Introduction to the
Theory of Integration
This is Volume 13 in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: PAULA. SMITHA N D SAMUEL EILENBERG A complete list of titles in this series appears a t the end of this volume
Introduction t o t h e
Theory of Integrat ion T . H . HILDEBRANDT Department of Mathematics University of Michigan Ann Arbor, Michigan
1963
ACADEMIC PRESS
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Preface
This book is an outgrowth of lectures given at the University of Michigan by the author on the subject of integration during a period of more than a quarter of a century. The assumption at the beginning of the course was that the student was familiar with the salient facts of functions of a real variable, that is had a basic knowledge of the topological properties of the real line, continuous functions, functions of bounded variation, derivatives, and Riemann integrals. This same background is assumed here. At no time was it possible to include in the course of lectures all the material included in the book. It represents what we would have liked to discuss if there had been time. The subject matter presented is essentially the classical theory of Stieltjes and Lebesgue integration. In that way it is reactionary relative to the present style of graduate mathematical instruction, which veers strongly toward the abstract and sometimes overlooks the concrete basic ideas. We have tried to include as far as possible the ideas which form the'basis of abstract procedures in the theory of integration, and hope that we have thereby made the latter a little more understandable. Since the book is essentially of a textbook character, we have been rather sparing with references and giving credit for various ideas. The difficulty is that so often one overlooks an important work and disappoints the originators of ideas by neglecting to mention them. Then, too, most of the ideas presented are by this time in the public domain. By way of acknowledgment, we are grateful to Professor R. E. Langer of the United States Army Mathematical Research Center at the University of Wisconsin for the opportunity of spending several fruitful months at the Center. Also, we appreciate having the chance to discuss certain problems with Professor P. Porcelli of Louisiana State University. We cannot overlook the patience of the graduate students who have formed captive audiences while some of the material here presented was under development. We have not burdened anyone with a critical reading of the manuscript. As a consequence all awkwardness of presentation and errors in reasoning are the sole responsibility of the author. T. H. HILDEBRANDT Ann Arbor, Michigan January 1963 V
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PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . .
V
Chapter I
A General Theory of Limits 1. Directed Sets. The Moore-Smith General Limit 2 . Properties of Limits . . . . . . . . . . . . . 3 . Values Approached . . . . . . . . . . . . . 4 . Extreme Limits . . . . . . . . . . . . . . . 5 . Directed Sets of Sequential Character . . . . . 6 . General Sums . . . . . . . . . . . . . . . . 7 . Double and Iterated Limits . . . . . . . . . 8 . Filters . . . . . . . . . . . . . . . . . . . 9 . Linear Spaces . . . . . . . . . . . . . . . .
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1 5 7 9 11 12 13 17 20
1 . Functions of Intervals . . . . . . . . . . . . . . . . . . . . . . . 2 . Integrals of Functions of Intervals . . . . . . . . . . . . . . . . . . 3 . Existence Theorems . . . . . . . . . . . . . . . . . . . . . . . . 4 . The Integral of an Interval Function as a Function of Intervals . . . . 5 . Integral as a Function of the Interval Function . . . . . . . . . . . 6 . Functions of Bounded Variation . . . . . . . . . . . . . . . . . . 7 . Continuity Properties of Functions of Bounded Variation . . . . . . . 8 . The Space of Functions of Bounded Variation . . . . . . . . . . . . 9 . Rieniann-Stieltjes Integrals . . . . . . . . . . . . . . . . . . . . . I0 . Existence Theorems for Riemann-Stieltjes Integrals . . . . . . . . . . 11. Properties of Riemann-Stieltjes Integrals . . . . . . . . . . . . . . . 12 . Classes of Functions Determined by Stieltjes Integrals . . . . . . . . 13 . Riemann-Stieltjes Integrals with Respect to Functions of Bounded Variation . Existence Theorems . . . . . . . . . . . . . . . . . . . . . 14 . Properties of f i g with g of Bounded Variation on (a,b) . . . . . . 15 . Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . . 16 . The Integral as Function of the Upper Limit . . . . . . . . . . . . . . 17 . The Integral as a Function of a Parameter . . . . . . . . . . . . . . . 18 . Linear Continuous Functionals on the Space of Continuous Functions . . 19 . Additional Definitions of Riemann Integrals of Stieltjes Type . . . . . . 20 . Stieltjes Integrals on Infinite Intervals . . . . . . . . . . . . . . . . 21 . A Linear Form on the Infinite Interval . . . . . . . . . . . . . . . .
25 27 28 33 35 37 40 41 47 49 52 56
Chapter II
Riemannian Type of Integration
sf:
58 61 69 76 78 81 86 97 98
Chapter Ill
Integrals of Riemann Type of Functions of Intervals in Two or Higher Dimension Interval Functions . . . . . . . . Subdivisions . . . . . . . . . . . Integrals . . . . . . . . . . . . Functions of Bounded Variation in 5 . Continuity Properties . . . . . . .
I. 2. 3. 4.
101 . . . . . . . . . . . . . . . . . 102 . . . . . . . . . . . . . . . . . 103 . . . . . . . . . . . . . . . . . Two Dimensions . . . . . . . . . 106 109 . . . . . . . . . . . . . . . . .
vii
viii
CONTENTS
The Space of Functions of Bounded Variation in Two Variables: BV2 . . . Functions of Bounded Variation According to Frkhet . . . . . . . . . Riemann-Stieltjes Integrals in Two Variables . . . . . . . . . . . . . Stieltjes Integrals in Two Dimensions with Respect to Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 . Relation between Double and Iterated Integrals . . . . . . . . . . . . 11. Double Integrals with Respect to Functions of Frkhet Bounded Variation 6. 7. 8. 9.
I14 116 123 128 132 136
Chapter IV
Sets 1 . Fundamental Operations . . . . . . . . . . . . . . . . . . . . . . 2 . Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . 3 . Properties of Classes of Sets . . . . . . . . . . . . . . . . . . . . . 4 . Extensions of Classes of Sets Relative to Properties . . . . . . . . . .
.
141 146 146 148
. .
155
Chapter V
Content and Measure 1. Content of a Linear Bounded Set . . . . . . . . . . . . . 2 . Properties of the Content Functions . . . . . . . . . . . 3. Bore1 Measurability . . . . . . . . . . . . . . . . . . 4 . General Lebesgue Measure of Linear Sets . . . . . . . . 5 . Lower Measure . . . . . . . . . . . . . . . . . . . . 6 . Measurability . . . . . . . . . . . . . . . . . . . . . 7 . Examples of Measurable Sets . . . . . . . . . . . . . . 8 . Sets of Zero Measure . . . . . . . . . . . . . . . . . 9 . Additional Properties of Upper and Lower Measure . . . 10. Additional Conditions for Measurability . . . . . . . . . 11. The Function a ( x ) Is Unbounded . . . . . . . . . . . . . 12. Lebesgue Measure . . . . . . . . . . . . . . . . . . . 13 . Relation between a-Measure and Lebesgue Measure . . . . 14. Relations between Classes of a-Measurable Sets . . . . . . 15 . Measurability of Sets in Euclidean Space of Higher Dimension I6 . Some Abstract Measure Theory . . . . . . . . . . . . . . 17 . Upper Semiadditive Functions . . . . . . . . . . . . . . .
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. . 157 . . 162 . . 163 . . 166
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168 171 172 . . 173 . . 173 . . 175 . . 179 . . 182 . . . 183 . . . 188 . . . 191 196 . . .
Chapter VI
Measurable Functions I . Semicontinuous Functions . . . . . . . . . . . . . . . . . . . . . . 2 . Measurable Functions . . . . . . . . . . . . . . . . . . . . . . .
3 . Properties of Measurable Functions . . . . . . . . . . . . 4 . Examples of Measurable Functions . . . . . . . . . . . . . 5 . Properties of Measurable Functions Depending on the Measure 6 . Approximations to Measurable Functions. Lusin’s Theorem . .
199 203 . . . . . 205 . . . . . 207 Functions 209 . . . . . 215
Chapter VII
Lebesgue-Stieltjes Integration 1. The Lebesgue Postulates on Integration
. . . . . . . . . . . . . . . 219
2 . The Lebesgue Method of Defining an Integral . . . . . . . . . . . . . 3 . A Riemann-Young Type of Integral Definition . The Y-Integral . . . . .
224 231
ix
CONTENTS 4 . Properties of the Integrals
. . . . . . . . . . . . . . . . . . . . . 239 . . . . . . . . . . . . . 239
5 . The Integral as a Function of Measurable Sets
6. 7. 8. 9.
L-S Integrals as Functions of Intervals . . . . . . . . . . . . . . . . Absolute Continuity . . . . . . . . . . . . . . . . . . . . . . . . Properties of the Integral Z(f, a, E ) = sEfda as a Function o f f . . . . Directed Limits and L-S Integration . . . . . . . . . . . . . . . . 10. Properties of the Integral Z(f, a, E) = s E f d a as a Function of a . . . I 1 . Integration with Respect to Functions of Bounded Variation . . . . . I2 . Integration with Respect to Unbounded Measure Functions . . . . . .
241 242 244 255 258 261 274
Chapter Vlll
Classes of Measurable and Integrable Functions 1 . The Class of a-Measurable Functions . . . . . . . . . . . . . . . . 2 . The Class of Almost Bounded a-Measurable Functions . . . . . . . . 3 . The Space L , . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . The Space L p with 0 < p < w . . . . . . . . . . . . . . . . . . . 5 . Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . The Space L2. Orthogonal Functions . Riesz-Fischer Theorem . . . . . .
285 288 290 291 296 299
Chapter IX
Other Methods of Defining the Class of Lebesgue Integrable Functions Abstract Integrals
.
I . The Space L1 as the Completion of a Metric Space by Cauchy Sequences 2 . Construction of the Space L1 by the Use of Osgood’s Theorem . . . . . 3. L-S Integration Based on Monotonic Sequences of Semicontinuous Functions 4 . Lebesgue-Stieltjes Integrals on an Abstract Set . . . . . . . . . . . . Chapter X
.
310 312 317 320
.
Product Measures Iterated Integrals Fubini Theorem 1. Product Measures . . . . . . . . . . . . . . . . . . . . . . . . . 2 . The Lebesgue-Stieltjes Integrals as the Measure of a Plane Set . . . . . 3 . Fubini Theorem on Double and Iterated Integrals . . . . . . . . . .
. .
327 333 335
I . Riemann Integrals and Derivatives . . . . . . . . . . . . . . . . . 2 . On Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Vitali Theorem . . . . . . . . . . . . . . . . . . . . . . . . 4 . Derivatives of Monotonic Functions and of Functions of Bounded Variation 5 . Derivatives of Indefinite Lebesgue Integrals . . . . . . . . . . . . . . 6 . Lebesgue Integrals of Derivatives . . . . . . . . . . . . . . . . . . . 7. Lebesgue Decomposition of Functions of Bounded Variation . . . . . . 8 . The Radon-Nikodym Theorem . . . . . . . . . . . . . . . . . . .
343 345 354 357 361 365 369 375
. . . . . . . . . . . . . . . . .
379
Chapter XI
Derivatives and Integrals
SOME
REFERENCE BOOKSON INTEGRATION
SUBJECT INDEX
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
381
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CHAPTER I
A GENERAL THEORY OF LIMITS
The theory of integration depends on limits of a variety of types. To avoid deriving properties of each of these limits separately, we present a general theory of limits as developed by E. H. Moore and H. L. Smith which includes as special cases the limits used in this book. 1. Directed Sets. The Moore-Smith General Limit
To set up a general theory of limits, we examine some of the limits which occur in analysis and note their common features. The simplest limit is that of a sequence {a,,}of real numbers and is defined as follows : the sequence { a , / }has a as a limit written lim,p, = a, if for every positive e, there exists an integer 1 2 , depending on e, such that if n > n,, then I a,, - a 1 < e, or a - e < a,, < a e. If we introduce the Peano symbolism where 3 = implies, 3 = there exists, 3 = such that, then this limit definition can be written
+
Next we have the limit of a function f(x) defined on an interval a 5 x 5 b, at a point xo of [a, b], viz., liml+2u f(x) = c if and only if for every e > 0, there exists a d , such that if 0 < I x - x,,I < d,, then I f ( x ) - c I < e , or
A slightly different type of limit occurs in connection with the Riemann integral. Iff(x) is defined on the interval a 5 x 5 b, 0 stands for the subdivision of [a, b] by the points a = xo < x,< x, < ... < x, = b and x L - lS x L ’ Sxl,then JR.f(x)dx is defined as the limit of the approximating sums L’:’=,f(x,’) (x,- x L p J as the maximum 1
2
1. A GENERAL THEORY
OF LIMITS
of the lengths of the subintervals [ x , xi-J or I xi- xiel I approaches zero. If we denote the maximum of I x i - x,, I for i = 1, ...n by 1 a 1, the norm of a, then
or
This differs from the preceding cases in that Z?=,f(x,') ( x i - xi-J considered as a function of a is a many-valued function. A fourth type of limit occurs in the definition of total variation of a function of bounded variation on an interval [a, b]. If f ( x ) is of bounded variation on [a, b], then the total variation V:f is defined as the least upper bound (1.u.b.) of ZC,. I f ( x J - f ( x i - J I for all subdivisions a of [a,b]. However, the function of a: g(a) =Zg-l I f ( x i ) f(x,,) I has a monotonic character relative to a, in the sense that if we add additional points to a, then g(u) does not diminish. Consequently, if we mean by a,> c2 that the subsdivision a1 contains all of the points of az, then we have
A similar situation occurs in connection with the upper and lower Darboux integrals r f ( x ) d x and -fa f ( x ) d x of a bounded function on [a, 4. Let us now examine these instances of definition of limits in order to deduce their common characteristics. We note in the first place that the notion of limit applies only to functions. A sequence is a function of the integers, the approximating sums in integration and variation are functions of the subdivision of the fundamental interval. Secondly, some sort of order is involved in each of these definitions, for sequences, the order of the integers, for functions of x , the order established by the distance I x - xo 1, for Riemann integration the order d e h e d by the norm I a 1, and in total variation the order a12 a2 defined by inclusion. In order to bring the second and third cases in line with the other two, we rephrase them as follows: (I
1.
-
DIRECTED SETS. MOORE-SMITH GENERAL LIMIT
3
lim f ( x ) = c x+xo
e> 0 3 3 x, 3 0 < I x
- x,,
I < I x , - xoI 3 I f ( x )
- c I <e
and
These considerations suggest the following generalization : Let Q be a general class of elements q and f ( q ) a real valued function on Q. In Q assume a relation R holding between some pairs of elements q1and qgof Q,i.e., for some q1and q2,we have qlRq2.For R we postulate the minimum condition associated with order, viz., R has the transitive property: if q,Rq, and q2Rq3,then qlRq,. We then define the general limit: lim, f ( q ) = a, if and only if for every e > 0 there exists a q, such that if qRq,, then I f ( q ) - a I < e, or lim, f ( q ) = a = e > 0 3 3 q, 3 qRq, 3 I f ( q )
-a
I < e.
It is fairly obvious how each of the instances of limits we have considered follows this pattern. Also we note that the same set of elements may be ordered in different ways, giving rise to different definitions of limit. Since in the sequel we shall have frequent occasion to use the statement involved in the definition of limit, we shall abbreviate it: (7' 4 e 7 qRqJ : I f ( q ) - a I < In any general theory of limits it is desirable to preserve as far as possible the basic properties valid in the simpler instances. Among these one of the most important facts is that if lim, f ( q ) exists, then this limit is unique. This means that if lim, f ( q ) = a and lim, f ( q ) = b, then a = b. Now if lim, f ( q ) = a, then (e, q,', qRq,') : I f ( q ) - Q I < e, and if lim, f ( q ) = b, then (e, q i , qRq,N) : I f ( q ) - b I < e. In order to draw any conclusions from these two statements, we need to know that for every e > 0 there exists at least one qo, such that q,,,Rqe' and q,,Rq," simultaneously. Then it follows that for every e > 0, a - b I < 2e, i.e., a = b. This suggests the following additional postulate on R, which we call the compositive property: for any two q1 and q2 there exists a q3 such that q3Rql and q3Rqg,which might be stated, any two elements of Q have a successor.
I
4
I. A GENERAL THEORY OF LIMITS
We have then proved that: if R has the transitive and compositive property and if lim, f ( q ) exists, then this limit is unique. It turns out that the transitive and compositive properties of the relation R are sufficient for setting up a theory of limits which contains the elementary cases as special instances. As a consequence we apply a special label to general classes Q with such an R and define: 1.1. Definition. A general set Q with a binary relation R holding between some pairs of elements of Q is called a directed set if the relation
R is transitive and compositive. As additional instances of directed sets we note the following: (1) Let P be any set of elements and let Q consist of all finite subsets of P, and let the elements of Q be directed by inclusion, i.e., if q1 and q2are any two sets in Q, then q1Rq2if and only if q1 contains all of the elements of q2.
(2) Let P be any set of elements and let Q consist of all finite or denumerable subsets of P, directed by inclusion. (3) Let P be any set of elements and Q be a collection of subsets of P which with any two sets q1 and q2 contains also the union or sum of the two sets, direction being by inclusion. Other instances will occur later in the book. It is possible to add 00 and - co as possible limits in the usual way, i.e., lim, f(q) = co is equivalent to the statement, for every e > 0 there exists a q , such that if qRq,, thenf(q) > e ; and lim,f(q) = -a if for every e > 0 there exists a q, such that if qRq,, then f(4) - e. In case we allow co and - co as values for the set of real numbers 91, we shall call the result the extended line denoted 92". It is apparent that in the definition of limit we can replace e by ke where k is fixed relative to e and positive, since ke covers the same ground as e. As a consequence the definition for finite limit could also read (ke, q,, 4RqJ : I f ( 9 ) - a I < ke. If Q is a directed set, if f ( q ) is on Q to the reals, and if the general limit lim,f(q) exists, finite or infinite, this limit is called a MooreSmith or directed limit.
