 237 V. (subsystem of 3)1, associated with point a) ........................... ........... 246 D(a, gyp, ,., V.),

2[t.), D(a, w, >.,

(upper and lower derivate and derivative

of c with respect to t4 at a on 9)t.) .............................................. 246

D(a, p, j., C), D(a, p, y', C), D(a) jo, 4,, C) (the same on C if C is an indefinitely fine

system) ........................................................................ 247

0' (ordinary system associated with C) ............................................ 280 Yt (Vitali system lof sets)). ......................................................... 247

48. (subsystem of all V e Q3 containing a) ........................................... 247 D(a, ap,

D(a, v, tG, 23), D(a, o, ¢, $3) (upper and lower Vitali derivate and Vitali

derivative of ,p with respect to t' on $3) ......................................... 247

¶ (paving system (of sets]) ......................................................... 254

D(a,,p, t4, 1)), D(a,p, 0,'t), D(a,,,, 4', ') (upper and lower paving derivate and paving derivative of m with respect to ,y on ')......................................... 254

Z (covering system (of a set A)). ................................................... 261 Z (tile system [of sets!) ............... ........................................... 268

D(a,,p, 4', $), D(a,,p, 4p, tive of

), D(a,,p, 4', Z) (upper and lower tile derivate and tile deriva-

. with respect to tk on T).

............................................. 269

d(a, A, y , £2), d(a, A, 7G, C), d(a, A, , 0) (upper and lower outer density and outer density of A at a for iy onf; ) ............................ ...................... 275

dx(a, A, i, 0), dx(a, A, , 0), dx(a, A, , ID) (upper and lower inner density and inner

density of A at a for !Z on C) ...................... ........................... 275

d(a, A), d(a, A), d(a, A); dx(a, A), dx(a, A), dx(a, A) (upper and lower outer density

and outer density, upper and lower inner density and inner density of A at a (in R.I) ........................................................................ 281

b(a, A), A(a, A), o(a, A); 3x(a, A), bx(a, A), dx(a., A) (densities (in R.) on the system of all open intervals) .......

.................................................. 282

Cam,- # M (the distinguished sequence ((Z,)) of interval systems converges to M for 4,).. 298 f(x, X, e),P")(x, x, e) ............................................................. 305

INDEX Borel sets 3, 82, 93, 112; B. system 3

(The numbers refer to pages)

Above, approximately continuous a. 287; continuous a. 144; somewhat continuous A. 286

Absolute-function 18; absolute G3 87

Bound, greatest lower b. 8; least upper b. 8 Boundary-segment (of an interval) 300 Bounded set 9; ,-bounded function 110 Bounding transformation 23

Absolutely additive set function 14; a. continuous 56 Accumulation, point of a. 5, 9 Adapted, sequence of decompositions a. to a function 184 Addition, closed with respect to a. 2

Additive set function 11; absolutely a. set function 14; completely a. set function 14; countably a. set function 14; totally a. set function 14 Almost all 2; a. everywhere convergent 121 Always of positive measure 83 Analytic set 82, 83, 93, 112; a. a. over fDt 71; non-a. sets 100, 102

Approximately continuous 286, 287; a. c. above (below) 287 Associated, measure function pp a. with 76; metric space a. with 9 ft by rp 32, 55, 57; ordinary system :O* a. withO 280; point

a. with X by p 32; point function a. with m 55; set function a. with an interval function 298, 308

Associative law of the multiplication of set functions 232 Asymptotic convergence 126, 133, 135, 213; a. c. for . 126; ip-asymptotic convergence 126 Asymptotically convergent subsequences 137

-Atom 45

Atomic set functions 48; decomposition of a totally additive set function in atomless and a. part 51 Atomless set functions 48 Axioms, metric a. 8; topological a. 4 Baire functions 120,122, 140; B. f. of let, 2nd class 146, 176; B. f. of let, 2nd order 146 Basis of the real numbers 100

Below, approximately continuous b. 287; continuous b. 144; somewhat continuous b. 288 Border of a set 4; b. points of a set 4; b. sets 4

317

Cantor discontinuum 99 Carath4odory linear measure 105, 108, 268, 286

Category, set of first (second) c. 5, 10; set of first (second) c. in itself 5, 10 Cauchy inequality 202, 208; Lagrange-C. inequality 208; C. sequence 9