+ +
+
2.
5
PROPERTIES OF LIMITS
2. Properties of Limits
We proceed to show that the properties of limits of sequences are essentially preserved in the more general setting. For the sake of convenience, we shall assume that the limits involved in this section are finite. 2.1. If lim,f(q)
= a,
2.2. If lim,f(q)
= a,
and b is any constant, then lim, bf(q) = ba. For (e, q,, qRq,) : I f ( q ) - a I < e implies I bf(q) - ba I < I b I e.
For IIf(4)
I
-
then lim, I f ( q )
I a II 5 Ifh)
2.4. If lim,f(q)
= a,
2.5. If lim,f(q)
=
-
a
I = I a 1,
I.
then the set of values of.f(q) is ultimately bounded, i.e., there exists a q, and a b > 0 such that for qRq, we have Ifh)I
lim,f(q)
*
a, and limqg(q)
=
b, then lim, f(q)g(q)
=
ab
=
lim,g(q).
This follows from the inequality
IfW - ab I S Ifh)II g(q) and the ultimate boundedness of f ( q ) . 2.6. If lim,f(q)
=a
-
bI
+ I b IIf(4)
and afO, then lim,( 1 /f(q))
=
1/ a
=
-
a
I
1 /lim,f(q).
I a1/2, then If(q) I > 1 a1/2 for qRq,, so l/a I = IfW a 111 a IIf(4) I < 2 Ifh) a 111 a IP.
For if a f O and if e < that I I l f ( q )
-
-
-
A combination of 1.2.5 and 1.2.6 yields the usual theorem on the limit of a quotient of two functions. All of these theorems are special cases of the general theorem:
6
I. A GENERAL THEORY OF LIMITS
2.7. If F(x,, x2,..., xJ is defined on the value space of fl(q), f 2 ( q ) ,...,
f,(q) and continuous at (al, ..., a,) where lim,fl(q) a2,..., lim,f,(q) = a,, then
= a,,
lim,fi(q)
=
F(f1h),f2(4),...,fn(4)) = F h , , a2”” a,) = F(1imqf1( 4 ) , lim,f, ( 4 ) ,..., lim,f, ( 4 ) ) .
lim,
For a directed set the notion of subsequence is replaced by that of cojnality with Q. We define:
Q’of Q is coJnaZ with Q if for every q of Q,there exists a q‘ in Q’such that q‘Rq. A set Q’cofinal with Q is also a directed set with the same R, as can easily be verified. 2.8. Definition. A subset
Q’is cofinal with Q,then lim,,f(q’) = a. This is a generalization of the statement that if a sequence a, converges, then every subsequence converges to the same limit. The proof is obvious. There is a converse:
2.9. If lim,f(q) = a and
Q‘ cofinal with Q,there exists a subset Q” cofinal with Q’ and therefore with Q, such that limq,, f ( q ” ) = a, then limf(q) = a. For suppose that f ( q ) does not have a as a limit. Then there exists an e > 0 such that for every q there exists a q, with q,Rq such that I f(q,) - a I > e. Now the set Q’consisting of q , is cofinal with Q, and so there exists a subset Q” cofinal with Q’and so with Q such that limq,, f(q”) = a, which obviously contradicts the inequality I f(q”) - a I > e for every q”.
2.10. If f ( q ) and a are such that for every
2.11. A necessary and sufficient condition that f ( q ) have a finite
limit is that the Cauchy condition of convergence hold, viz., for e > 0 there exists q , such that qlRq,, q2Rq, implies f ( q l ) - f ( q J 1 < e. The necessity follows in the usual way. For the sufficiency take e = l / n (or any monotonic sequence en+ 0). Select q,, so that qnRqn-l and so that q‘Rq, and q“Rq, implies -f(q”) I < l/n. Then from the transitive property of R it follows that I f(q,+J - f(q,,+,) 1 < l / n for all n and m, so that f(q,) is a Cauchy sequence of real numbers and consequently lim,f(q,,) exists. If this limit is a, then (e, n,, n > n,) : 1 f(q,) - a 1 < e. Further, I f(q,,) - f ( q ) I < e if n - 1 > l/e and qRq,-,. If we set q , = q,-, for n greater than n, and 1 l/e, then (e, q,, qRq,) : I f ( q ) - a I < 2 e, i.e., lim, f ( q ) = a.
I
)@(fI
+
.
3.
7
VALUES APPROACHED
In these properties of limits, no mention has been made of the possible multiple valuedness of the functions involved. An examination of the proofs will show that the properties hold even if the functions are multiple valued, provided it is understood that if f ( q ) belongs to the class C and g(q) belongs to the class D , then f ( q ) g(q) belongs to a subset of the sum of any element in C with any in D. Similarly for f ( 4 ) g ( d .
+
2.12. Definition. We define a function f ( q ) on a directed set Q
monotone nondecreasing (nonincreasing) if f (q) is single valued and q,&, implies f h 1 ) 2 .fhJ ( f ( 4 , ) 2 f(42))‘ As in the case of monotonic sequences we have: 2.13. T H E O R E M . If .f(q) is monotonic nondecreasing and bounded above, then lim,f(q) exists as the least upper bound (1.u.b.) of f ( q ) in q. Otherwise, lim,f(q) = co. A similar statement holds for nonincreasing functions. For if a = l.u.b.,f(q), then for e > 0 there exists q, such that a - e
+
3. Values Approached
Obviously lim, f(q) for a function on a directed set Q need not exist. However, as we shall see, for any function on a directed set there exists at least one number a on the extended real axis - co 5 a 5 co, which is a value approached (or quasi-limit), which notion we define as follows: 3.1. Definition. a is a value approached of f(q) on the directed set Q if for every e > 0 and q, there exists a q,, such that q,,Rq and
+
I f(q,,)
- a I < e, if a is finite, f(4,J > e if a = co, and f(q,,) < if a = - co. Or ( e , q ; q e q ) : qegRq,1 f(q,,) - a I < e for a finite, f(q,,) > e for a = co and f(q,,) < - e for a = - co. The notion of value approached is a close relative of the inaccurate definition of limit of a sequence a,, which states that a is a limit of a,, if a, can be made to differ from a by as little as one pleases by making n large enough. The set of values approached (abbreviated v.ap.) of a function f ( q ) on a directed set Q can be obtained as follows: Let E, be the set of values .f(q’) for q’Rq, and F, the closure of E, obtained by adding to E, all of its limiting points. Obviously the sets F, are monotonic - e
+
8
1. A GENERAL THEORY OF LIMITS
nonincreasing in q in the sense that q,Rq, implies F,, is contained in Fp,. Then: 3.2. T H E O R E M . The set F, = II,F,, the set of numbers contained in all of the F,, is the set of values approached off(q) and is not vacuous. For suppose a is a v.ap. and suppose, if possible, that a is not in F,. Then there exists a qo such that F,, does not contain a. Since F,, is closed there exists a vicinity V ( a ) containing no point of F,,. Since the sets F, are monotone nonincreasing in q, V ( a ) will have no point in common with F, for qRq,. This contradicts the definition of v.ap. On the other hand, suppose that a is in F,. Then for every q there exists a q, = q' with q'Rg such that either a belongs to Eg,or is a limiting point of E,. . In either case, (e, q, qe,J : q,,Rq, 1 .f(qe,) - a I < e, for a finite and similarly for a = & co, i.e., a is a v.ap. of f ( q ) . To show that F, is not vacuous, we proceed contrapositively, or suppose that F, is empty. If a is any value - co 5 a 2 00, then as above there exists a vicinity V ( a ) and a q, such that qRq, implies that F, and V ( a ) have no points in common. The Bore1 theorem holds on - co 5 a 2 GO and consequently there exist a finite number of points al,...,aTLand corresponding q ...,q, such that L':=l V(aJ 1: covers - co 5 a I co. Consequently, if q,Rql, ... qoRql,,which qo exists by the compositive property of R, then for qRq, the sets F, are vacuous, which is contrary to their definition. Hence F, is not vacuous. Fo being the product or common part of closed sets is closed. As an alternative way of obtaining v.ap. by f(q) we have:
+
+
+
3.3. T H E 0 R E M . a is a v.ap. off(q) on the directed set Q if and only
if for every q there exists a monotone increasing sequence of q, such that q,,Rq and limnf(q,) = a. For the necessity we need only apply the definition of v.ap. with e = l/n, n = 1, 2, ..., q,Rq, and q,,Rq,-,. For the sufficiency we note that under the conditions of the theorem a belongs to all F, and so is a v.ap. In the case of sequences of real numbers a,, any v.ap. is also the limit of a subsequence of a,. If we define a as a sublimit of f ( q ) if there exists a subset Q' cofinal with Q such that lim,, f(q') = a, then we see at once that any sublimit is also a v.ap. But we do not know whether conversely every v.ap. is also a sublimit.
4.
9
EXTREME LIMITS
4. Extreme Limits
Since the class F, of v.ap. for a functionf(q) is closed, it has both a maximum and a minimum on - m 5 a 5 co. These two special v.ap. are designated the greatest of the limits of f ( q ) (K,f ( q ) ) or limit superior (lim sup) and least of the limits (ljm,f(q)) or limit inferior (lim inf), respectively. Since the second designations are frequently referred to as lim sups and lim infs, which have a rather nonmathematical suggestion, we shall usually speak of the greatest and least of the limits. From the extremal characterization of these notions we find at once:
+
4.1. T H E O R E M . If f ( q ) is defined on the directed set
Q, then
= g.l.b.[l.u.b.(f(q')
for q'Rq)
for q in Q]
M , f ( q ) = l.u.b.[g.l.b.(f(q')
for q'Rq)
for q in Q].
G,,f(q)
and Here g.1.b. is the abbreviation for greatest ower bound and 1.u.b. for least upper bound. If we set g(q) = l.u.b.[,f(q') for q'Rq], then g(q) is the maximum of the set F, and is monotone nonincreasing in q, so that E , f ( q ) = lim,g(q). From this we obtain the following alternative definition of lim,f(g), if it is finite: 4.2. K , f ( q ) = a if for every e > 0 there exists a q, such that for qRq, we have .f(q) < a e, and for every e and q there exists a q,, such that q,,Rq and .f(q,,) > a - e. Or (e, q,, qRq,) :f ( q ) < a e and (e, 4, q e g ) : q,gRq, . f f q p q ) > a - e* This definition combines one inequality from the definition of limit with the reverse inequality from the statement for v. ap. For if lim,ffq) = a, then lim,g(q) = a, so that (e, q,, qRq,) :f ( q ) 5 g(q)< a e. On the other hand, since g ( q ) 2 &,f ( q ) = a, for all q, and g ( q ) is the maximum of the set F,, then for any e and q there exists q,,Rq such that f ( q , , ) 2 g ( q ) - e > a - e. Conversely, if (e, q,, qRg,) :f(q) < a e, then for qRq, we have g(q) 5 a e ; and if (e, q, q eg-) : q,,Rq, f ( q ) > a - e, then for all q we have g(q) 2 a - e. Hence lim,f(q) = lim,g(q) = a. Similar considerations show that if E,f ( q ) = co, only the = a3 if and only if for second statement is needed, i.e., l=,f(q) every e > 0 and q there exists a q,, such that q,,Rq and f(q,,) > e .
+
+
+
+
+
+
+
10
I. A GENERAL THEORY OF LIMITS
The other condition is automatically fulfilled. For Z , , f ( q ) = - co, only the first typeof condition is needed, i.e., (e, q,, qRq,), f ( q ) <- e. It follows that if lim, f ( q ) = - co, then lim, f ( q ) = - co. The modifications in the above considerations needed for lim -P f(q) are obvious, or can be deduced from the observation that: 4.3. &,f(q)
lim, (-
f(q)). From the various definitions of L , , f ( q ) and lim, f ( q ) , when compared with those for lim,.f(q), it follows at on= =-
4.4. A necessary and sufficient condition that lim,,f(q) exist is that
-
lim,f(q) = @,f(q). We note the following properties of the extreme limits: 4.5. The statement lZ,f(q) 5 a, for a finite is equivalent to (e, q,, qRqJ : f ( 4 ) < a e* For if l z , f ( q ) = b and b 5 a, then (e, q,, qRq,) : f ( q ) < b e 5
+
+ a + e. On the other hand, if &,f(q) = b, then (e, q, q e J : q,,Rq, f ( q ) > b - e. This combined with (e, q,, q'Rq,): f ( q ) < a + e for q' = qeq= q gives b - e < a + e for all e > 0, so that b 5 a. eqe
4.6. If f ( q ) and g ( q ) are any two functions on the directed set Q,
then lim,f(q)
+ lim,g(q) 5 l&,(f(q) + d q ) ) 5 G , f ( q ) + lim,g(q) 5 h,(f(q) +g ( d ) 5 L , . N q ) + G , g ( q ) .
We demonstrate this set of inequalities for the case when all of the limits are finite. If G,f ( q ) = a and K , g ( q ) = b, then (e, q,, qRq,): f ( q ) < a e and (e, q,', qRq,'): g ( q ) < b e. It follows that (e, q,", qRq,") :f ( q ) g ( q ) < a b 2e, where q,"Rq, and q,"Rq,'. Consequently, by 4.5 &,(f (q) g ( q ) ) 5 a b = E , f ( q ) -f i , g ( q ) . From this inequality, we obtain G , ( f ( q ) g(q)) lim,f(q) 5 K , g ( q ) . On setting f ( q ) g ( q ) = h(q) and k ( q ) = - f ( q ) , so that g ( q ) = h(q) + k(q), and using the relation be-lim P k(q) 5 E , ( h ( q ) k(q)). tween L a n d lim, we get G , h ( q ) The other twozequalities in the series follow from those already derived by using the relation fi, f ( q ) = - limg(- f ( q ) ) . The verification that these inequalities s t r h o l d if co and - co are permitted as values for the limits, provided the ambiguous expression co - 03 does not occur is left as an exercise.
+
+ + + +
+
+
+
+
+
+
+
+
+
5.
DIRECTED SETS OF SEQUENTIAL CHARACTER
II
exists, i.e., lim,f(q) = &,,f(q), then as a result of these inequalities lim and lim are distributive relative to addition, for example, E, ( f ( q ) + g ( q ) ) = lim,f(q) G,g(q). 4.7. We note that if lim,f(q)
+
5. Directed Sets of Sequential Character
A directed set Q is said to be of sequential character if there exists a sequence (4,) of elements of Q cofinal with Q. The directed sets of sequential character are important because limit of a function on such sets is essentially a sequential limit. Of the instances of directed sets considered, those based on a metric or norm, i.e., where there exists a positive valued I q I such that qlRq, is equivalent to I q1 I 5 I q2 1, are of sequential character, any sequence {q?!}such that I q, I + 0 monotonically will serve as the sequence of the definition. Special instances are limJ+zof ( x ) and the definition of Riemann integrals. However, if Q consists of all finite subsets of a general set P directed by inclusion, then usually Q will not be of sequential character. The important property of limits on directed sets Q of sequential character is contained in the following theorem : 5.1. THEOREM. If Q is a directed set of sequential character, and f ( q ) is on Q to real numbers, then lim,f(q) = a if and only if for every monotone sequence (4,) cofinal with Q we have lirn,,f(qn) = a. The sequence q, is monotone if for every n: q,lRq,-l. If (q,] is any sequence cofinal with Q,then there exist monotone sequences cofinal with Q, the sequence {q?,'},which is determined serially by the conditions q,,'Rq,, and qlL'Rqll'-l, being such a sequence. That the condition stated is necessary follows from the definition of lim,f(q) = a. Observe, however, that the condition that the sequence (4,) be monotone is essential here. For the sufficiency, suppose if possible that for every monotone sequence (4,) cofinal with Q we have limJ(q,) = a, but that lim,f(q) # a. Then there exists e > 0, such that for every q, there is a q, with qqRq and I f(q,) - a I > e. Let (4,) be a sequence cofinal with Q.We can then determine for each n : q,,' such that q,'Rq,, and qn'Rqr,'-l,such that I f(q,,') - a I > e, since there exists a q for which qRq, and qRqnrM1 and a q, with qqRq such that [ f(q,) - a I > e. Then q,,' is monotone and cofinal with Q, but lim,f(q,') # a, leading to a contradiction.
12
I. A GENERAL THEORY OF LIMITS
We have assumed that a is finite. The case when a similarly.
=
f co follows
6. General Sums
As a very useful application of the theory of limits on directed sets, we define the notion of Z pf ( p ) , for a function f on a general set P to real numbers. We note that the sum of an infinite series ZJZ, is defined as the limit of s,, = Z:=la,, a finite number of terms, when this limit exists. Similarly for Zpf ( p ) , we denote by q any finite number of elements: p l , ... p , of P and consider g ( q ) = CP f ( p ) = 2:=1f ( p , ) . Now the class of finite subsets of P is a directed set ordered by inclusion. Hence we define:
,
Zpf ( p ) = lim, Zqf ( p ) when this limit exists as a finite number. We then have the following theorem:
6.1. Definition. The sum
6.2.
= lim,
q=, f(p,),
If f ( p ) is a real valued function on P, then Z pf ( p ) exists if and
only if f ( p ) vanishes except for a denumerable set of elements: pl, ...,p,,, ... of P and ZnI f ( p , ) I is convergent. Then Cnf ( p , ) is absolutely convergent and is the value of C pf ( p ) . If lim, Zq f ( p ) exists as a finite number, then the Cauchy condition of convergence applies and we have (e, q,, q1 1q,, q2 2 q J : I Zqlf ( p ) - Zq2f ( p ) I < e. If we take e = l/n, qg = q , and q1 = (q,, p ) , where p is any element of P not in q,, then 1 f ( p ) 1 < l/n. Then the set of points of P for which I f ( p ) I 2 l / n belong to q , with e = l/n and is finite. Hence the set of points for which f ( p ) f 0 is denumerable. If we use the same inequality, set q1 = (q2,pl', ... pk') where p l ' , ... pk' are any points not in q1 for which f ( p ) > 0, then C2f ( p , ' ) < e. It follows that Zn.f ( p ' ) for p' ranging over the points for which f ( p ' ) > 0 is convergent. Similarly one shows that Splf ( p " ) for p" the points for which f ( p " ) < 0 is also convergent. Then if p l , ..., p , , ... are the elements of P for which f ( p , ) f 0, we have L,, I f ( p , ) I convergent, so that L,, f ( p , , ) is absolutely convergent. Because of the convergence of CllI f ( p , , ) I , we have (e, n,, m > n,) : 22 I f ( p , ) I < e. If we take q , = (pl,...,p,, ) and q 2 q,, then
This says that lim, Zqf ( p ) = ZrLf ( p , ) .
7.
13
DOUBLE A N D ITERATED LIMITS
We note that we have here a means of defining absolute convergence of an infinite series ZajLwithout recourse to absolute values, via the property that lim, Cqalrwith q = (n,,,..,nJ exists. The theorem is immediately extensible to the case when f(p) is complex valued. In the case where ,f(p) has values in a more general space with linear properties, the existence of lim, .Y, f(p) defines unconditional convergence rather than absolute convergence. 7. Double and Iterated Limits
If we have two directed sets Q , with relation R, and Qe with relation R, (Q, and Q, may be the same sets with the same or different relations), we can- make the product set Q , x Q, into a directed set by assuming that ql'q2'Rq,''q.,'' is equivalent to q1'R,qlrrand q2'R2q2'', as a check on the postulaies for directed sets shows at once. [If Q, are any collection of directed sets with relations R,,, then the product set Q = IIcLQ,lis a directed set if q'Rq" is equivalent to q,'R,Iq,'' for all a . ] We shall limit ourselves to two directed sets P and Q each with an order relation R of its own. For a function ,f(p, q ) defined on the product set P x Q, we can then define the double limit: 7.1. Definition.
me)I
I
: .f(P, 4 ) - a < e, limp,f(p, 4 ) = a = (e7 P,, q,, PRP,, and similarly if a = co. In addition there are also the iterated limits: limJim,.f(p, q) and lim,lim,f(p, q ) , where we assume that all the limits indicated exist. Of constant occurrence in analysis is the relationship between the double limit and the iterated limits, and the interchange of order of of limits in the iterated limits. We have the following: 7.2. THEOREM. If limJ,qf(p, q ) exists as a finite number and
lim,f(p, q) exists for pRp,,, then limJim,f(p, q) exists and is equal to lim,,,.f(p, 4). For suppose lim,f(p, q ) = g(p) for pRp,. Then (e7 PRP,, 4,,, clRq,,): If(P, 4 ) - g(P) < eIf limI,,f(p, q ) = a, then (e, P,, q,, PRP,, 4RqJ: .f(P, 4 ) - a < e. Consequently, (e, per,pRp,') : I g(p) - a < 2e, where p,'Rp, and p,'Rp,. For, for any pRp,', we can select a q such that qRq,, and qRqe* As an immediate consequence we have :
I
1
I
I
14
I. A GENERAL THEORY OF LIMITS
7.3. Corollary. If limp, f ( p , q) exists as a finite number, lim, f ( p , q) exists for pRp, and limpf ( p , q ) exists for qRq,, then limJim, f ( p , q)
and lim$impf ( p , q) both exist and are equal to each other and to lirnpQf(p?4). An easily remembered and useful condition for the interchange of order of limits in iterated limits is embodied in what we call: 7.4. The Iterated Limits Theorem. If limpf ( p , q) exists $nitely for all q, and lim, f ( p , q) exists-finitelyfor all p and uniformly on P, then limJim, f (p, q), limJimp f (p, q ) , and limp,f (p,q) all exist and are equal. Let limpf ( p , q) = g(q) and lim, f ( p , q ) = h(p). Then since lim, f ( p , q) exists uniformly on P, the corresponding Cauchy con-
dition of convergence also holds uniformly on P, i.e., (e, q e , qfRqe, qflRq,, P ) ; If(P, 9') - f ( P , 4") I < e* On taking limits as to p , we have ('9 q e , q'Rqe, q''RqJ : I g(q') - glq") I < e, so that lim,g(q) or limJim, f ( p , q) exists. Call this limit a. By combining ('9 q e ' y qRq,'> P ) : I f ( P , 4) (e, 4, Peq, PRPeQ) : I f ( P , 4) 3'(
qe''>q'qe'')
I <e - g(P) I < e
- '(P)
: I g(q) - a I
(1)
(2)
<e
(3)
we get (e*Pe,pRPJ : \ h ( p ) - a l < ' e
(4)
provided we take p , = p,,, with q any such that qRq,' and qRq,". Then limph(p) = a, or hmJim, f ( p , q) = limJimp f ( p , q), so that both iterated limits exist and are equal. By combining statements (1) and (4), we get (e, P,,
PRPe, qRqe') : I f ( P , 4)
-a
I
< 4e
(5)
so that the double limit limp, f ( p , q) also exists and is equal to the iterated limits. This completes the proof of the theorem. We might note that by combining statements (3) and (5), we get (e, P,, q,, PRP,,
I f(PY
4) - g(q)
I <5 e
(6)
where q,Rq,' and q,Rq,". This statement is a kind of ultimate uniformity as to q and is equivalent to lim,,(f(p, q ) - g ( q ) ) = 0. In
7.
DOUBLE A N D ITERATED LIMITS
15
the case where Q is the set of positive integers N in their natural order, we can deduce actual uniformity as to N = Q, i.e., we have: 7.5. Corollary. If f(p, n) =f,,(p) is on P x N , where P is a directed
set, lim,f,(p) exists for each 17, lim,,f,(p) exists uniformly in p, then 1imT,limJf(p), limJimr, f , ( p ) and lim,,J,(p) all exist and are equal. Moreover, lirn,f,(p) is uniform as to n. For statement (6) in this setting reads (e, n,, p,, n > n,, pRp,) : I f,(p) - g , I < 5 e. Now there are only a finite number of n's less than or equal to n, for any given e and for each of these n's there exists a pen,such that pRp,,, implies If,,(p) - g,, I < 5 e. If we take p,' so that p,'Rp,, p,'Rp,,,, n = l,..., n,, then we have (e, p,', pRp,', n) : I f,,(p) - g , ) 1 < 5 e, which means that f,,(p) converges uniformly to g , in n. A careful examination of the proof of the iterated limits theorem shows that the uniformity of one of the inner limits is stronger than necessary and that a condition of the form of statement (6) will carry the argument. As a matter of fact, we have: 7.6 THEOREM. If limpf(p, q )
for each q, lim, f(p, q) = h(p) for each p and if lim,,(f(p, q) - h(p)) = 0, then limplimgf ( p , q ) , 1imJimpf ( p , q), and lim,,f(p, q) all exist and are equal. For from lim,,(f(p, q) - h(p)) = 0 it follows that (e, p,, q,, - h(P) I PRP,, q'R4,, q"R9,) : I f ( P , 4 ' ) - h(P) I < e, and I f(P, < e. Then I f(p, q') - f(p, 4") I < 2 e. If we take limits as to p, we deduce that 1 g(q') - g(4") I < 2 e for q'Rq,, q"Rq,. Then lim,g(q) exists. The remainder of the proof requires only slight alterations in statements (1) and (4) in the proof of the iterated limits theorem. Many theorems of analysis are consequences of the iterated limits theorem. For instance: "if f , , ( x ) on [a, b] are continuous at x = x,, and converge to f ( x ) uniformly in some vicinity of x,, then f ( x ) is continuous at x,,"is such a theorem. We order the integers n in their natural order, and the x by the condition X'RX''= I x' - x, I < I x' - xoI . By Corollary 1.7.5 we can conclude, moreover, that the measure of continuity d , , such that I f , , ( x ) - f,(x,) I < e for e > 0, 1 x - x,,I < d,,, is independent of n, i.e., the functions f , ( x ) are equally continuous at x,,.This suggests also the conjugate theorem: '' if the functions f , ( x ) converge to f ( x ) for 0 < I x - x, 1 < d, and f , , ( x ) are equally continuous at x,, then limx+rof ( x ) = lim,Lf,(x,)." If we further assume convergence and equal continuity at all points of the closed interval [a, b] and then apply statement (6) =g(q)
16
I. A GENERAL THEORY OF LIMITS
above with P = N , Q = 0 < I x - xo I < d for each xo, we obtain (e, nex0,d,,”, n > nez,, 0 I I x - xo I < deXo): I fJx) - f ( x ) I < e, x being permitted to be equal to xo because of the convergence at xo.Applying the Bore1 theorem to the intervals 0 5 I x - xo I < dex0 yields a finite number of points xl,...,xTn,a finite number of intervals 0 5 I x - x i I < d,, L covering [a, b], a finite number of n’s: neX1,..., ‘ex,,,? of which we can chose the largest n,. As a consequence for n > n , and a I xI b, we have I f,(x) -f(x) I < e, giving us the well known : 7.7 THEOREM. If f , , ( x ) is a set of equally continuous functions on [a, b] converging to f ( x ) at all points of [a, b],then the convergence
of f,, to f is uniform on [a, b] and f(x) is continuous. The conditions of uniformity in the hypothesis of the iterated limits theorem, while sufficient are not necessary for the existence of the iterated limits and the double limit and their equality. A situation which under certain conditions makes the uniformity condition necessary is covered by the following: 7.8. THEOREM. If
P and Q are directed sets, if f ( p , q) on P x Q
is monotonic nondecreasing in q for each p , and if limJim, f ( p , q ) = limglimpf ( p , q) where the limits exist as finite numbers, then limpqf ( p , q) exists and is equal to the iterated limits. In particular, if either P or Q,Q say, is the set of positive integers N in their natural order, then limpf ( p , q) is uniform as to Q = N . Obviously, “nondecreasing” can be replaced by “nonincreasing.” Suppose limpf ( p , q ) = g(q) and lim, f ( p , q) = h(p) with lim,g(q) = lim,h(p) = a. Then obviously g(q) is also nondecreasing in q and lim,g(q) = 1.u.b. g(q). Because of the monotoneity of f ( p , q ) in q, we have f ( p , q) 5 h ( p ) for all p and q. If we adjoin the fact that lim,h(p) = a, then (e, p , , pRp,, q) :f ( p , q) 5 h(p) < a 2e. On the other hand, since lim,g(q) = a, we have (e, q,, qRq,) : g ( q ) > a - e. Fix q‘ so that q‘Rq,. Then (e, q’, p,‘ = p , , , , pRp,‘) : f ( p , q’) > g(q‘) - e. Consequently, for qRq‘ and pRp,‘ we have: f ( p , q) 2 f ( p , q’) 2 g(q‘) - e > a - 2 e. If now we assume that p,”Rp, and peNRp,‘,then a combination of these statements yields : (e, p e t ’ , q,‘, pRp,”, qRq,‘) : a - 2 e
+
+
8.
FILTERS
17
As a special case we have: if a,,,, is a double sequence which is monotone nondecreasing in n for each rn, and if lim,lim,,am,, = lim$immamn,then both the inner limits are uniform. If we examine the proof of Theorem 1.7.8, we note that we use only the fact that lim,f(p, q) 5 h(p) for all p. This gives the following: 7.9. Corollary. If f(p, q) is monotonic nondecreasing in q for eachp,
and if lim, f ( p , q) 5 h(p) for each p, and lim,h(p) = IimJimpf(p, q ) then lim,,f(p, q) exists and is equal to lim,lim,f(p, q). 8. Filters
The notion of filters was introduced into mathematical thinking by the Bourbaki group and has gained widespread recognition as a tool in analysis, even to the extent of displacing the more easily understood concept of directed set. We consequently devote this section to the relation between directed sets and filters. 8.1. Definition. If P is a fundamental class of elements, then a filter
8 is a class of subsets of P which satisfies the following conditions: (1) if F belongs to 5, and the set E contains F then E belongs to 5 ; (2) if F, and Fz belong to 5, then F,, their intersection or the set of all elements in both F, and F2, belongs to 5 ; (3) the null set does not belong to 5. The conditions on a filter are strongly reminiscent of the conditions on a set of vicinities of a point in topological space. In particular, if no vicinity of p consists of p only, then the vicinities of a point p with p omitted form a filter, provided condition (1) is satisfied. If f(p) is a function on P, then limit of f(p) relative to the filter 5 is defined: = a if and only if for every e > 0,there exists a set F , in the filter 5 such that 1 f(p) - a I < e for all p in F,, or (e, F,, p in F,) : I f(p) - a I < e. A class 9 of subsets B of P is called a base for 5 if 9 is a subclass of 5, and if for every F of 5 , there exists a B of 9 contained in F. Then :
8.2. Definition. limsf(p)
8.3. T H E O R E M . If 9 is a base for the filter 5, then limsf(p) = a
if and only if lima.f(p)
=
a.
18
1. A GENERAL THEORY OF LIMITS
I
I
For suppose limg f ( p ) = a, i.e., (e, F,, p in F , ) : f ( p ) - a < e. If B e is contained in F,, then obviously (e, be,p in Be): 1 f ( p ) - a I < e, i.e., limBf ( p ) = a. Since 9 is contained 5, the converse is obvious. If we desire that 9 generate 5 by adding to 9 all the subsets of P which contain a set B of 9, then 5 will be a filter if we assume that 9 has the properties: (2') if B , and B, belong to 9, then there exists in 9 a set B, contained in the intersection of B, and B,; and (3) 9 is not vacuous and does not contain the null set. For if F, contains B, and F, contains B,, then the intersection of F, and F, will contain any B, contained in both B, and B,. In order to apply the notion of filters to the convergence of a sequence of real numbers a,, we let P be the set N of positive integers. The filter 5 consists of all subsets of N obtained by omitting a finite number of elements from N . A base 9 of 5 consists of all sets Nk defined by the integers n > k.
Q with relation R gives rise to a filter 5 on Q. We define the sets B, = [the set of all q' such that q'Rq]. Then the class 9 of B, satisfies the conditions (2') and (3). For B,, and Bq2will contain the set Bgawhere the q, such that q,Rq, and q J q , exists by the compositive property. Moreover, B, is not vacuous since the compositive property applied to q and q assures the existence of an element q, such that q,Rq. As a consequence, if we add to 9 all subsets of Q containing a set B, then the resulting class of sets is a filter. Moreover, lim,f(q) = a is by definition equivalent to limef(q) = a and so to limsf(q) = a.
8.4. THEOREM. Any directed set
8.5. Conversely, a filter is a directed set.
For suppose 5 is a filter on P. If we call 8 the set Q and define q,Rq, by inclusion, i.e., F, C F,, then Q is a directed set. The function f ( p ) on P to reals gives rise to the many-valued function f ( F ) on Q or the filter 5,where f (F) includes all f ( p ) for p in F. Then limsf(p) = a is obviously equivalent to limg f ( F ) = lim,f(q) = a. We see then that limits via filters or directed sets are in a sense equivalent. We shall restrict ourselves to directed sets, since we consider them a simpler concept than filters. The contrast between directed sets and filters is illustrated by considering the definition of unconditional convergence of an infinite series Znanas being equivalent to the existence of the lim, Zqan,where
8.
19
FILTERS
q = (n,,...,n J , finite subsets of the integers N . One might perhaps expect that the filter connected with this limit would be determined by the complements of finite subsets of the integers N . This is not the case. Our basic Q is the set of all finite subsets q of N . Consequently, the equivalent filter has as its base 9,the sets B,, where B, is the class of all finite subsets of N containing the set q. To characterize the sets F of the filter 5 adds additional complications.
EX ERClSES 1. Show that if f co are allowed as values of lim, f ( q ) and lim,g(q), where f and g are defined on the directed set Q, then lim, ( f ( q ) g(q)) = lim, f ( q ) lim,g(q), excepting for the ambiguous case cc -a.
+
+
+
2. Show that if f00 are allowed as values of &, and,&I functions f ( q ) and g ( q ) on the directed set Q we have:
lim,f(q)
+ lirn,g(q)
provided
+ co
-
then for two
+ g(9)) 5 G,f(q)+ lim,g(q) 5 G , ( f ( q ) + g ( d ) 5 lim,.f(s) + Z , g ( q ) , 5 lim,(f(q)
co is not involved.
3. Under what conditions would relations similar to those of Exs. 1 and 2 hold for limits of products and products of limits of two functions? 4. Given f ( p , q) on the product P x Q of the directed sets P and Q. Let g(q) be the many-valued function v.ap.,>f ( p , 9). Show that if limp, f ( p , q ) exists as a finite number, then lim,g(q) = limp, f ( p , q ) . 5. Deduce properties of a generalized iterated limit of a function f ( p , q ) on the product P x Q of the directed sets P and Q, defined: for every e > 0, there exists p e such that if pRp,, then there exists q p psuch that if qRq,,,
then I f ( P , 4) - a
I < e.
6 . If .f,(x) is a sequence of continuous functions on the closed interval [a, b] such that f n + l ( x ) 2 f n ( x ) for every n and x , and if limnf n ( x ) = . f ( x ) is continuous, then the convergence of f , ( x ) to , f ( x ) is uniform on [a, b].
7 . If f , ( x ) is a sequence of monotonic nondecreasing functions on the closed interval [a, b],and if lim,,f,(x) = , f ( x ) and f ( x ) is continuous, show that the convergence of f , ( x ) to f ( x ) is uniform on [a, b].
I
8. Suppose the double sequence anlnis such that (a) 2, I a,, converges for each m, (b) lim,a,, = a, for each n, and (c) lim,,,Z, 1 a,nn = Cn an 1. Show that 1im7,ZnI a,,, - a, I = 0.