Characterization of the set functions that are gyp-integrals 168, 255

Class, Baire functions of first, second c. 146, 176

Closed cube 96; c. interval in R, 91, in R. 91;

c. set 4; c. in a set 5, 6; c. sphere 8; c. with respect to addition, intersection, subtraction 2 Closure of a set 4, 6 Combinatory product of sets 91

Compact set 6, 9; c. sets of R-measurable functions 135; c. in a set 6; selfcompact 6 Complement of a set 4 Complete cover of a field 76; c. field 34; extension of a set function to a c. field 34; c. inverse image 111; c. set 10; c. system of sets 34

Completely additive set function 14; c. ip-integrable sequence 214; convergent in a c. tv-integrable manner 214 Content function 84, in R. 90; methods for the construction of a c. f. 90, 104

Content-like 88; c.-1. determining function 175,188, 251,264, 289, 296; strongly c.-1. 89

Continuous measure function 83; c. point function 139; measurability of c. functions 139; c. set function 43; c. above (below) 144;c-continuous above (below) 144; f-continuous point function 141; 4-continuous interval function 297; ,y-con-

tinuous set function 54, 56; sequences of 0-c. a. f. 56; absolutely c. 56; approximately c. 286, 287; approximately o. above (below) 287; equi-continuous

INDEX

318

point functions 57; equi-4--continuous set functions 56; semi-continuous 144; upper, lower semi-continuous 144; somewhat c. above, (below) 286; totally c. 56

Continuity-element of p 44; c.-function of w 43; c.-point of v 44; yG-continuityfunction of p 56 Convergence, asymptotic c. 126, 133, 135, 213; a. c. for ' 126;,-asymptotic c. 126; mean c. 208; "c. en mesure" 135; c. of

p-defined 110

S-decomposition .183; 5-field 3; S-ring 3; 3-scale

180; S-system 3; smallest

5-

system over 9)1 3

Dense 5; dense-in-itself 6; n9where dense 5 Density 274, 275, 281, 282; d. theorem 277, 278, 280, 281, 282, 286, 290; inner d. 275,

a distinguished sequence of interval

281, 282; outer d. 275, 281, 282; upper, lower d. 275,281,282; upper, lower inner d. 275, 281, 282; upper, lower outer d. 275, 281, 282

systems to a set 298; set of c. 121 Convergent in a p-integrable manner 213; e.

Denumerable set I Derivate(s) 246, 302; upper, lower d. 246,

in a completely ,p-integrable manner 214; c. in the mean (of order p) 208, 211, 214, 216; c. sequence of points 6, of point functions 121, of set functions

302; limit functions of a d. 294; normal

23; c-convergent sequence of functions 121; to-convergent subsequences 132,

Vitali d. 247, 294, 296, 302; upper, lower Vitali d. 247, 302

d. 274, 294; orderly d. 261, 293; paving d.

254; upper, lower paving d. 254; tile d. 268, 269, 296; upper, lower tile d. 269; .

121;

Derivative of an interval function 302; d.

asymptotically c. see Convergence; asymptotically c. subsequences 137; regularly c. 258; uniformly c. sequences of point functions 124, 135, of set func-

of a set 6; d. of a set function 246; paving d. 254; regular d. 257, 258, 273, 291; tile

134, 139; almost everywhere c.

tions ' 23; uniformly c. subsequences 135, 137; quasi-uniformly c. 142; simplyuniformly c. 142

Converging, sequences of sets c. to a point 246.' Convex function 116, 117; discontinuous c. function 117; c. metric 198, 203; c. set 198

d. 269; Vitali d. 247, 302

Determining function of an integral 150; content-like d. f. 175, 188, 289, 296

Diameter of a set 9 Differ, two sets d. only as to zero-sets 30 Difference of sets 1

Differentiation 246; integration and d. 289 k-dimensional measure in R 106, 107; k-d.

inner measure in R. 107; k-d. outer measure in R. 106; n-dimensional closed

(half-open) cube 96; n-d. Euclidean

Countable set 1 Countably additive set function 14 Cover 7, 261; complete c. of a field )jr with

inner measure in R. 93; n-d. outer

respect to r 76; measure-cover 68, 84 Covering system 201; Vitali's C. Theorem 266, 267, for intervals 266

k-dimensionally measurable (in R.) 107; n-dimensionally measurable (in R.)