I
I
20
I. A GENERAL THEORY OF LIMITS REFERENCES
On Moore-Smith general limit: E. H. MOORE:Definition of limit in general analysis, Proc. Natl. Acad. Sci. 1 (1915) 628. E. H. MOOREand H. L. SMITH:A general theory of limits, Am. J. Math. 44 (1922) 102-121. H. L. SMITH:A general theory of limits, Natl. Math. Mag. 12 (1938) 371-379. G. BIRKHOFF:Moore-Smith convergence in general topology. Ann. Math. (2) 38 (1937) 39-56. E. J. MCSHANE: Partial orderings and Moore-Smith limits, A m . Math. Mon. 59 (1952) 1-10, On Filters:
N . BOURBAKI: Elements de Mathematiques XVI. Pt. I, livre 111: Topologie Generale, Actualites sci. et ind. No. 1196. (1953) p. 8 ff.
9. Linear Spaces
Although not immediately connected with the notion of general limits, the idea of linear spaces has so many of its roots in the theory of integration that it seems desirable to devote a paragraph of this introductory chapter to a summary of some of the important basic concepts connected with such spaces.
space is any collection of elements usually subjected to certain conditions or having certain properties.
9.1. Definition. A
9.2. Definition. A linear or vector space X of elements x, y, z,... is
a set of elements subjected to the following conditions: (a) There is an operation + between any two elements of X to an element of X,under which X forms an Abelian group, that is: (1) for every x,y of X there exists a z of X such that z = x y , (2) for every x, y of X : x y = y x, (3) for every x,y , z of X : x .(y z) = (x y ) z, (4) for every x, y of X there exists a z in X such that x z = y . As a consequence there exists a unique element of X its zero denoted 0 such that for all x : x 0 = 0 x = x.
+
+
+ + + +
+ +
+
+
(b) To every number a and element x of X there corresponds an element ax such that
(5) 1 x = x for all x, (6) for all numbers a and b, and x of X : (ab)x = a(bx), 9
9.
21
LINLAR SPACES
+ y ) = ax + ay, (8) for all numbers a and b and x of X: ( a + b ) x = ax + bx. (7) for all numbers a and s,y of X:a ( x
It follows that 0 * x = 0 for all s. Here numbers may be limited to the field of real numbers or may be the field of complex numbers. We shall limit ourselves to the real numbers, i s . , to real linear spaces. Most simple instances of linear spaces are classes of functions on some range P. If f ( p ) and g ( p ) are two such functions, then usually f f g ) f p ) = f ( P ) .!?(P), and ( o f ) ( P i a . f ( P ) for all P of P. Instances of linear spaces are : ( I ) the set of all functions on a range P ; (2) the set of all bounded functions on a range P ; (3) if Q is a directed set, then the set of all functions f ( q ) for which lim,f(q) exists forms a linear space (this is another way of stating the first part of Theorems 1.2.1 and 1.2.3); ( 3 ’ ) the set of all functions f(q) on the directed set Q such that lim, f ( q ) = 0 is a linear space; (4) the set of all functions on f ( p ) as defined i n 1.6 exists. This set is identical a range P for which L,, with the set of all functions for which L,,I f ( p ) I < cc;( 5 ) the set of all sequences a,! such that Ljpjtexists; (6) the set of all continuous functions on an interval [a, b]. The linear spaces which parallel the class of real numbers in their properties are the normed linear spaces. We define :
+
+
9.3. Definition. X is a iiorniecl linear s p c e if there exists on
1 1 x 11
(1) for every x of X :
(2) for every x and y lently :
11 x 11 2 0; of X : 1 1 x + y I /
I I/ x
(2’) for every x, J’, z of X : 1 1 x - z I / 5 1 1 which is usually called the triangle property; (3) for every number a and x of X: 11 ax
(4)
X a norm :
satisfying the conditions:
11 x 11
=
11
I / + / I y 11, .‘c
=
-y
11
or equiva-
+ 11 y
-
z
11,
I a [ . 1 1 x I];
0 if and only if x = 0 .
If X is the set of all bounded functions f ( p ) on P, then 11 f / l can defined as the 1.u.b. I f ( p ) I for p on P. If X is the set of all functions f ( p ) onPsuchthat.YPlf ( p ) I < x,thenIIf(/canbedefined a s L p l f ( p ) 1. In terms of the norm, it is customary to define the notion of limit: lim,!x,, = x is equivalent to litnj,1 ) .x,~- x
11
=
0,
22
1. A GENERAL THEORY OF LIMITS
or if Q is a directed set lim,x,
=x
is equivalent to lim,
I( x,
-
x
11
= 0.
However, limits in linear spaces need not be limited to such norm limits. For instance, if X is a space of functions on P,we can define “lim”l,f,r by pointwise convergence: lim,,A,(p) = f ( p ) for all p of P. Or we can define “lirn”)(f,, = f as lim,!X,(p) = f ( p ) uniformly on P. Or we may define “lim”?!f,, =fprovided there exists a set of subsets of P: Pu such that P = .ZuP,and lim,,f,,(p) = f(p) uniformly on each P,l. Similar statements hold for directed limits. Of particular importance among linear spaces are the linear normed complete spaces. We define : 9.4. Definition. X is a linear normed complete space if it satisfies
the Cauchy conditon of convergence relative to the norm, i.e., if for every sequence of elements x, such that lim,,n 11 x, - x,,[I = 0 there exists an element x in X such that lim,! 11 x,! - x 11 = 0. Such spaces are frequently called Banach spaces. We prefer the terminology “linear normed complete” because these spaces were considered by many mathematicians ‘before Banach, and also because the terminology adequately describes the character of these spaces in contrast to other linear spaces. In addition to completeness relative to sequences one might also have completeness relative to any or some directed sets Q, in the sense that if x ( q ) is on the directed set Q to X and such that lim, 1 2 1 I x ( q J x(q,) I( = 0, then there exists an element x of X such that lim, 11 x ( q ) - x 11 = 0. Following the proof as given for the special case when x ( q ) are real numbers in 1.2.11, it is easy to show that:
,
9.5. THE ORE M . If X is a linear normed complete space, then it has
the completeness property also relative to any function x ( q ) on any directed set Q to X . For limits in a linear space, it is not always true that a sequential type of completeness when it exists and holds carries with it completeness relative to all directed sets. A functional on a linear space is in fact a function; f on the space X assigns to elements x real numbers. Of particular interest are the linear functionals, defined:
is a linear functional or form on the linear space X if it assigns a real number to each x of X and if for a and b real 9.6. Definition. f ( x )
9.
23
LINEAR SPACES
+
+
numbers and x and y elements of X we have f ( a x by) = af(x) bf (Y)9.7. Definition. In a linear normed space, a linear functional is continuous if lim,z11 x, - x 11 = 0 implies lim,f(x,) = f(x) or lim,f(x, - x) = 0. It is an easy matter to prove: 9.8 THEOREM. A necessary and sufficient condition that a linear functional on a linear space X be continuous is that there exists a constant M such that 1 f ( x ) I 5 M 11 x 1 I for all x of X , or that f(x) be bounded (or limited) on X . As an example of a linear functional, we might note that if X is the class of real valued functions f on the directed set Q, such that lim,f(q) exists, then lim,f(q) is a linear functional on this linear class, which is another way of stating Theorems 1.2.1 and 1.2.3.
This Page Intentionally Left Blank
CHAPTER I I
RIEMANNIAN T Y P E OF INTEGRATION
In this chapter, we take up some types of integrals suggested by the Riemann definition of integral. We consider first functions defined on a subinterval of the real line, and consider later the extension to two or higher dimensions. As a basis we develop first the properties of integrals of functions of intervals, using the theory of the Riemann integral as a guide. Properties of functions of bounded variation are summarized as an instance of integrals of functions of intervals. Then Riemann-Stieltjes integrals are developed first for the case of any two functions, and then particularly for the case where the function with respect to which the integral is taken is of bounded variation. Finally the integration theory for one-dimensional space is in part extended to two or more dimensions in the next chapter. 1. Functions of Intervals
We assume a basic finite interval [u, b] = a 5 x 5 b, denoted by X . Subintervals [c, d ] with a 5 c 5 x 5 d 5 b will be denoted by I. Although we have indicated that Z = [c, d ] means the closed interval c 5 x 5 d, for most of the considerations of this chapter, it is immaterial whether Z be closed, open, or half-open. 1 .l.Definition. An interval .fuiiction, or function of intervals f(Z) is assumed to be defined for all subintervals of [a, b] and may be many valued. We mention some examples of functions of intervals. Suppose g(x) is a bounded point function on [a, b]. Then g(x) gives rise to the following interval functions :
(1) f(Z) = g(x) for c 5 x 5 d ; (2a) f ( I ) = 1.u.b. g(x) for x on [c, d ] ; (2b) f(Z) = g.1.b. g(x) for x on [c, d ] ; 25
26
11. RIEMANNIAN TYPE OF INTEGRATION
(3) f (1) = g ( d ) - g ( c ) ; (4) f ( Z ) = I g ( d ) - g(c) 1; ( 5 ) f ( Z ) = oscillation of g ( x ) on [c, d ] = 1.u.b. [Ig(x,) - g(x,) I for x,, and x, on [c, d ] ] = w(g; Z ) = 1.u.b. of interval function (4) for all Z' contained in I ;
(6) f ( Z ) (6a) f ( Z ) (6b) f ( Z )
( d - c) = g(x)Z(Z) for all x of [c, d ] ; (1.u.b. f ( x ) on [c, d ] ) ( d - c ) = M ( d - c); (g.1.b. f ( x ) on [c, d ] ) ( d - c ) = m ( d - c ) ;
= g(x) = =
( 6 4 f ( Z ) = g(c) (d - c ) or f ( I ) = g ( d ) ( d - c) ; (7) f ( Z ) = (g(d) - g ( c ) ) / ( d - c ) of interest in connection with derivatives;
(8)
f(Z)
=
( d d ) -g(c))21(d-
c).
If we have two point functions g ( x ) and h ( x ) we get: (9) f ( Z ) = g( x ' ) (h(d) - h ( c ) ) with c 5 x 5 d, which leads to Riemann-Stieltjes integrals. Some modifications of this are : ( 9 4 f ( Z ) = g(c) ( h ( d ) - h ( c ) ) or f ( Z ) = g ( d ) (h(d) - h ( c ) ) ; (9b) f ( I ) = (g(c) g(d)L) M d ) - h ( c ) ) P ; (9C) f ( Z ) = ( g ( x l ) ...g(Xk)) (h(d) - h ( c ) ) / k , where X l , x k are points, possibly restricted, of [c, d ] ; where h(c) f h(d) (10) f ( I ) = ( d d ) - g ( c ) ) / ( h ( d ) for all c, d of [a, b];
+
+
(11) f ( Z ) = (g(d) - g(c))'/(h(d) theory of Hellinger integrals.
- h(c)),
which occurs in the
A function of intervals in turn can give rise to other functions of intervals, e.g., if f ( I ) is a function of intervals then I f ( Z ) I ; l.u.b.f(I) ; g.1.b. f ( I ) ; 1.u.b. [f ( Z ' ) for I' contained in I]; g.1.b. [f(Z') for Z' contained in r] are functions of intervals. 1.2. Definition. A single-valued function of intervals is said to be
additive if for every division of any I into a finite number of adjacent intervals 11, ..., Zk, so that I = Z, + Z, ... Ik, we have .f(I) = ,Zf=l f ( I i ) . A single-valued function f ( Z ) is said to be upper semiadditive if under the same conditions f ( I ) I Zfzl f ( Z i ) , lower semiadditive if f (I) 2 Zf=,f (Ii ). If f ( Z ) is additive, then obviously I f ( Z ) [ is upper semiadditive. The indefinite Riemann integral is an additive function of intervals.
+ +
2.
INTEGRALS OF FUNCTIONS OF INTERVALS
27
Example (6a) above is lower semiadditive, and (6b) is upper semiadditive. If f ( I ) is an additive function of intervals of [a, b], then f ( I ) is completely determined by the point function g ( x ) = f [a, X I , since fk, dl = f [a, dl - f b , CI = g(d) - d c ) . 2. Integrals of Functions of Intervals
We call a subdivision (or partition) of [a, b] a division of [a, b] into a finite number of adjacent intervals by the points a = xo < x , < ... < x,,= b or by the intervals I,: [xtP1 5 x 5 x,],i = 1 ... n. We shall denote such a subdivision by a, where a can mean either (xo,x,, ...,xn) or (Il, ..., I?,).We shall say that a1 isfiner than or a refinement of a2, denoted by a, 2 a2, if every x,of a2 is contained in a, or every interval of a, is a subinterval of some interval in az.For one-dimensional space, it is usually more convenient to assume a given by the points of subdivision. We shall denote by a, a2, the totality of dividing points in both a1 and az arranged in linear order. As indicated in 1.1, it is possible to make a directed set out of the subdivisions of [a, h] in two distinct ways: (a) by introducing a metric or norm and defining I a I = maximum of lengths of I , or of x, - xZ-,, the norm order relation (T] Ra, being equivalent to I a1 I 5 I a2I ; (b) by defining a,Ra2 by set inclusion, i.e., a, 2 a, if a, contains all of the points of a2, or ( T ~is finer than a2. Any interval function f ( I ) and a subdivision (T of [a, b] give rise to a function of g ( a ) = ZT=, f ( I J , where a = (Il, ..., I J . The two methods of ordering subdivisions then give rise to two types of integrals for interval functions.
+
2.1. Definition. We shall say that: the interval function f ( I ) on [a, b]
has a norm integral on [a, b] (denoted N lim f (Ii) exists. 2.2. Definition.
f, f ( d I ) ) ,
i f liml,140g(a)=
The function f ( I ) on [a, b] has a a-integral (denoted
a J: f ( d I ) ) , if lim,g(a) = limoZi f ( I J exists in the sense of succes-
sive subdivisions or refinements. The Riemann integral J : g ( x ) d x is a norm integral based on the interval function (6): f ( I ) = g ( x ) ( d - c), c 5 x 5 d, the Darboux upper and lower integrals J b g ( x ) d x and f' g ( x ) d x are a-integrals based on the interval functions (6 a) and (6 bL'viz., f ( I ) = (g.1.b. g ( x ) on
28
11. RIEMANNIAN TYPE OF INTEGRATION
[c, d ] ) (d - c ) and f ( I ) = (1.u.b. g(x) on [c, d ] ) (d - c), respectively. Obviously, if f ( I ) is additive, then for all c, Zuf ( I ) = f ( X ) , so that J : f ( d l ) exists in either sense and has the value f ( X ) . If Zuf ( I ) does not have a limit, we have values approached and greatest and least of the limits. As a consequence we define N f ( d I ) = G,u,-o Z5f ( I ) and c f ( d I ) = K u Z 5f ( I ) and similarli for the lower integrals N r f ( d i ) and a f ( d I ) . The Darboux upper and lower intergrals foraa point func&n g(x) are the upper and lower a-integrals for f ( I ) = g(x) (d - c) with c 5 x 5 d.
sb
sb
Sb
3. Existence Theorems
Since the two types of integrals we are considering are special cases of limits of real valued functions on directed sets, we are able to apply the general theory. We therefore have : 3.1. T H E O R E M (Cauchy condition of convergence). A necessary and sufficient condition that N J : f ( d I ) or a J : f ( d I ) exist is that the
corresponding Cauchy condition of convergence holds, i.e., for the norm integral:
for the a-integral:
These statements can be replaced by and
respectively. Also since from I Zo:l f ( I ) - Zu,f ( I ) I < e and 1 Zu3 f ( I )Z u , f ( I ) I < e, it follows that I Zulf ( I ) - So,f ( I ) 1 < 2 e, an equivalent statement can be obtained by replacing the condition on a2 in each case by a2 2 a,, so that for instance for the norm integral, the Cauchy condition of convergence might read: (e, d,, 1 al 1 < d,, a2 2 a J : I zluzf ( I ) - Zulf ( I ) I < e. For we can compare Z5,f ( I )
3.
and Zuz f ( I ) with Zul+u2 f ( I ) since I a, I < d, and 1 g 1 “ 2 I < d,. A second existence theorem is the following:
+
29
EXISTENCE THEOREMS
I a2I < d,
3.2. THEOREM. A necessary and sufficient condition that
implies
[:f(&)
sb
exist is that the corresponding upper and lower integrals: f ( d Z ) and f ( d I ) be finite and equal. NG;e that these upper and lower integrals are defined in terms of extreme limits, not as in the case of Riemann integrals as least upper and greatest lower bounds. For the norm integral there exists a reduction of the norm limit to sequential limits, so that we have:
Sb
A necessary and sufficient condition that N J: f(dZ) exist is that for every sequence of subdivisions ajLof [a, b] such that lim?jI o?,[ = 0 the approximating sums 2an f(Z) shall converge to a finite limit. such that lirnTZ The necessity is obvious since any sequence I I = 0, is cofinal with I a I --f 0. On the other hand, if limnZun f(I) exists for every (T,~ such that I aIjI + 0, then this limit is the same for all sequences, since any two sequences a,!and could be combined into a single sequence all’’ having the same property, for which limn Zur,f ( I ) exists uniquely. Suppose then l i m , , - o Zuf ( I ) does not exist or is not equal to this coinmon limit a. Then there exists e > 0, such that for every n there exists n,, with I a,/I < l / n and [ ZU*f(Z) - a I > e. Consequently, limjt Zon f(Z) f a, which leads to a contradiction. We observe that we have demonstrated essentially that if a directed set Q is directed via a norm: [ q 1 so that lim, f ( q ) is equivalent to lim,q,+o,f(q),then a necessary and sufficient condition that lim, f ( q ) exist is that lim j , f ( q , l )exist for every sequence {q,,}such that q,, 0 (compare 1.5.1.). For the case when f ( I ) is single valued and upper semiadditive (lower semiadditive) the sums Xof ( I ) are monotonic nondecreasing (nonincreasing) as functions of a. Consequently : 3.3. THEOREM.