Cube, (n-dimensional) 96; closed c. 96; halfopen c. 96

Darboux sums 190; upper, lower D. a. 190 Decomposition of a totally additive func-

tion in atomless and atomic part 51,

space R. 8; n-d. measure in R. 9e; n-d.

measure in R. 80, 93 93

Discontinuity-element of .p 43; d.-function of v 43; d.-point of. 44; 4-discontinuityfunction of p 56

Discontinuous convex function 117; purely d. set function 43; decomposition of a

sequence of d. adapted to a function

totally additive set function in continuous and purely d. part 43; purely #-discontinuous set function 54 Discontinuum, Cantor d. 99 Disjoint 2; d. for v 28

184; distinguished sequence of d. 186; singular d. of a set 39, 48, 257 Decreasing, monotone d. sequence of sets 2; monotone d. set function 12

Distance of two points 8; d. of a point from a set 8; d. of two sets 8 Distinguished sequence of decompositions 186; d. a. of interval systems 297; con-

in continuous and purely discontinuous

part 43, in regular and singular part 40, 43; d. of set of integration 183; Sdecomposition of set of integration 183;

INDEX

319

vergence of a d. s. of interval systems to a set M 298; d. system of sets, open in A, 7 Distributive laws 1 Divergence, set of d. 121

ous convex f. 117; p-defined f. 110; de termining f. of an integral-ISO; content-

Edge of a cube 96

176; interval f. 90, 297; derivation of

Egoroff, theorem of E. 124 Element of a set 1; sequence of elements 2; continuity- and discontinuity-element

interval f. 302; set f. associated with an interval f. 298, 308; upper, lower limit f. 187; upper, lower limit f. neglecting the zero-sets 295; limit f. of a derivate

Empty set 1

294; p-measurable f. 110, 113, 135; com-

Equation, functional e. f(z, + x2) - f(x,) +

pact sets of p-measurable f. 135; sequences of p-ineasurable f.. 119; non-

ofp43,44

f (x:) 116

Equi-continuous point functions 57; equi-

like determining f. 175, 188, 251, 254,

289, 296; discontinuity-f. of p 43; kdiscontinuity-f: of p 56; gyp-equivalent f. 118; p-finite f. 110; 4 -integrable f. 149,

r1-measurable f: 115, 116, 120; measure

p-Equivalent functions 118 Euclidean space 8

f. 61, 84; measure f. p associated with >- 76; continuous measure f. 83; ordiuary measure f. 81, 84; regular measure

Everywhere, almost e. convergent 121

f. 72, 74; negative-f. 18; ordinate set of a

Extension of a set function to a complete

f. 237; partial f. restricted to a subset 18, 144, to a partial system of sets 17;

t&-continuous set functions 56

field 34; e. of a totally additive set function from a field to a or-field 74, 76, 80

Extremes of a totally additive set function 16

F,-set 5 Field 3; complete f. 34; 5-field 3; o-field 3; product field 226, 232 Fine, indefinitely f. system 247; ordinary i. f. s. 247 Finite set 1; f. subdivision 91; simple f. system of intervals 91, 297;p-finite function 110

First, set off. category 5,10; Baire functions of f. class 146, of f. order 146; f. mean value theorem 194 Frontier 4; f: point 4 Fubini's theorem 238

Function(s). (see also Set function), absolute-function 18; approximately continuous f. 286, 287; f. approximately continuous above (below) 287; faire f. 120, 122, 140, of 1st, 2nd class 146, 176, of 1st, 2nd order 146; p-bounded f. 110;

content f. 84, in- R. 90; continuityfunction of p 43; J,-continuity-function of p 56; continuous measure f. 83; continuous pt. f. 139; continuous set f. 43; measurability of continuous f. 139; f. continuous above (below) 144; se-continuous point f. 141; f. p-continuous above (below) 144; t'-continuous set f. 54, 56; convex f. 116, 117; discontinu-

point f. 11; point f. associated with p 55;

positive-f. 18; semi-continuous f. 144; upper, lower semi-continuous f. 144; set f. 11, see also Set function; f. some-

what continuous above (below) 286; p-sotnmable f. 165

Functional equation f(x, + x:) - f(x,) + J(x,) 116 Gs-set 5; absolute G, 87

Geometric interpretation of the integral 237

Greatest lower bound 8

Half-open cube (n-dimensional) 96; h.-o. interval in R, 91, in R. 91 Holder inequality 202, for integrals 202 Image, complete inverse i. 111

Imperfect, totally i. set 87 Improper (p-) integral 176, 177, 182 Increasing, monotone i. sequence of sets 2;

monotone i. set function 12 Indefinitely 'me system 247; ordinary i. f. s. 247; ordinary system .a associated with the i. f. s. CO 280

Inequality, Cauchy i. 202, 208; Holder i. 202; Lagrange.-Cauchy i. 208; Minkowski i. 200, 205; Schwarz 1..203; triangle

i. 8 Infimum 8 Inner (upper, lower) density 275, 281, 282;

1JW

320

i. Lebesgue measure 93; k-dimensional i. measure (in R.) 107; n-dimensional i. measure (in R.) 93; i. p-measure 65; i. point 4 p-Integrable function 149, 176; convergent in a go-i. manner 213; convergent in a

Law, asociative 1. of the multiplication of set functions 232; distributive laws 1 Least upper bound 8 Lebesgue measure 93, 100; inner, outer L. measure 93; measurable in the L. sense

completely p-i. manner 214; so-i. sequence 213; completely so-i. sequence

Length of the edge of an n-dimensional

214

Lexicographical order 304

So-Integral (s)

150; characterization of set

functions that are so-i. 188, 255; determining function of an i. 150; inequalities for i., Holder 202, Minkowski 206, Schwarz 203; geometric interpretation of the i. 237; improper (p-) i. 176, 177, 182; iterated i. 241; limits of integrals 154, 156, 168, 213; first, second mean value theorem for i. 194, 195, 197; proper i. 176; transformation of i. 296; upper, lower p-i. 171, 190 Integrand 160

Integration 149; i. and differentiation 289; set of i. 150 Intermediate values 51; i, value theorem 51, 53

Interpretation, geometric i. of the integral 237

Intersection of sets 1; closed with respect to

93; L. sums 180

cube 96 Limes of a sequence of sets 2; Limes inferior, superior 2 Limit, lower, upper 1. function 187; 1. func-

tions of a derivate 294; lower, upper 1. function neglecting the zero-sets 295; 1. point 6, 9; limits of set functions 13, 14, 22, 23, 60, of integrals 154, 156, 168, 213

Line segment (of R.) 198 Linear 93; 1. form 199; Carathdodory 1.

measure 105, 108, 268, 286

Lower density 276, 281, 282; 1. inner, outer density 275, 281, 282; 1. derivate 246, 302; 1. p-integral 171. 190; 1. limit function 187; 1. 1. f. neglecting the zerosets 295; 1. paving derivate 254; 1. semicontinuous function 144; 1. (Darboux) sum 190; 1. tile derivate 269; 1. Vitali derivate 247, 302

i. 2 Interval(s), closed i. in R, 91, in R. 91; half-

Maximum of a totally additive set function

open i. in R, 91, in R 91; open i. in R, 91, in R 91; boundary-segment of an i.

Mean convergence 208; convergent in the m.

17

300; covering system of i. 266; finite sub-

(of order p) 208, 211, 214, 216; first,

division of i. 91; simple (finite) system of i. 91, 297; Vitali Covering Theorem for i. 266; i. function(s) 90, 297; '-con-

second at. value theorem 194, 195, 197 Measurability of continuous functions 139, of Baire functions 140

tinuous i. f. 297; derivates, upper and

Measurable in the Lebesgue sense 93; µrmeasurable (in R.) 107; K,-measurable

lower derivates, derivatives of i. f. 302; set function associated with an i. f. 298, 308; Vitali derivates, Vitali upper and lower derivates, Vitali derivatives of i. f. 302; distinguished sequences of i. systems 297 Inverse, complete i. image 111 Isolated point 6 Iterated integrals 241

1. dimensional (inner, outer) measure in R. 108, 107; k-dimensionally measurable (in R.) 107

93; non-analytic µ.-measurable sets 100, 102; so-measurable function 110. 113,135;

compact sets of c-m. functions 135; sequences of m-m. functions 119; p-m. set 61, 110; k-dimensionally m. (in

R.) 107; n-dimensionally measurable (in R.) 93; non-pl-measurable functions 116, 116, 120; non-p1-m. sets 102, 103; none-m. sets 62, 87; onedimensionally non-m. sets 102, 103 Measure 61; p-measure 61, 110; inner, outer

p-m. 65; Carathtodory linear in. 105,

Kernel, open k. 4; measure-kernel 68, 83, 84

108, 268, 286; k-dimensional inner, outer

Lagrange-Cauchy inequality 208

m. in R. 108, 107; k-dimensional m. in R 106, 107; (inner, outer) Lebesgue m.