Tl
I I-+
3.4. THEOREM. If f ( I ) is single-valued and upper (lower) semiadditive and Zaf ( 1 ) is bounded above (below) as to n, then a f ( d I ) exists and is equal to l.u.b.aZuf ( I ) (g.l.b.uZo,f(Z)).
s:
30
11. RIEMANNIAN TYPE OF INTEGRATION
As an instance, we note that if g ( x ) is a point function on [a, b ] and f ( 1 ) = g ( d ) - g ( c ) I then f(Z) is upper semiadditive, so that if g ( x ) is of bounded variation, the total variation of g is the a J: f(dZ). We shall consequently write V : g ( x ) = J: 1 dg I. Further, the Darboux integrals of a bounded point function g ( x ) are a-integrals of the interval functions (6a) and (6b) listed in Section 1 above. For a slightly more sophisticated existence theorem, we define the oscillation function: cu(Sf; I ) , which is an interval function, as follows:
I
I ) = 1.u.b. [I Xu f - ZU2 f 1, for all a1 and a9 of Z]. Here Za f stands for the sum of f(Z’) over all the intervals I’ belonging to a. This oscillation functions w (Sf; I ) is positive and lower semiadditive. For if Z = I , I,, with I, and Z, adjacent, then for e > 0, there exist subdivisions aI’ and a,” of Zland a,’ and a,’’of I, such that
3.5. Definition. w(S’
+
+
+
where a’= al’ a,’ and a‘’= all‘ a,”. Since the inequality holds for all e, we get w(Sf; Z,) u ( S f ; I,) 5 o(S’ I ) , the lower semiadditivity of w(Sf; Z). With the aid of this oscillation functiori, we obtain a third existence theorem for integrals of interval functions.
+
3.6. THEOREM. A necessary and sufficient condition that N w ( S f ; dZ) = 0 ; that n J: f(dZ) exist is that
Jl
exist is that N
a J b u(S’
dZ)
= 0,
or g.l.b.u
C bu(S’
I)
J’: f(dZ)
=0.
0
We shall prove only the first half of this theorem, i.e., for the norm integral, the case for the a-integral follows the same line of argument. We show that the conditions of the theorem are equivalent to the Cauchy condition of convergence. Suppose N J: w ( S f ; dZ) = 0. Then
3.
31
EXISTENCE THEOREMS
Suppose I o1 1 < d , and o2 2 a f‘ Then by rearrangement of terms, so as to bring together the terms in each subinterval of C J ~we , find that
so that the Cauchy condition of convergence holds in its second form. On the other hand suppose for e > 0, d , is determined by the Cauchy condition of convergence, and select a so that I CJ I < d,. Let CJ consist of the n points a = x o < x , < ... < x , = b. Then on [x,-,, x , ] , we can find 0%‘ and C J ~ ” so that
c .f- c f 4%[xz--l’4 )5 c f - c f + eln. 2 0,
(I*’
and
U,“
0,’
U/
If we set a’ = 2, 0%’and a“ = Zlat“ then and 05 W(Sh [ X i - , , x,l) 5
I CJ’ I < d , and I a’’I < d ,
c U
I -
Then N
s”, w(S’dI)
= 0.
3.7. Definition. We shall call an interval function pseudoadditive
at a point x
if
limg,#+ (o,o) (f[ x - d‘, x
+ d”] - f [x - d ‘ , x ] --
f [x, x + d”]) = 0 ,
with d‘ > 0 and d” > 0. Then the Cauchy condition of convergence gives rise to the following necessary condition for the existence of the norm integral of an interval function f ( I ) :
Jl
3.8. T H E O R E M . A necessary condition that N f ( d I ) exist is .that J ( I ) be pseudoadditive at every interior point of [a, b ] . This results from a comparison of ZU, f and ZUJ where o1 and a2 agree excepting that in 0 2 , the interval [x - d‘, x d”] of C J ~is red”]. placed by the two intervals [ x - d‘, x] and [x, x For the a-integral, since we can always include any given point x in a subdivision, the corresponding necessary condition is one sided, i.e., we have:
+ +
32
11. RIEMANNIAN TYPE OF INTEGRATION
3.9. T H E O R E M . A necessary condition for the existence of a J : f ( d I ) is that for every x: limLUdt(f[x, x”) -f[x, x’]-f[x’, x”]) = 0, x < x’< x” and limrt,+ (f[x”,x] - f [x”,x’]-f[x‘, x]) = 0, x” <
x’< x. We could call these two properties pseudoadditivity on the right and left, respectively. It is to be noted that f ( I ) can be pseudoadditive both on the left and right without being pseudoadditive. The importance of the pseudoadditive condition is that it is the connecting link between the norm and a-integrals. We have: 3.10. T H E O R E M . A necessary and sufficient condition that the norm integral of an interval function f ( I ) exists is that the a-integral exist and that f ( I ) be pseudoadditive at every interior point of [a, b]. Since a, 2 u 2 implies that I crl I 5 I u 2 1, it follows that if the norm integral exists, the a-integral exists and the values agree. Conversely, suppose a J]:,f(dI)exists. Then (e, ac, a 2 ae) : I Z0f ( I ) - a J: f ( d I ) 1 < e. Let a? consist of the n 2 points: a = x,, x,,..., x,,,x ? , +~ b. Since .f(I) is pseudoadditive at x,, ..., x,,, there exists a d, such that if 0 < d,’, d,“ < d,, then I f[x, - d?‘, x! dZ”]-f[x, - d,’, x,].f [x,,x1+ d l ” ]I < e/n. For the d, so determined consider any subdivision a with I a I < d,. Let u, = a a, and compare Z0f and Z0,J using the same value o f f for any interval of cr which contains no point of a e as an interior point. If x,’- d?’,x, dt” are the end points of intervals of a containing a point of u e , then
+
+ +
+
This means that N J : f ( d I ) exists and is equal to cr Cf(d1). The interval function f ( I ) = g(x) (d - c), c 5 x -I d, with g(x) bounded is pseudoadditive since f(Z) 5 (1.u.b. g(x)) ( d - c). It follows that in the definition of the Riemann integral, limits of the approximating sums can be taken either as 1 a I 0 or as cr becomes finer. This is the import of the Darboux theorem on the existence of a Riemann integral. --f
4. INTEGRAL
33
OF A N INTERVAL FUNCTION
4. The Integral of an Interval Function as a Function of Intervals
JB
4.1. T H E O R E M . If f ( Z ) is an interval function on [a, b] and f ( d Z ) exists, and if a 5 c < d 5 b, then f ( d I ) exists. We apply 11.3.6. Suppose a is a subdivision of [a, b] containing the points c and dsuch that ZBw(Sf;I ) 5 e. Then obviously Zns w(S’ I) t e , if or is a subdivision of [c, d ] agreeing with a on this interval. If follows that Jf f(dZ) defines a single-valued interval fuqction on [a, b] which we denote: J I f ( d Z ) .
Jp
4.2. T H E O R E M . If J : f ( d Z ) exists and a
< c < b, then J : f ( d I )
+
S : : f W )= S I : f ( d V Since all of the integrals in the formula exist, for e > 0 and for the same d,, we determine a’ = a(a, c ) , a” = a(c, b ) , and a = a(a, b ) such that I a‘ 1, I a” 1, 1 (r I < d,, and I J: f(dZ) - .ZB1 f ( Z ) I < e, J,”f(dZ) - ZOnf ( Z ) I < e, and I J: f(dZ) - ZBf ( Z ) I < e. If a = a’ a“, then I J: f(dZ) J,”f(dZ)- J: f(dZ) I < 3 e, from which we deduce the equality in the theorem. Since the interval [a, b] can be replaced by any subinterval I = [c, d ] , we conclude that:
I
+
+
+
J: f(dZ) exists and if I = I , I,, where I is any subinterval of [a, b] and I , and I., are adjacent, then J I f ( d Z ) = SI, f ( d Z ) J12 f ( d I ) , or J I f(dZ) is an additive function of intervals on [a, b]. In particular there exists then a point function g(x) so that g(x) = J: f ( d Z ) , which defines the interval function J I f(dZ). The reasoning for the case where we deal with a a-integral instead of a norm integral, is similar. For the converse, we have separate results for the norm and the a-integrals. 4.3. Corollary. If
+
4 . 4 ~ .T H E O R E M . If a J:f(dZ) and a
J: f(dZ) exist, then a J: f ( d Z )
exists and is equal to the sum of these two integrals. For (e, a,(a, c),
and
0’
2 ae(a,c ) ) : CB, w(Sf; I) < e
(e, a,(c, b ) , a” 2 ae(c, b ) ) : imply (e, a = a‘
xOrJ w(Sf; I ) < e
+ o r ’ ) : xBw ( S f ; 1) Cotw ( S f ; I ) + Cot, w(S’
Then a J: f(dZ) exists.
=
I ) < 2e.
34
11. RIEMANNIAN TYPE OF INTEGRATION
The additional condition needed for the corresponding result in the case of norm integrals is suggested by the fact that if the norm integral of f ( I ) exists, then f ( I ) is pseudoadditive at every interior point of [a, b]. We have: 4.4N. T H E O R E M . If N
JI f(dZ) and N f f ( d I ) both exist and f(Z)
is pseudoadditive at the point c, then N J: f ( d I ) exists and is equal to the sum of these two integrals. For under the hypothesis of the theorem, f( I ) will be pseudoadditive at every interior point of [a, b]. Moreover, the u J I f ( d I ) and u J: f ( d I ) will exist, so that u J : f ( d I ) exists also. Hence by Theorem 11.3.10, N J: f ( d I ) exists. In view of the additive properties of the integral we have: 4.5. Approximation Theorem. If u is any subdivision of [a, b] and if
Jl f(dZ) exists, then I J: f ( d I ) - Zof ( I ) 1 S Zgw(Sf,I ) .
For if u = ( I l , ..., I*, ..., I n ) , then J: f ( d I ) = Zi JI f ( d I ) and obviously 1 JI.f(dI)- f f I J I 5 w(sf; Ii). For the case of Riemann integrals, the indefinite integral J;T g(x)dx is a continuous function of x. For integrals of interval functions, this property becomes : 2
4.6. T H E O R E M . If N J: f ( d I ) exists, then (e, d,, l(I) < d,) : N S I f f d I ) - f f O < e, or lim~(~)-.~l N J I f f d I ) - f f I ) = 0, and
I
I
I
uniformly on [a, b]. Here Z(I) is the length of the interval I. This theorem is an immediate consequence of the existence theorem 11.3.6 and the approximation theorem 11.4.5. The corresponding theorem for a-integrals reads : 4.7. T H E O R E M . If
(T
J:f(dI) exists, then for every point x of [a, b]
and d > 0:
fim,+,ff[x = limd-o
- d, X I - 0
( f [ x ,x
+ d]-
For the interval [a, x] we have
f
I-d
ffdl)l
Xtd (T I
f(dI))
= 0.
5.
INTEGRAL AS FUNCTION OF INTERVAL FUNCTION
35
Let x - do be the point of 0 preceding x. Then obviously w(Sf; [x - d, X I ) < e for 0 < d < do. Consequently, for d < do
I J'
f(dI)
- f [x -
d, x] 1 I w (Sf, [x
- d,
XI) < e.
r-d
Similarly for the right of the point x . These two theorems can be interpreted as saying that in a certain sense, the single-valued additive interval function s I f (dI) is dzflerentially equivalent to f ( I ). If we base the continuity of the interval function on its behavior as l(I) + 0, then these theorems assert that the continuity properties of the point function determined by JIf(dI) are governed by those of f ( I ) . The differential equivalence also shows up in a substitution theorem. If f ( I ) is an interval function on [a,.b] and h(x) is a bounded point function, then in an obvious way h(x') . f ( I ) for x r on I defines an interval function and may have an integral. We have: If f(1) is an interval function on [a, b] such that J: f ( d I ) exists and g ( I ) = J I f ( d I ) ; if further h(x) is a bounded function on [a, b], then J: h(x)f(dI) exists if and only if h(x)g(dI) exists, and the two integrals are then equal. The same type of integral (norm or 0 ) is assumed throughout. For if r~ is any subdivision of [a, b] and I h(x) I < M on [a, b],then by the approximation theorem 11.4.5, and the existence of $:f(dI) we have
4.8. Substitution Theorem.
$I:
5. Integral as a Function of the Interval Function
The following two theorems are immediate consequences of corresponding theorems on limits : 5.1. THEOREM. If J:.fl(dI) and J:.f,(dI,J
exist and the values of
&(I) are contained in the class of valuesf,(I) +f,(I), then J:f,(dI) exists and f f , ( d I ) == (f, +fJ (dI) = Jtf,(dI) J:f,(dI).
$:
+
36
11. RIEMANNIAN TYPE OF INTEGRATION
5.2. T H E O R E M . If
J: f(dZ) exists, then J:cf(dZ) exists and is equal to
c s:f(do.
The following theorem on the convergence of integrals of a sequence of interval functions is an immediate consequence of the iterated limits theorem I. 7.4. and f(Z) are interval functions on [a, b] such that for every value f ( Z ) there exists a sequence of values Al(Z) converging to f(Z), and if the sequence J:.fn(dZ)exists uniformly in the sense that lim ZU~(5’’~; I ) converges uniformly to zero in n, then J: f(dZ) exists and is equal to lim?/J:f,,(dZ). It is assumed that the integrals (norm or a) are the same throughout this theorem. A special convergence theorem holds for a-integrals, for the case when the functions f n ( Z ) are upper semiadditive : 5.3. T H E O R E M . If f,(Z)
5.4. T H E O R E M . If the interval functions f n ( I ) are. single-valued and
upper semiadditive on [a, b], and have a finite integral, and if 1imTLJl(Z)= f(Z) for each I, then f ( 1 ) is also upper semiadditive and 0 fy(dZ) I J:’,(dz). The inequality still holds if f(Z) does not have a finite (integral. If l i + J:f,,(dl) = co, there is nothing to prove. Let then limn J, f,(dZ) = c < co. There exists then a subsequence of A)(Z) such that lim, J: j., (dZ) = c, and consequently (e, in,, m > me): J:&,m (dZ) < c e. From this it follows that for m > me and all 0, we have Zo f nnt ( I ) < c e. By taking limits as to m, we get: Zu f(Z) 5 c e for all e > 0 and all a. This implies that J:f(dZ) I c= -n lim J: f,(dI). We observe that we have proved incidentally:
+
+
+
+
5.5 Corollary 1. If AL(Z) are upper semiadditive and converge to f(Z) for all Z and a J : f , ( d I ) I M for all n, then a J”,(dZ) exists 5 M . Another consequence of the theorem is: 5.6. Corollary II. If x z ( I ) are upper semiadditive interval functions with finite integrals on [a, b],if f,/(I) converges to f(Z) for all Z, and if lim,a J: fn(dZ) = a J: f ( Z ) , then lim,,a Jzf,,(dZ) = a J: f(dZ) for all x of [a, b]. For by the additive property of integrals, and the properties of greatest and least of limits (see 1.4.6) we have:
6.
FUNCTIONS OF BOUNDED VARIATION
37
EXERCISES 1. If the interval function f(Z) has an integral on [a, b], does the integral function 1 f(I) I also have an integral?
2. If a sequence of interval functions At(Z) converges uniformly on the set of subintervals of [a, b] to f ( I ) and if J: f,,(dl) exist, does J: f(d1) exist and is it equal to the possibly existing limn J: f,(dZ)?
3. Let Q be a directed set and ,A,(Z) a set of upper semiadditive interval functions on [a, b]. If o J: .f;,(dZ) exist as finite numbers and lim,fy(I) = f ( I ) for all Z of [a, b], show that o S: f ( d I ) 5 lim, a J b f,(dl). If in addition l i m p J: ,fq(dl) = a J]:f(dZ) is l i m p J,"fqpZ) =b J,"f ( d I ) for all x of [a, b]? 6. Functions of Bounded Variation
The class of functions of bounded variation on a linear interval [a, b] plays a n important role in real analysis. Since the total variation is a n instance of a n integral of an interval function, it seems justifiable to devote space to the consideration of these functions and a collection of their properties.
f(x) on [a, b] is of bounded variation on [a, b] if the function of subdivisions g(a) = ZoI f ( x i ) - f ( ~ ~ I-is~ ) bounded as t o a. The least upper bound of g(a) as to a is called the total variation o f f o n [a, b]. 6.1. Definition. A function
38
11. RIEMANNIAN TYPE OF INTEGRATION
Since the interval function F(Z) = I f(d) - f(c) 1 = I 0:f I is upper semiadditive, it follows that the total variation when it exists is the a-limit of g(a), i.e., the total variation is the a-integral of the interval function I f 1, so that the properties of the a-integral apply. We shall consequently represent the total variation off on [a,b] by f I df I. It follows that iff is of bounded variation on [a, b] it is of bounded variation on every subinterval of [a, b], and that the point function v(x) = J,”1 df I is monotonic nondecreasing. It also follows immediately from the definition that if f(x) is of bounded variation so is cf(x) for all constants c and J: ldcf I = I c I f I df I. Further, if f and g are of bounded variation, so is f g and J: 1 d ( f g ) 1 5 J: I df I Jf:1 dg 1. These statements include the fact that the space of functions of bounded variation is linear. Any function of bounded variation on [a, b] is also bounded since IfW -f(a) I 5 v(x) 5 s“, I df I. A monotonic nondecreasing (nonincreasing) function is obviously of bounded variation with J: I df I = I f ( b ) - f ( a ) I. As a consequence any linear combination of monotonic nondecreasing functions Etatf,(x) is also of bounded variation. This statement is in a sense reversible, in that we have:
+
+
+
6.2. THEOREM. Every function of bounded variation on a linear
interval [a, b] can be expressed as the difference of two monotonic nondecreasing functions. For if v(x) = J: 1 df 1, then v(x) (f(x) - f ( a ) ) and v(x) (f(x) - f ( a ) ) are both monotone nondecreasing. For 0: (v(x) f (f(x) - f ( a ) ) = J: I df I f (f(d) -f(c)) 2 0 for all c > d. If we set P(X) = Q (v(x) +f(x) - f(a)) and n(x) = Q (v(x) - f ( x ) f ( 4 )then f(x) -f(a) = p(x) - n(x) and I df I = P(X) n(x). The functions p(x) and n(x) can be obtained in another way. Suppose that for any a of [a,x] we set P ( a ; a, x) = Zi(f(xi’) ~ ( X ‘ ~ - Jwhere ) the ( x ’ ~ - xi’) ~ , are the intervals of a for which Of 2 0, and let N ( o ; a, x) = - Zi(f(xi”) - f(x”{J), where the ( x ” ~ - xi”) ~ , are the intervals of a for which O f 5 0. Then P ( a ; a, x) and N ( a ; a, x) are both monotone nondecreasing in a and iff is of bounded variation on [a,b] the least upper bounds as to a of P ( a ; a, x) and N ( a ; a, x) both exist as a-limits converging say to P(a, x) and N(a, x). Now for each a, we have P ( a ; a, x) - N ( a ; a, x) = f(x) f ( a ) and P(a, a, x) N ( a ; a, x) = zlb I O f 1. Taking limits as to CT we find P ( a ; x) - N(a, x) = f(x) - f ( a ) and P(a, x) N ( a , x) =
+
+
s:
+
+
+
6.