DM= 93, 100; n-dimensional m. in R. 93; ndimensional inner, outer in. in R. 80, 93; always of positive in. 83; m.-cover 68, 84; in. function 61, 84; in. f. v associated with >, 76; continuous in. f. 83; ordinary in. C. 81, 84; regular m. f. 72,

321 lexioographioal o. 304; convergent in the mean of o. p 208, 211, 214, 216

Orderly derivate 261, 293; o. system (of sets) 261, 293 Ordinary measure function 81, 84; o. indefi-

nitely fine system 247; o. system £1`

74; m.-kernel 68, 83, 84 Metric axioms 8; in. space 8; in. space associated with 9)1 by m 32, 55, 57; in. space

associated with la 280 Ordinate set Z, of the function f 237 Outer density 275, 281, 282; lower, upper o.

of partial systems 31; convex M. 198,

density 275, 281, 282; o. Lebesgue measure 93; o. So-measure 65; k-dimensional

203; polar in. 199, 204

Minimum of a totally additive set function 17

Minkowaki inequality 200, for integrals 205 Monotone decreasing, increasing sequence of sets 2; m. decreasing, increasing set function 12; product of m. (non-monotone) set functions 223, 234; in. Suslin scheme 71

Multiplication of (monotone, non-monotone) set functions 223, 234; associative law of the in. of set functions 232

n-dimensional closed (half-open) cube 96; n-d. Euclidean space R. 8; n-d. measure in R. 93; n-d. inner, outer measure in R. 80, 93 n-dimensionally measurable 93 Negative-function 18 Neglecting zero-sets 28; limit-functions, a. z.-s. 295; subset of a set, n. a zero-set 29

Neighborhood of a point 4; n. p of a set 9 p-Net 9 Nikodym, Radon-N. theorem 170, 171, 255 Non-analytic sets 100, 102; non-,.1-measurable functions 115, 116, 120; non-pi-m. sets 102, 103; non-so-m. sets 62, 87; one-dimensionally non-m. nets 102, 103

Norm (of a decomposition) 186 Normal derivate 274, 294; n. system (of seta) 274, 294

Nowhere dense 5 Nucleus of a set 6; a. of a Suslin scheme 71

O. measure in R. 106; n-dimensional o. measure in R. 80, 93

Parameter of regularity 258 Part, regular and singular p. of a totally additive set function 38, 43; scattered p. of a set 6 Partial function, restricted to a p. system of sets 17, to a subset 18, 144; p. system of sets 17; metric space of p. systems 31; representative of a p. system 31 Paving derivate 254; lower, upper p. derivate 254; p. derivative 254; p. system 254

Perfect set 6

Point of accumulation 5, 9; p. associated with X by v 32; distance of two points 8, of a p. from a set 8; p. function 11;

p. function f associated with w 55;

border p. 4; continuity-p. 44; discontinuity-p. 44; frontier p. 4; inner p. 4; isolated p. 6; limit p. 6, 9 Polar metric 199, 204 Positive-function 18; always of p. measure 83

Product field (p. of fields) 226, 232; p. of sets

91, 223; p. of totally additive set functions 223, 228, 233, 234 Proper integrals 176; p. subset I

Purely discontinuous set function 43; p. #-discontinuous set function 54 Quasi-uniform convergence 142

Numbers, basis of the real n. 100 Onedimensionally non-measurable sets 102, 103

Open interval in R, 91, in R. 91; o. kernel 4; o. set 4, 8, 10; o. in a set 4; distinguished system of sets, o. in A, 7; see also Halfopen Order, (Baire) function of let, 2nd order 146;

Radon-Nikodym theorem 170, 171, 265 Rest, basis of the r. numbers 100 Regular convergence 258; r. derivative 257,

258, 273, 291; r. measure function 72, 74; r. part of a totally additive set function 38, 43; decomposition of a totally additive set function in its r. and singular part 40, 43; r. set 86; 4-regular set

INDEX

322

54; r. set function 40; method for construction of a r. set function 74; r. system (of sets) 2.58, 280 Begu)Krity, parameter of r. 258 Representative (of a partial system) 31 Hien,ann sums 183 Ring (of sets) 3; 6-ring 3; or-ring 3 p-net 9

tered part of a s. 6; sequence of sets 2, see also Sequence; (proper) subset of a a. 1; subset of a a., neglecting a zeroset 29; sum of sets 1; system of sets 2, see also System; analytic a. 71, 82, 83, 93, 112; Borel s. 3, 82, 93, 112; border a. 4;. bounded s. 9; clpsed a. 4; compact s. 6, 9, 135; complete s. 10; convex s. 198; countable s. 1; dense a. 5; a. dense-

Scale 180; s-scale 180

Scattered part of a set 6.; a. set 6 Scheme, Suslin s. 71; monotone S. s. 71 Schwarz inequality 203 Second, set of s. category 5, 10; a. class 146, 176; s. mean value theorem 196, 197; a. order 146 eguient, boundary-a. (of an interval) 300; line a. (of R,,) 198 Self-compact set 6 Send-continuous function 144; lower, upper s.-c. f. 144 Separable set 7 Sequence of decompositions adapted to a function 184; a. of elements 2; s. of rpmeasurable functions 119; a. of sets 2;

in-itself 6; denumerable a. 1; disjoint sets 2; sets disjoint for p 28; empty a. 1; finite a. 1; Fe-set 5; G1-set 5, 87; A.-measurable a. 93; non-analytic measurable s. 100, 102; c-measurable a. 61, 110; r on-, -.measurable s. 62, 87; non-µ,-measurable a. 102, 103; nowhere dense s. 5; open a. 4, 8; ordinate set Gf

of the fuuction..f 237; perfect a. 6; regular a. (for v) 36; tG-regular a. 54; scattered a. 6; self-compact a. 6; separable s. 7; singular s. 36; tG-singular a. 54; totally imperfect a. 87; Young a. 87; zero-set 27; z.-s. in the wider sense 33

Set function (s) 11; a. f. associated with an interval function 298, 308; characteriza-

s. of sets converging to a point 246;

tion of the set functions that are ip-

limes and limes inferior, superior of a a. of sets 2; s. of set functions 13, 23; a. of +y-continuous set functions 56; almost everywhere convergent a. 121; asymptotically convergent s. 126, 133,

integrals 168, 255; derivates of a. f. 246; limits of a. f. 13, 14, 22, 23, 60; multiplication of s. f. 223, 234; associative law of the multiplication of s. f. 232; product

135; Cauchy s. 9; completely integrable a. 214; convergent s. of points 6, of point

f. 18, 23; convergent sequences of 9. f. 23; uniformly convergent sequences of a. f. 23; absolutely additive s. f. 14; absolutely continuous a. f. 56; additive a. f. 11; atomic a. f. 48; atomless s. f. 48;

functions 121, of set functions 23; nconvergent s. 121; distinguished a. of decomposition 186, of interval systems 297; convergence of a distinguished a. of interval systems to a.set 298; integrable e. 213; monotone decreasing, in-

creasing s. of sets 2; quasi-uniformly convergent a. 142; simply-uniformly convergent a. 142; uniformly convergent

a. of point functions 124, 135, of set functions 23 Set of first, second category 5, 10; a., of convergence 121; density of a a. 274, 291; derivative of a a. 6; diameter of a a. 9;; difference of sets 1; distance of a point from a s. 8, of two sets 8; s. of divergence

121; element of a a. 1; s. of integration 150; intersection of sets 1; neighborhood

p of a a. 9; nucleus of a s. 6; product of sets 91, 223; relations for sets 1; scat-

of a. f. 223, 228, 233, 234; sequences of a.

completely additive a. f. 14; contentlike a. f. 88; continuous s. f. 43; 'continuous a. f. 54, 56; sequences of 4continuous a. f. 56; countably additive s. f. 14; equi-¢-continuous s. f. 56; monotone decreasing, increasing a. f. 12; purely discontinuous e. f. 43; purely #-discontinuous a. f. 54; regular a. f. 40;

singular a. f. 40; strongly content-like a. f. 89; totally additive a. f. 14; decoin-

position of a totally additive s f. in its atomless and atomic part 51, in its con-

tinuous and purely discontinuous part 43, in its regular and singular part 40, 43, 56; extension of a totally additive a. f. 34, 74, 76, 80; extremes, maximum,

minimum of a totally additive s. f. 16;

INDEX

regular, singular part of a totally additive s. f. 38, 43; totally continuous a f. 66

o-field 3; s-ring 3; a-system 3; smallest osystem over 9)2 8 Signum 199

323 simple (finite) s. of intervals 91, 297;