J: I df
I.
39
FUNCTIONS OF BOUNDED VARIATION
From this it follows that P(a, x )
=p(x)
and N ( a , x ) =
n (x). This approach to functions of bounded variation is due to Jordan, who first called attention to these functions. Consequently, the formulas f(x) - f ( a ) = p ( x ) - n ( x ) with Ji 1 dfl = p ( x ) n(x) give what is usually called the Jordan decomposition of a function of bounded variation. The functions p ( x ) and n ( x ) are called the positive and negative variations of f( x ) . The above approach suggests still another method of defining the functions p ( x ) and n ( x ) , as well as the bounded variation of a function. Let n stand for a finite number of disjoint subintervals: I , = [ x z r x,”], , i = 1 ... m of [a, XI. Then p ( x ) = 1.u.b. Zz(f(x,’) - f ( x , ” ) ) and n ( x ) = - g.1.b. Z l ( f ( x , ‘ ) - f ( x t ” ) ) for all possible sets n of disjoint subintervals of [a, X I . This leads to the following definition of a function of bounded variation :
+
6.3. T H E O R E M . f ( x ) on [a, b] is of bounded variation if and only if
G(n) = Z l ( f ( x , ‘ ) - f ( x , ” ) ) is bounded as a function of n = (Il, ..., I J , finite sets of nonoverlapping intervals of [a, b ] . The total variation off on [a, b] is equal to l.u.b.TG(n) - g.l.b.,G(n), l.u.b.nG(n) being the positive variation on [a, b] and g.l.b.nG(n) the negative variation. The function p ( x ) and n ( x ) of the Jordan decomposition enjoy the following minimal property : 6.4. T H E O R E M . If f ( x ) is of bounded variation on [a, b] and if f ( x ) = p , ( x ) - n , ( x ) , p , ( a ) = n,(a) = 0, and p , ( x ) and n,(x) monotonic nondecreasing, then p , ( x ) - p ( x ) and n,(x) - n ( x ) are both positive or zero monotonic nondecreasing functions on [a, b ] . This is equivalent to proving that Apl 2 Ap and An, 2 An for every subinterval of [a, b ] . By the preceding result, for every interval [c, d ] of [a, b] and every e > 0, there exists a set n of disjoint intervals of [c, d] such thatp(d) - p ( c ) - e iZnAf5 p ( d ) - p ( c ) . Sincef ( x ) f ( a ) = p , ( x ) - n,(x) we have
Consequently, p , ( d ) - p , ( c ) 2 p ( d ) - p ( c ) - e for all e > 0, or Ap, 2 Ap for all intervals [c, d ] . Similarly An, 2 An. It follows that: 6.5. T H E O R E M . If f ( x ) is of bounded variation on [a, b], then p(x) is the minimal monotonic nondecreasing function such that
40
11. RIEMANNIAN TYPE OF INTEGRATION
p ( a ) = 0 and Op 2 Offor all intervals [c, d ] , and n ( x ) has the same property relative to - f ( x ) . For if Op, 2 O f for all intervals [c, d ] , then On, = d ( p , - f ) 2 0 for all intervals, so that n,(x) is monotone nondecreasing andf(x) f ( a ) = p , ( x ) - n,(x). Consequently Ap, 2 Op. 7. Continuity Properties of Functions of Bounded Vatiation
Since the discontinuities of a monotonic function are all of the first kind, i.e., f ( x 0) and f ( x - 0) exist for each x, and are at most denumerable in number, it follows that any function of bounded variation is continuous excepting at a denumerable number of points at each of which f ( x 0) and f ( x - 0) exist. Moreover, since the total variation function v(x) = J: I df I is an integral of an interval function, it reflects the discontinuities of I Of\,i.e., by 11.4.7 we have v(x 0) - v ( x ) = I f ( x 0) - f ( x ) I and v(x) - v(x - 0 ) = 1 f ( x ) - f ( x - 0 ) 1. It follows that if f ( x ) is a continuous function of bounded variation, then v ( x ) , p ( x ) , and n(x) are also continuous. We define a simple break (or step) function B(x, x o ; c, d) as follows: B(x, x,; c, d) = 0 for a 5 x < x o ; B ( x , x o ; c, d) = c for x = xo and B(x, x,; c, d ) = d for x , < x 5 b. If f ( x ) has a discontinuity of the first kind at x, then f ( x ) B(x, x,; f ( x o ) - f ( x , - 0), f ( x , 0 ) - f ( x , - 0)) will be continuous at x , with value f ( x , - 0 ) . Now if { x , ) are the points of discontinuity of a function of bounded variation f ( x ) , then it follows that (I f f x , , ) - f ( x , - 0) I I . f ( x , 0 ) -f (x,)l) 5 J: I df IConsequently, the series Z,B(x, xr7;f ( ~ , , -) f ( x , - O ) , f ( x , 0 ) f ( x , - 0)) is absolutely and uniformly convergent on [a, b], since the total variation of B(x, x , ; c, d) is at most I c 1 d-c The series will consequently converge to a function whose only discontinuities are at x = xrL,the discontinuities matching those of f ( x ) . It follows that the function f , ( x ) =f ( x ) - ZrlB(x,x,; f ( x , ) f ( x , - 0), f ( x , 0 ) - f ( x , - 0)) is continuous on [a, b]. This gives rise to another decomposition of a function of bounded variation and we have :
+
+
+
+
+
+
+
+
+I
I.
+
7.1. THEOREM. Iff(x) is of bounded variation it can be written in the form f ( x ) = f , ( x ) fb ( x ) , where f,( x ) is continuous on [a, b] and &(x) is a pure break function, a uniformly convergent sum of simple break functions.
+
8.
41
SPACE OF FUNCTIONS OF BOUNDED VARIATION
It is possible to define f , ( x ) directly in the form:
f,fx)
c
=
(f(Y
a 5 y t x
where f ( a
-
0)
+ O!
-f(Y
- 0))
+f(x) - f f x
-
0)
=f ( a ) .
8. The Space of Functions of Bounded Variation
We have already seen that the space of functions of bounded variation on [a, b] is linear. In addition it has the multiplicative property, viz. : If f , ( x ) and f 2 ( x ) are of bounded variation on [a, b], then fl(x)f,(x) has the same property and J: I d(f, * J 2 1) 5 M , I df2 I M 2 I dfl I, where M , = 1.u.b. [ I f i ( x ) 1, x on [a, b]], i = 1,2. For I f,(dlf,(d) - f , ( C ) . f i ( C ) I 5 I f l ( d ) I . 1 f 2 f 4 - . f 2 W I 8.1. T H E O R E M .
s:
IfJC)
+ Jl
I - I f , f 4- f $ )
+
I.
Further, the space of functions of bounded variation has the absolute property, viz. : 8.2. T H E O R E M . If f ( x ) is of bounded variation on [a, b], then
I f ( x ) 1 is also of bounded variation on [a,b] and J: I df I 5 J: I d( 1 f I) I. For IIfW I If(4 II 2 If(4 - f ( c ) I. -
This absolute property together with linearity assures us that with any two functions f , ( x ) andf,(x) the space of functions of bounded variation contains also the greater f l V f 9 ( x ) and the lesser f , A J 2 ( x )of these functions. Here f , V f 2 ( x ) =f , ( x ) if f , ( x ) 2 f J x ) and =JL(x) if f 2 ( x ) 2 f , ( x ) , while for f , A f 9 ( x ) the inequalities are reversed. For, as can easily be verified f , v . f , ( x ) = Q (f,(x) + J L ( x ) I f , ( x ) f,W I) and f , A f , ( x ) = $i + f , ( x ) - If,(.) -fib)I). These statements can be reformulated:
(fib)
+
8.3. T H E O R E M . The space of functions of bounded variation on
[a, b] form a lattice if the order relation f , 2 f, means f , ( x ) >=.f2(x) for all x. 8.4. Definition. A lattice is defined as a set of elements having an or-
der relation ( 2 )defined between some of its pairs of elements, and with any two elements x, y contains also the least upper bound x V y and the greatest lower bound x A y. Here x V y is an element z such
42
11. RIEMANNIAN TYPE OF INTEGRATION
that (a) z 2 x and z 2 y , and (b) if u L x and u L y , then u 2 z, the inequalities being reversed for x A y . It happens that the space of functions of bounded variation is also a lattice if order is defined in a different way, viz., f , ( l ) j , if fi - f, is a monotone nondecreasing function. If we define the interval function F(I) = f ( d ) - f(c) for I = [c, d ] , then f , ( Z ) f . if and only if F,(I) 2 F,(I) for all I of [a, b]. Since the addition of a constant to the function f does not affect the value of the corresponding interval function, it follows that in this ordering two functions are equivalent, if they differ by a constant. We might therefore limit ourselves to functions f for which f ( a ) = 0. The function f is monotonic nondecreasing if f ( 2 ) O . Consequently, in this ordering f V 0, if it exists, is the least monotonic nondecreasing function f,such that LIL 2 Of, for all intervals. As we have seen in 11.6.5, when f is of bounded variation then this functionf, is the positive variation p of f. Similarly, - n = f A 0, so that the total variation Jf I df I = f V 0 - f A 0. For any two functions f, and f, we can show that f, V f,= f, (fl - f,) V 0, and f, A fi =f, (f,-f,) A 0, from which we conclude that the space of functions of bounded variation under this second type of ordering is also a lattice.
+
+
8.5. The space of functions of bounded variation becomes a normed
+
space if we define 11 f 11 = I f ( a ) I J: I df I, so that the distance between two functions f, and f, can be defined d(f,, f,) = I f,(a) f,(a) I J: I d(fl -f,) I. It can easily be verified that the norm and metric properties are satisfied under these definitions. The norm gives rise to a topology in the space of functions of bounded variation via the condition “lim”, f, = f is equivalent to
+
lim,
Ilf, -f II = lim,(If,W - f ( a ) I
+ 1’I a,-f)
II
= 0.
This norm convergence is a rather strong type of convergence. In the first place it is obvious that limn 11 f, - f 11 = 0 implies limnf,(a) = f ( a ) . Then from
I f , ( X ) - f (x) (f,(a) - f (a) I 5 If,W -f(a) I + J; I d(f, - f l I 5 Ilf, -flL -
it follows that lim, f , ( x )
=f(x)
uniformly on [ a $ ] . Also from
8.
43
SPACE OF FUNCTIONS OF BOUNDED VARIATION
JI
I
Jz
1
it follows that lim, Idf, = Idf uniformly on [a, b]. However, it is possible to have lim, f,(x) = f ( x ) and lim, J: I df, I = J: I df I uniformly on [a, b] without having limn s“, d(fn - f) I = 0. For if on [0,1] we take f,(x) = m/n for ( m - l)/n < x 5 m/n, m = 1,...,n, and f,(O) = 0, then the f,(x) are monotone nondecreasing, I df, I = f,(x), and limnf,(x) = x uniformly on [0, 11 but J,’ I d ( f , - f) I = 1 for all n. Under the norm as defined, the space of functions of bounded variation is complete, i.e., we have:
I
8.6. THEOREM. If for the sequence of functions fn(x) of bounded variation on [a, b] we have lim ] I f , - f , 11 = 0, then there exists a
function f ( x ) of bounded variation such that lim, ( 1 f - f n 11 = 0. Reasoning as above, we find that lim,,f,(x) exists uniformly on [a, b] and is equal to f ( x ) say. Also, lim, I df, I exists. Now by 11.5.4 we have J: I dfl 5 limn I dfn I = lim, I df,, I, since we are dealing with integrals of upper semiadditive interval functions. Then f is also of bounded variation. By the same theorem we have d ( f - f,) I 5 grn J: I d ( f , - f,) I. Now by the hypothesis of our theorem we have (e, n,, m 2 me, n 2 n,) : J: I d ( f , -f,) I 5 e, so that for n 2 n,: I d(f, - f) I 5 e. As a result of these considerations we have
Jl
Jl
Jfl
Jl I
Jl
limn Ilf, - f l l
= lim,(IAzW
-ffa)
I + I d ( f , - f l I)
= 0.
8.7. From the above considerations, it is apparent that in the space of
functions of bounded variation, there is a variety of modes of convergence. We list the following:
+
(1) Norm convergence: limn 11 f, - f 11 = limn (1 f,(a) - f ( a ) I J: I d f f , -f) I) = 0. (2) Double uniform convergence: lim,f’(x) = f(x) uniformly on [a, b] and lim, I df, I = J,”I df I uniformly on [a, b]. It should be noted that uniform convergence of a sequence of functions of bounded variationf,(x) to f(x) does not necessarily imply thatf(x) is of boundx 5 l/n; ed variation. For instance, the functions f,(x) = 0 for 0 I = x sin (n/x) for l / n 5 x 5 1 are each of bounded variation on [0, 11, converge uniformly to f ( x ) = x sin (n/x) which is not of bounded variation on [O, 11. (3) limnf,(x) = f ( x ) for all x of [a, b] and limn J: I df, I = 1 df I and SO limn I df, = J,”[ df I for all x of [a, b]. The last remark is
JI
JE
I
s:
44
11. RIEMANNIAN TYPE OF INTEGRATION
a special instance of 11.5.6, since the interval function F(I) f(c) I is upper semiadditive on [ a,b].
=
If(d) -
(4) lim,f,(x) = f ( x ) for all x of [a, b ] ; J: 1 dfn I bounded in n. The second half of this condition is usually stated: the sequence f, is uniformly of bounded variation, where :
5 is uniformly of bounded variation if there exists a constant M such that I dfl 5 M for all f of 5. Since by theorem J: I dfl 5 limn d f , it follows that in this type of limit, the limit function f ( x z necessarily also of bounded variation, i.e., this type of convergence operates within the space of functions of bounded variation. It is sometimes referred to as weak convergence. 8.8. Definition. A set of functions
Jl
JI
(5) lim,f,,(x) = f ( x ) except at a denumerable set of points of [a, b ] ; J: I df,( I is bounded in n and J: 1 df 1 < 00. In connection with the limit (4)we have an important theorem due to E. Helly which is in a sense an extension of the Weierstrass-Bolzano theorem on the space of real numbers, to functions of bounded variation. 8.9. Helly’s Theorem. If f , , ( x ) is a sequence of functions, uniformly
of bounded variation on [a, b] such that f,(a) is bounded in n, then there exists a subsequence f,,(x) and a function f ( x ) of bounded variation such that lim, f,,(x) = f ( x ) for all x of [a, b]. The proof of this theorem depends on the following lemma, which is basic in theorems of the Weierstrass-Bolzano type : Let G be the space of infinite sequences of real numbers: S = (al, ..., a,, ...) = {a,}. Then if S,, = (a,,,, ..., a,,p,...) is any sequence of such sequences such that for every p the a,rpare bounded in n, then there exists a subsequence S,,m and an S such that lim,S, rn = S in the sense that for every p : limma,wL,= up. The proof of this lemma is based upon the use of the WeirstrassBolzano theorem in the form: if a,, is any bounded sequence of real numbers then there exists a subsequence a,,,n and a number a, such that lim,anm = a. This form is equivalent to the topological statement : any bounded set of real numbers has at least one limiting point. is bounded, there exists a subsequence Since the sequence a ( l J 1 such that limm a?L”)L( 1 1 1 = a,. For the sequence ~ , m, I ~ I there ~ , exists a subsequence n::’ of TI:,:’ such that limma7L;]2 = a2. Continuing in this hl
,111
8.
SPACE OF FUNCTIONS OF BOUNDED VARIATION
45
way we get a succession of subsequences of the integers:
... , ...
ny) , nil , ..., n:,:) , (01
ny’ , n‘,“’, ..., n , ,
...
ny’ , nil , ..., n, , (TI
...
... each row a subsequence of the preceeding row and such that lim7nan9’p = a, for all r 5 p . If we let r increase indefinitely, this set of sequences might have no sequence of the integers in common. In order to get a sequence effective for our purpose, we use the diagonal procedure, i.e., we consider the sequence of integers nfl, r = 1, 2, ... Then for r 2 m, nf) will be a subsequence of ni;). Consequently, limr a p p = a,, since the convergence will depend only on the terms for which r > p . This statement is equivalent to limlS,,(71 = S in terms of coordinater wise convergence. Returning to Helly’s theorem, we consider first the special case when the functions are monotonic nondecreasing, i.e., assume that f , , ( x ) is a uniformly bounded sequence of monotonic nondecreasing functions on [a, b]. Let {x,,} be a denumerable dense set of points of [a,b] including a and b. Then { f , , ( x l , ) }= S,,is a sequence of sequences, satisfying the conditions of our lemma so that there exists a subsequence n, of the integers / I such that lim, f (x,) exists for all p . Denote these limits by f(x,,). Now ,f(x) is monotonic nondecreasing on the xl,, i.e., x, 5 x,‘ implies f(x,) 2 f ( x , ’ ) . Consequently, we know that the function f can be extended to a monotonic nondecreasing function defined for all points x for which f ( x - 0) = f ( x 0) = . f ( x ) as determined by the x,. It now develops that if f(x, 0) = f ( x , - 0) = f(x,,), then f,,,(x,) converges to f ( x , ) . For we have (e > 0, d,, 0 < x, - xg < d,) :f ( x , ) 2 f ( x , ) - e. For a particular such p , we take m 2 m e p so that f(x,) >f,,,,(xp) - e. Then for the same m’s: J1711
+ +
46
11. RIEMANNIAN TYPE OF INTEGRATION
This observation assures us that limmfinm(x) = f(x) except possibly at the points of discontinuity of f(x). Let us denote by xp’ this at most denumerable number of discontinuities of f(x). Then by applying the lemma to f,m (x,‘) we get a subsequence f,.,of functions such that limkfn.,(xp’) exists for all p . Consequently lim,fn,7t(x) exists for all x of [a, b]. The final step is almost self-evident. We write A,(.) =f,(a) + P,(x) - N,(x), where P,(x) and N,(x) are the positive and negative variations of the functions f,(x), respectively. If f,(x) is a sequence of functions uniformly of bounded variation, P,(x) and N,(x) will be uniformly bounded sequences of monotonic nondecreasing functions. We can then pick a subsequence n , of the integers so that f,m (a), Pnm(x), and Nflm(x) converge for all x and consequently f,m (x) = f,m (a) P, m (x) - Nnm(x) will converge for all x. In some of the appliations of this theorem it is sufficient to know that the subsequence fnrn (x) converges to a function of bounded variation f(x) excepting at a denumerable set of points. As a corollary we have an alternate form of this theorem: =f(x,),
+
8.10. T H E O R E M . If
5 = [f(x)] is an infinite set of functions of bounded variation on [a, b],which are uniformly of bounded variation, then there exists a sequence of distinct elements {f,} of 5 and a function g of bounded variation such that lim,.f,(x) = g(x) for all x of [a, b]. 8.11. Remarks on Compactness. The word “compact” as applied to
a set of elements in an abstract topological space was introduced by FrCchet in his thesis [Sur quelques points du Calcul Fonctionnel, Rend. Circ. Palermo 22 (1906) 1-74] in connection with spaces in which the notion of a limit of a sequence is defined. According to FrCchet limits of sequences of elements have three properties: (1) lim,x,, if it exists, is unique; (2) if x , ~= x for all n then limnx,,= x ; and (3) if lim,x, = x and nK is a subsequence of the integers n then lim,x,,, = x. In such a sequential-limit space, a limiting element of a set E has a sequence of distinct elements of E converging to it. A set E is compact if every infinite subset has at least one limiting element. As a consequence a set E of elements is compact if and only if from every sequence {x,} of elements of E we can extract a subsequence {x,m } having a limit. If we define convergence of a sequence of functions of bounded
9.
47
RIEMANN-STIELTJES INTEGRALS
variation by means of pointwise convergence, then the Helly theorem asserts that a subset of the set of functions of bounded variation is compact if it is uniformly of bounded variation. The lemma used in the proof is equivalent to the statement that in the space of all sequences of real numbers, where limit is defined by coordinatewise convergence, a set of such sequences is compact if each coordinate value is bounded (or compact). Recently the word " compact " has been applied to sets in a topological space, provided a form of the Borel theorem applies, i.e., any covering of the set by open sets can be reduced to a finite subset of the covering sets. To distinguish between these two types of compactness, one might label the one tied to the existence of limiting elements W-B (Weierstrass-Bolzano) compactness and the covering type B (Borel) compactness. We state without proof the theorem: In a metric space, in which convergence is related to a metric 6(x, y ) in the sense that lim,,x, = x is equivalent to limn6(x,, x) = 0, a set is B-compact if and only if it is W-B-compact and closed.
EXERCISES 1. Show directly from the definition, that if f ( x ) is of bounded variation on [a, b], then- f(x + 0) and f(x - 0) exist for all a < x < b, .f(a 0) exists at a and f(b - 0) at 6.
+
2. Show that if f ( x ) has only discontinuities of the first kind on [a, b], i.e., f ( x 0) and . f ( x - 0) exist for all x of [a, b] as finite numbers, then f ( x ) is bounded on [a, b] and has at most a denumerable number of discon-
+
tinuities. 3. Show that the indefinite integral of a Riemann integrable function f ( x ) : g ( x ) = f ( x ) d x is of bounded variation and that I dg(x) I = I f(x) I dx. 4. If f q ( x ) is a directed set of functions of bounded variation such that lim,f,(x) = f ( x ) for x on [a, b], is it true that I df I 5 lim, - J: 1 df, I ?
s:
s,"
S:
S:
9. Riernann-Stieltjes integrals
The Riemann-Stieltjes (R-S) integral is an integral of functions of intervals depending on two point functions: f(x) and g ( x ) with F(I) = F[c, d! = f(x) (g(d) - g ( c ) ) , c 5 x 5 d. Stieltjes, following
48
11. RIEMANNIAN TYPE OF INTEGRATION
Cauchy, considered the case where f ( x ) is continuous and g ( x ) o f bounded variation on [a, b] and was content to note that for this combination of functions the function of subdivisions
converges as I a I + 0. Since Riemann, the emphasis has been more in the direction of determining for what combinations of functions f ( x ) and g ( x ) a given mode of convergence leads to an integral. As in the case of functions of intervals, we have for Z t , f ( x L '()g ( x z ) g ( x t - J ) , the two modes of convergence, viz., as I a I + 0, leading to the norm integral: N J fdg, and by successive subdivisions or refinements, leading to the a-integral: a J fdg. Both integrals will be called Riemann-Stieltjes (R-S) integrals. In the early part of this section, no restrictions beyond the existence of the integrals will be placed on the functions f and g . Later we shall consider the special case when g ( x ) is limited to being a function of bounded variation on [a, b]. We note a few examples of Stieltjes integrals. Iffix) = c on [a, b], then for all functions g ( x ) , J: cdg = c ( g ( b ) - g ( a ) ) . If g ( x ) = c on [a, b], then J: f d g = 0 for all f ( x ) . If g ( x ) = x , then the RiemannStieltjes integral becomes a Riemann integral. If g ( x ) = B ( x , x,,; c, d ) the simple break function with value c at x = x , and break d at x = x,, then the only contributions to 2 J A g are the terms which come from points in the vicinity of x,. If a does not include xo, then G(a) = f ( x ' ) d , where x 1 5 x' 5 x 2 , x, and x , being the points of a immediately adjacent to x, on either side. If a includes x,, then G(a) = f ( x " ) c f ( x " ' ) ( d - c), where x , 5 x'' 5 x, and x , 5 x'" 5 x,,. If the norm integral exists these two expressions must have the same finite limit: f ( x , ) d as x',X I ' , and x"' approach x,. If f ( x ) is discontinuous at x , i.e., lim,,,o f ( x ) ;tf(x,), then d = 0, and so also c = 0, since x'' and x"' are independent of each other. Then g ( x ) is continuous at x = x,. Consequently, if g ( x ) is discontinuous, then f ( x ) must be continuous at xo and N J l f d g = f ( x , ) d = f ( x , ) ( g ( x , 0 ) - g ( x , - 0)). If the a-integral exists, then x,, can eventually be included in a, so that
+
+
fimi, -,,+off(x")c + f ( X ' " ) ( d - c)) ,"'+z"-O
+
must exist and be equal to f(x,)c f ( x o ) ( d - c) = f ( x , ) d . If f ( x ) is discontinuous on the left, then since x'' and x'" are independent, it follows that c = 0, i.e., g ( x ) is continuous on the left. Iff(x) is
10. EXISTENCE THEOREMS FOR R-S INTEGRALS
49
discontinuous on the right, it follows that c - d = 0, i.e., g ( x ) is continuous on the right. Consequently, if a fdg exists, then f ( x ) and g ( x ) = B(x, x o ; c, d ) do not have discontinuities on the same side of xg and Syk =f(x,)d =f(x,) M x , 0) - g ( x , - 0)). A kind of geometrical interpretation of the Stieltjes integral is as follows: In three dimensions, = f ( x ) is a cylinder on the xy-plane with elements parallel to the z-axis, and z = g ( x ) is a cylinder on the xz-plane with elements parallel to the y-axis. Consequently, J ydz is the (signed) projection onto the yz-plane of the area on the cylinder y = f(x) cut off by the cylinder z = g ( x ) .
s:
+
JJ
9.1. We note the following variants of the basic interval function for
R-S integrals, which in some cases lead to effective integrals of Stieltjes type : (1) F ( I ) = f ( c ) ( g ( d ) - g ( c ) ) ; left Cauchy integral; (2) F(I) = f(d) ( g ( d ) - g ( c ) ) ; right Cauchy integral; (3) F ( I ) = 8 (f(c) +f(d)) ( g ( d ) - g ( c ) ) ; mean integral; (4) F(I) = f ( x ) ( g ( d ) - g(c)),c < x < d ; modified integral; (5) if g ( x ) has only discontinuities of the first kind (or breaks): F ( I ) = f ( c ) (g(c 0) - g ( c ) ) f(x) ( g ( d - 0) - g ( c 0)) f(d) ( g ( d ) - g ( d - 0 ) ) , with c < x < d. Both modes of convergence can be considered in each of these cases.
+
+
+
+
10. Existence Theorems for Riemann-Stieltjes Integrals
The theorems of 11.3 for integrals of interval functions yield the following : 10.1. If for two functions f(x) and g ( x ) on [a, b],N a J: f d g exists.
J’:
10.2. A necessary and sufficient condition that N
f d g exists, then
J: f d g ( a S: f d g )
exist is that the corresponding Cauchy condition of convergence hold for the approximating sums Z, fdg. 10.3. A necessary and sufficient condition that N
J: f d g
(0
7:
f,f d g )
exist is that the corresponding upper and lower integrals fdg and S3 f d g be finite and equal. -a Note that these upper and lower integrals are defined as greatest
50
11. RIEMANNIAN TYPE OF INTEGRATION
and least of limits, and not in terms of upper and lower bounds of the function f on intervals of subdivision. 10.4. A necessary and sufficient condition that N J: fdg (cr J: f d g ) exist is that N J: w ( S f d g ; I ) = 0 (0 w ( S f d g ; I ) = 0 ) . Since w ( S f d g ; I ) 2 w ( f , I ) I g ( I ) I, where w ( f ; I ) is the oscillation off on I = [c, d ] and g ( I ) = g ( d ) - g ( c ) , we have:
Jl
J: f d g exists, then N J: w ( f ; I ) Ig(I) I = 0 ; if cr J: f d g exists, then cr J: w ( f ; I ) I g ( I ) I = 0 J or] equivalently 10.5. Corollary 1. If N
g.l.b.,Z, w ( f ; I ) Ig(I)
1 = 0.
This leads to: 10.6. Corollary 11. If N Jf:f d g exists, then f and g have no common discontinuities on [a, b]. If u Jf:f d g exists, then f and g have no common discontinuities on the same side of any point of [a, b]. We prove the second part of this corollary first. Suppose u Jffdg exists. Then by Corollary I (e, ere, u 2 ae) : Zuw ( f ; I ) g ( I ) I < e. If x, is a point of discontinuity of f ( x ) on the left, then by including x, in cr it follows that w(f; [x', x,]) I g ( x ' ) - g(x,) I < e where x' is the point of cr immediately to the left of x,. Since we can add to cr any point between x' and x,, it follows that ~ ( f[x,, ; - d, x,]) I g(x,) g ( x , - d ) I < e for 0 < d < x,, - x'. Since xo is a point of discontinuity o f f on the left, there exists e' > 0 such that 0 < d < x, - x' implies w ( f ; [x, -. d, x,']) > e', so that I g(x,) - g ( x , - d ) I < e/e'. Consequently, limx4xo-,, g ( x ) = g(x,), or g ( x ) is continuous on the left at x,. If then g is discontinuous on the left, f ( x ) must be continuous on the left at x,. The reasoning for the right hand is similar. If N J: fdg exists, then (e, d,, I u I < d,) : Zuw(f, I ) g ( I ) < e. Let x, be a point of discontinuity of f ( x ) . Since cr J: fdg exists if N f d g exists, it follows that g ( x ) is continuous either on the left or the right of x,. Further, w(f; [x, - d, x, d ' ] ) Ig(x, d ' ) g ( x , - d ) I < e if 0 < d, d' < d,. Since f ( x ) is discontinuous at x,, there exists e' > 0 such that w ( f ; [x, - d, x , + d ' ] ) > e' for 0 < d, d' < d,, and so also 1 g ( x , d ' ) - g ( x , - d ) I < e/e'. Consequently,
I
I
Jl
+
1
+
+
linI(d,&)+(+o, + 0) k f x ,
+ d ' ) - g ( x , - d ) ) = 0.
Since g ( x ) is continuous on either the left or the right, it follows that lim,+,og(x) = g(x,). Then if g ( x ) is discontinuous at x,, f ( x ) must be continuous.
10. EXISTENCE THEOREMS FOR
51
R-S INTEGRALS
We note that: 10.7. If either f ( x ) or g ( x ) is continuous at x,, then the interval func-
tion F(I) = F[c, d ] =f ( x ) (g(d) - g ( c ) ) is pseudoadditive at x , provided f and g are bounded in the neighborhood of x,. This follows at once from the identity: F[c, dl - ( F k , XOI F[x,, dl) = ( f ( x / ) - f ( x " ' ) ) ( g ( d ) - g(x0)) (f(x') - f ( x " ) ) M X , ) - g ( c ) ) with c 5 x' I d, c 5 x" 5 x,, c, 5 x"' S d.
+
+
10.8. Conversely, if F(I) = f ( x ) ( g ( d ) - g ( c ) ) is pseudoadditive at x,, then either.f(x) or g ( x ) is continuous at this point. For if g ( x ) is discontinuous on the right, then there exists a sequence xTL+x o 0 such that I g(x,) - g(x,) I > e, for some fixed e > 0. If in the expression for F[c, x J - (F[c, x,] F[x,, x,]) we take x" = x r = x,, and then let c + x , - 0 and x,+ x , 0, we conclude that limzL."f+l,+o f ( x " ' ) = f ( x o ) . If we take x r r r= xo, and x" = x', then we conclude that liml, r,--O f ( x ' ) = f ( x ) . Thenf(x) is continuous at x,. The same reasoning, by parity, applies if g ( x ) is discontinuous on the left. If we take into account 11.3.10, we obtain:
+
+
+
-
10.9. T H E O R E M . If f ( x ) and g ( x ) are bounded on [a, b], then a necessary and sufficient condition that N JBfdg exist is that a J: fdg exist and f and g have no common discontinuities on [a, b]. If either f ( x ) or g ( x ) is a continuous function on [a,b], then N J: fdg exists if and only if a Jlfdg exists, and they are equal. In particular it is immaterial whether the Riemann integral J: f ( x ) d x is defined as a norm or a a-integral. This is essentially the burden of the Darboux theorem. As an additional necessary condition for the existence of J: fdg we have :
10.10. T H E O R E M . If f ( x ) and g(x) are such that J: fdg exists (in either sense), then there exists a finite number of closed intervals of [a, b ] : [Il,...,I,] such that f ( x ) is bounded on each Ik, and g ( x ) is constant on the closed complementary intervals : 11',..., Consequently, f f d g is zero on the intervals Ik', and the value of fdg is independent of the values assumed by f ( x ) on these intervals. Suppose a fdg exists. Then by 11.10.5: (e, ae, a 2 ae) : E0w(f; I ) I g ( I ) I < e , so that w ( f ; I ) I g ( I ) < e for each I of a. Let ue
s:
Jl
I
52
11. RIEMANNIAN TYPE OF INTEGRATION
consist of the points a = x , < x , < ... < x,, = b, and suppose that f i s unbounded on [xk,xk+,].Then ~ ( f I ;) is cc on this interval so that g ( I ) = g ( x k i 1 ) - g(xk) = 0. Let x' be any interior point of [xk,xk+,]. Then either w ( f i [xk,x']) or w(f; [x',xk+!])is infinite. Consequently, g ( x ' ) = g(xJ or g(xk,.,), so that g ( x ) is constant on [x,, xk+J Since there are only a finite number of intervals in ae,this proves the theorem for a J:fdg and consequently also for N J:fdg. The difference between the cr and norm integral is that for the latter any point of unboundedness off(x) [i.e., in every neighborhood of which f ( x ) is unbounded] must be interior to one of the intervals of constancy of g ( x ) , while for the a-integral it may be an end point. Since the value of the integral J:fdg is unchanged iff(x) is altered on an interval of constancy of g ( x ) , we might assume that f(x) is linear on such intervals, and restrict ourselves to the case where f ( x ) is bounded on [a, b]. The function g ( x ) is subject to the same boundedness conditions as f ( x ) . This can be shown easily by using the integration by parts theorem below.