Suslin B. of sets 71, 73, 112; tile s. 268, 278, 280, 284, 286, 290, 294, 296; Vitali 5. 247, 289, 293, 294, 296, 298

Theorem, density th. 277, 278, 240, 281, 282,

Similarity transformation 98 Simple (finite) system of intervals 91, 297 Simply-uniformly convergent 142 Singular decomposition of a set 39, 48, 267; a. part of a totally additive set function 38, 43; decomposition of a totally addi-

286, 290; th. of Egoroff 124; Fubini's th. 238; intermediate value th. 51, 5.3; first, second mean value th. 194, 195,

tive set function in its regular and a. part 40, 43; s. sets 36; 4-singular sets

Tile derivate 268, 269, 280. 296; lower, upper t. derivate 269; t. derivative 269; t. system 268, 278, 280, 284, 286, 290, 294,

54; a. set function 40 Smallest 6-system, v-system over 9)2 3 Somewhat continuous above, below 286

Space, n-dimensional Euclidean s. R 8; metric a. 8; metric a. associated with TI by to 32, 55, 57; metric s. of partial systems 31; topological a. 4 Sphere 8; closed s. 8 Strongly content-like 89 Subdivision, finite s. (of I or of t) 91 Subsequences, asymptotically convergent s. 137; ,p convergent s. 132, 134, 139; uniformly convergent a. 136, 137 Subset of a set 1; a, of a a., neglecting a zero-

set 29; proper s. of a s. 4 Subtraction, closed with respect to a. 2 Successor (of a set of a Suslin scheme) 71 Sum of sets 1; Darboux sums 190; lower,

197; Radon-Nikodym th. 170, 171, 255;

Vitali's Covering Th. 265, 267, for intervals 266

296

Topological axioms 4, t. space 4

Totally additive set function 14; decomposition of a t. a. s. f. in its atotaless and atomic part 51, in its continuous and purely discontinuous part 43, in its regular and singular part 40, 43; extension of a t. a. a. f. 34, 74, 76,80; extremes,

maximum, minimum of a t. a. S. f. 16; product of t. a. a. functions 22.3, 228, 233, 234; regular and singular part of a t. a. s. f. 38,43; totally continuous (with respect to 4') 56; totally imperfect set 87 Transformation of integrals 296; bounding t. 23; similarity t. 98 Translation 97 Triangle inequality 8

upper (Darboux) s. 190; Lebesgue a.

180; Riemann a. 183 ,y-Summable function 165 Supremum 8; ,o-supremum 204 Suslin scheme 71; monotone S. scheme 71; S. system of sets 71, 73, 112

Uniformly convergent sequences of point functions 124, 135, of act functions; u. c. subsequences 135, 137; quasi-uniformly convergent sequences 142; simply-uniformly convergent sequences

System(s) of sets 2; Borel a. (over 192) 3; complete e. of sets 34; covering a. (of a set A) 261; 1-system 3; smallest 6. system over 192 3, disjoint systems 2, d. a. for ,y 28; distinguished a. of sets, open in A. 7, indefinitely fine s, 247, distinguished sequence of interval s. 297, convergence of a d. a. of i, a. to a

Upper density 275, 281. 282; u. inner, outer density 275, 281, 282; u. derivate 246, 302; u. w-integral 171, 190; u. limit function 187, neglecting the zero-sets :2X15; u. paving derivate 254; u. semi-continuous function 144; u. (Darboux) suns 190,

set 298, normal system 274, 294; orderly a. 261, 293; ordinary indefinitely fine s.

247; ordinary s. C* associated with ,Q 280, partial s. of sets 17; metric space of partial s. 31; representative of partial s.

31; paving s. 254; regular s. 258, 280; c--system 3; smallest ,r-s. over 9)2 3:

142

u. tile derivate 269; u. Vitali derivate 247, 302

Value(s), intermediate v. 51; intermediate v. theorem 51, 53; first, second mean v. theorem 194; 195, 197 Vitali's Covering Theorem 265, 267, for intervals 266; V. derivates 247, 294, '296,

LNDEX

324

302; V. lower, upper derivates 247, 302;

Zero-rat(s) 27; two sets differ only as to s.-s.

V. derivative 247, 302; V. system (of

30; neglecting s.-s. 28; limit functions, neglecting the s.-a. 296; subset of a set, neglecting a z.-s. 29; s.-s. in the wider sense 33

sets) 247, 289, 293, 294, 296, 298

Young set 87