+
11. Properties of Riemann-Stieltjes Integrals 11.l.The integral
J: fdg is a bilinear function in f and g , in the sense
that if for f , ( x ) , ...A,(x ) and g , ( x ) , ...g,(x) the integrals J:f,(x)dgi(x) exist, and c ,,...,cr,; d ,,..., d , are constants, then forf(x) = ,c,f,(x) and g ( x ) = q = , d , g z ( x ) the integral J:fdg exists and is equal to zzj czdi .f,dgi. This follows from 11.5.1 and 11.5.2.
c=
Sl
11.2. If S:fdg exists, then for a 5 c 5 d 5 b: J:fdg exists and J,fdg =
J:fdg is an additive function of intervals. This follows from 11.4.1. 11.3. The continuity properties of the function h(x) = J:fdg are determined largely by those of g ( x ) . In particular, if g ( x ) is continuous at x,, then h ( x ) is continuous at x,. If g ( x , - 0) exists, then h(x, - 0) exists and h(x,) - h ( x , - 0) = f ( x , ) (g(x,) - g ( x , - 0 ) ) . Similar statements hold for g ( x , 0) and h(x, 0). For if x , < x,, then
+
I
s"% J1
-ffx') MX,)
- g(x,))
+
I 5 4SfAg;
[ X I , X"1A
and the left-hand side approaches zero as x l + x , - 0. If g ( x ,
- 0)
11.
53
PROPERTIES OF R-S INTEGRALS
exists and is not equal to g ( x , ) , then g ( x ) is discontinuous on the left and f ( x ) is continuous on the left at x,. Then To
limL,+To-oJ * p g = limll
11.4. Approximation Theorem. If
1 J: f d g
-
11.5. Convergence Theorem. If limltAl(x) = f ( x ) for all x of [a, b],
and J:f,,dg exists uniformly in the sense that lim C G i w ( S f 4 g ;I ) = 0 uniformly in n, then limp!J:f,,dg and fdg both exist and are equal. The integrals and corresponding limit are assumed to be the same throughout. This is an immediate consequence of the iterated limits theorem 1.7.4.
Jt
11.6. Substitution Theorem. If f ( x ) is bounded on
[a, b] and k ( x )
=
J : g ( x ) d h ( x ) , then f f ( x ) d k ( x ) exists if and only if J: f ( x ) g ( x ) d h ( x ) exists and the two integrals are equal. This is a special case of the substitution theorem for integrals of interval functions 11.4.8 and can also be proved directly. The integrals are assumed to be of the same type throughout. 11.7. Integration by Parts Theorem. If
Js
f d g exists, then gdf exists and J: fdg = f( b ) g ( b ) - f ( a ) g ( a ) - gd!: The proof of this theorem depends on the following identity:
J:
54
11. RIEMANNIAN TYPE OF INTEGRATION
provided I a I = max ( x i - xi...,) < dJ2, since then max (xi' - xliPl) 5 2 max ( x i - xi-,) < d,. Then N fdg = f(b)g(b) -f ( a ) g ( a ) N s", gdf. If a J fdg exists, then (e, a,, a 2 a,) : I J: fdg - ZafAg I < e. Select a a 2 a,. Then
Jz
=
I h2-0
=
I i; ( f f x , ) (gfx'*+J- g f x , ) ) +fW(g(x,)
,
) fgfx',+J - g(x,'))
-p
d g
I -
2=0
gfx,'))- f f d g
I<
e
since a' = (a = xo5 xl' 5 x , I x,' I ... 5 x',+~= x, = b) includes t~ and so a' 2 ae.Consequently, a s", gdf exists and is equal to f ( b ) g ( b ) - f ( a ) g f a ) - Stfdg. There is a slightly more complicated proof for the case of the a-integral. From ZafAg = f ( d ) g ( d ) - f(c)g(c) - Zarg d f , with a = (c = xo,x,,..., x, = d) and a' = (c = xo', x,', ..., XI,+, = d), it follows that ZalfAg - Zaz fAg = Za,z gAf - COg1 gAf, where 0, and ol'and az and u2' are related subdivisions of the interval I = [c, d]. Then o(SfAg; I ) 5 w(SgAf; I ) 5 w(SfAg; I ) for every I, and equality holds. If then a s", w(SfAg; I ) = 0, then a s", w(SgAf; I ) = 0 and the existence of a s", fdg carries with it the existence of a J:gdf. Let (T = a,+ a,', where (e, ae, a 2 ae) : I f fdg - ZafAg I < e, and (e, ae', a 2 ue') : I J:gdf - ZagAf I < e. If in Za fAg, we take x,' = xi-,, then n
C f ( x , - , ) M X , ) - g(x,-J) = f ( b ) g f b ) - f(a)g(a) 2-1
-
2 g f x , ) f f f x , ) -f(x,-,)). 2-1
Then for every e > 0:
Ip
g
+ j;g4f - ( f f b ) g f b )- f f a ) g f a ) )
I < 2 e.
The integration by parts formula follows. Since for any function f ( x ) and any x of [a,b] we have f ( x ) = c d f , we have
-f ( a )
11.
55
PROPERTIES OF R-S INTEGRALS
Then the integration by parts formula can be written f b
or :
. a
( f ( x ) -f(b))dg(x)
=
-c" ( d a ) - g(x))df(x) 0
jhdf(x) jxM Y ) = j:&(Y) jhdf(x).
11.8.
If we assume that J: f,(x)df(x) and substitution theorem gives us
1; jxfl(xidf(x)gl(Yids(Y)
=
Jg
g,(y)dg(y) each exist, then the
jbJ' g,(Y)dg(Ylfi(x)df(x). (I
Y
Because of the linearity properties of the Stieltjes integral, we can extend this linearly in the sense:
11.9. THEOREM. If J:f,(x)df(x) and J: g,(y)dg(y) exist for i ..., n and if h(x, y) = 2rE1 f,(x)g,(y), then
j'df(x) 1: h(x, Y)&(Y)
=
?':
ddY)
Jb
=
1,
h(x, Y)df(X).
This is a special case of the so called Dirichlet formula.
EXERClSES 1. Which of the theorems or properties holding for Riemann-Stieltjes integrals are also valid for variants of the interval functions for Stieltjes integrals mentioned in 11.9.1? 2. Show that if
J: fdg exists, then g ( x ) is bounded on a set E of a finite
number of nonoverlapping subintervals of [a, b], f ( x ) being constant on the complement of E relative to [a, b]. 3. Is the substitution theorem still valid if the boundedness condition on f ( x ) is dropped? 4. Show that if f ( x ) is bounded and
l: f ( x ) h ( x ) d g ( x ) Jb
exist, then
f(x)g(x)dh(x)
+
Jb
S:
s:
s:
g ( x ) d h ( x ) , f ( x ) g ( x ) d h ( x ) and g(.-c)h(x)df(x) exists and
f(x)hfx)dg(x)
+ jbg ( x ) h ( x ) d f ( x )
= f ( b ) g ( b ) h ( b ) - f(a)g(a)h(a). 5. Show that if for a function .f(x) on [a, b],
sg ,fdf exists, then J: fd'=
1 2 ( ( f ( b ) ) 2 - (f(a))2). If n > 2 , and an integer, is it true that if
exists then
S:
f "df = ( ( f ( b ) ) " + ' - ( f ( a ) ) " + ' ) l ( n
+ I)?
c, f " d f
56
11. RIEMANNIAN TYPE OF INTEGRATION
12. Classes of Functions Determined by Stieltjes Integrals
Any given function g(x) on [a,b] determines a class 5 of functions f(x) such that $:fig exists for all f of 5. For instance, the function g(x) = x determines the class of Riemann integrable functions, the break function p(x, xo;d,, d,) with d, # 0 and d, # d, determines the class of functions f(x) continuous at x = xo.Any such class of functions is linear. In the same way, a fixed function f(x) determines a linear class @ of functions g(x) such that fdg exists for all g of B. A class B of functions g(x) determines a class 5 of functionsf(x) such that J: fdg exists for every g of B and every f of 5. The class of functions 5 is linear and Jf:fdg is bilinear on the product of the two classes 5 x Bl,where Bl is the linear extension of B, or the smallest linear class containing @. A similar statement is possible if 5 and B are interchanged. In the history of the theory of integration, the class of continuous functions has had an honored role. As a matter of fact, it is often taken for granted that any integration process on a linear interval should make every continuous function f(x) integrable. This raises the question of characterizing the class B of functions g(x) such that fdg exists for every function f(x) continuous on [a,b] and every g of B. Since f(x) is continuous, it is immaterial whether the norm or the u-integral is involved. The following theorem is an answer to the question :
$1
$:
12.1. THEOREM. A necessary and sufficient condition that $:jag exist for every function f(x) continuous on [a,b] is that g(x) be of bounded variation on [a, b]. The sufficiency part of the theorem goes back to Stieltjes. We apply Theorem 11.10.4. If g is of bounded variation, then a brief calculation shows that w(Sfdg; I ) 5 w ( f , I ) - v ( I ) , where v ( I ) = J I I dg I. The continuity and consequent uniform continuity of f(x) gives (e, d,, 1(I) 5 d,) : w ( f , I ) I e. Then if I u I 5 d,,Zow(SfAg;I ) 5 e Zmv(I)= e * v[a,b], and f f d g exists. The necessity part of this theorem is a little more complicated. We have the important lemma: 12.2. Lemma. If a, h 0, and Znun is divergent, then there exist constants c, > 0 such that lim,c, = 0, and Z,c,a, is still divergent. Crudely expressed, this lemma states that there is no last divergent series of positive terms.
12.
57
CLASSES OF FUNCTIONS
If s, = Pm = a,, then c, ditions of the lemma. For
=
l/s, is a sequence satisfying the con-
Since s,,+co, for any n we can select m so large that s,-l/s,+,, < 9. It follows that Znan/s, violates the Cauchy condition of convergence and is divergent.. Our lemma is a special case of the Abel-Dini theorem, which states that if a,, 2 0, and Zna, is divergent, then Z,La,/s~ is divergent if k 5 1 (which follows at once from the above lemma), and convergent if k > 1. The harmonic series with a, = 1 for all n, is a special case. A similar theorem is true if a, 2 0 and Znan is convergent; if r , = Cz=,a,, then Znan/riis convergent if k < 1 and divergent if k 2 1. So there is no last convergent series of positive terms. For details the reader is referred to K. Knopp: “Infinite Series,” 1928, 6 39. We also need the following local property of functions of bounded variation : 12.3. THEOREM. A function g ( x ) is of bounded variation on [a, b]
if and only if for every x of [a, b], there exists a vicinity [ x - d,, x d,] such that f(x) is of bounded variation on [ x - d,, x d,], ( d , = 0 if x = a, and d , = 0 if x = b ) . The “only i f ” is obvious. The “ i f ” follows from the Bore1 theorem. Returning to our theorem, we proceed contrapositively and assume that J: fdg exists for all continuous functions f ( x ) but g ( x ) is not of bounded variation on [a, b ] . Then there exists a point x, of [a, b] such that g ( x ) is of infinite variation on every interval [x, - d,, x, d,], and consequently of infinite variation either on [x, - d,, x,] for all d , or on [x,, x, d,] for all d,. Assume that the former case holds. We can then find a monotone nondecreasing sequence of points x , approaching x, on the left, such that 2 , l g ( x n + J -g(x,)l =co. Consequently, by the lemma above, there exist c, such that lim,c, = 0 and Zncn1 g(x,+J - g ( x , ) 1 = co. Construct f ( x ) as follows: (a) f(x) = 0 for a 5 x S x,, and for x 2 x,; (b) f ( x , ) = 0 for all n ; (c) f f + , x , + J ) = c,sgnk(x,+,) - g ( x , ) ) ; and (4f(x) is linear between x, and $(x, x,,,) and between $(xn x,+J and x , + ~ .Then f(x) is continuous on [a, b]. If ~7is any subdivision of [a, b] with x i < x, S xlkil, we can add x , and points of x , between
+
+
+
+
+
+
+
58
11. RIEMANNIAN TYPE OF INTEGRATION
xkrand x, to o so that
+ %+,)) (g(%+,) - g(x,,)) >
Af(8(x,
n=m
for any given M . Consequently, limJo fdg does not exist, contradicting the assumption that fdg exists. There is also a theorem if the role off and g is interchanged, i.e., we have :
Jt
12.4. T H E O R E M . If J:fdg exists for all functions g(x) of bounded variation on [a, b], thenf(x) is continuous on [a, b].
For if we set g(x) = p(x, x,; d,, d,), the break function with discontinuity at x = x, and d , f 0, d, f d,, then J: fdg exists if and only f(x) is continuous at x = x,. From these two theorems we conclude that the classes of continuous functions and functions of bounded variation are complementary or adjoint with respect to the Stieltjes integral fdg.
Jt
13. Riemann-Stieltjes Integrals with Respect t o Functions of Bounded Variation. Existence Theorems
In view of the above considerations, Stieltjes integrals with respect to functions of bounded variation play a prominent role, and this is frequently the only case considered. In the following sections we derive additional properties of these integrals when g(x) is of bounded variation. 13.1. T H E O R E M . If g(x) is of bounded variation on [a, b ] , for
Z = [c, d ] , g(Z)
= g(d)
-
I
w ( f , 1) g(1) 15 4 S f A g ; 1)
I
I 1,
g ( c ) and v(Z) = J I dg
then
s 4.L 1) ' v ( l ) ,
I
Here w ( f ; I) g(Z) = w ( f , I ) * v(Z) = 0 if v(Z) = 0. The left-hand inequality is obvious and holds whether g(x) is of bounded variation or not. For the right-hand inequality take any two subdivisions u1 and u, of Z and set o3 = u1 u2. Then
+
=
w(f, I)
- V(Z).
13.
59
FUNCTIONS OF BOUNDED VARIATION
From these inequalities and 11.10.4, we conclude at once: 13.2. T H E O R E M . A necessary condition that J z f d g exists is that J: w ( f , I ) Ig(I) = 0 and a sufficient condition is that cu(J I ) *
Jl
I
v(I) = 0. A stronger result holds: 13.3. T H E O R E M . A necessary and sufficient condition that J: fdg exist is that J: w ( f , I ) v(I) = 0. To prove the necessity, we consider first the case of the a-integral, and show that if a w(" I ) I g ( I ) I = 0, then a J: w ( f , I)v(I) = 0. We can assume that f ( x ) is bounded on [a, b]. Then (e, oe, a 2 ae): Sotr,(f,I ) I g ( I ) 1 I e. Now by definition (e, ae', a L at') : I dg I 2 CJg ( I ) 1 e. As a consequence for a 1a e ae':
J6
+
Jg
+
This proves that a J: o ( f , Z) * v ( I ) = 0. For the norm integral, we note that if N J: fdg exists, then cr J: fdg exists and f and g have no common discontinuities. As a consequence a J: w I ) * ~ ( l=)0 and f and v have no common discontinuities. But then the interval function oJ(f, I ) * v(I) is pseudoadditive at every point, so that N J: I ) . V(I) = 0 (o(f, 1) . V ( I ) = 0. Since for g ( x ) monotonic nondecreasing to(Sf3g; I ) = w ( f , I ) * g ( I ) , and v ( x ) is monotone, the import of this theorem is:
(L
(li(f,
s:
13.4. Corollary I. A necessary and sufficient condition that f: fdg exist for g of bounded variation is that J: fdv exist. Since, moreover, P ( x ) = $ ( v ( x ) g ( x ) - g ( a ) ) and N ( x ) = (~(x) ( g ( x ) - g ( a ) ) ) , we have:
4
+
13.5. Corollary II. If g is of bounded variation, then
Jl
J: fdg exists
s"
if and only if J: f d P and J: f d N exist and then J: fdg = f d P - fdN. It follows that the definition of Stieltjes integrals with respect to functions of bounded variation could be based on that of integrals with respect to monotonic nondecreasing functions. For the case where g ( x ) is a monotonic nondecreasing function, and f ( x ) is bounded, we have for any fixed subdivision a : (I ,
60
11. RIEMANNIAN TYPE OF INTEGRATION
This means that if g ( x ) is monotone then the upper and lower a-integrals off with respect to g, are definable in terms of upper and lower sums and their respective greatest lower and least upper bounds, that is, are of the nature of Darboux integrals. An additional corollary of our existence theorem is: 13.6. Corollary 111. If g , and g , are of bounded variation on [a, b] and if for every subinterval [c, d ] , we have J: I dg, [ I I dg, 1, and if f d g , exists, then J:.fdg, exists also. For functions of bounded variation, the approximation theorem takes the form:
Jl
Jl
13.7. Approximation Theorem. I f f is of bounded variation and u
is any subdivision-of [a, b] and if J: f d g exists, then
I
Jb
fdg
-
C f ( X i ’ , (g(x,)
-
a
gk-,)) I5
c 4f, I ) v(I)
’
Obvious is: If g is of bounded variation on [a, b ] and if J: fdg exists, then J-: f d g IS M J: dg where M is the 1.u.b. of f on the totality of intervals complementary to those for which g ( x ) is constant. The decomposition of a function of bounded variation in the form g ( x ) = g,(x) g,(x), where g,(x) is the continuous part of g ( x ) and g,(x) is the break or saltus function of g leads to:
I
I 1,
+
13.8. THEOREM. If g ( x ) is of bounded variation on [a, b ] and
exists, then jlfdg
Slfdg, and
= ffdg,+lbfdg,
where g ( a
-
0)
= g(a)
S: f d g
J: f d g , both exist and =ffdg,+ C , f ( X ) (g(x and g ( b
+ 0)
= g(b).
+ 0) - g ( x
-
O)),
13.
61
FUNCTIONS OF BOUNDED VARIATION
As a consequence of 11.10.10 we can assume that f ( x ) is bounded on [a, b ] . Since JI dg, 15 JI dg for every interval I, it follows by 11.13.6 that the existence of J,", f d g carries that of f fdg, with it. Consequently, by the linear properties of the integral, J: f d g , exists also. Set P(x, X J = P(x, q,; g(x,) - g ( x , - O), g ( x , 0) g ( x , - 0 ) ) , where P(x, x,; d,, d2) is the simple break function for x,, d,, d,. Then for each point of discontinuity x, of g ( x ) , J: f ( x ) d P ( x , x,) exists also and is equal to f ( x , ) ( g ( x , 0 ) g ( x , - 0 ) ) . If g,(x) = Z J ( x , x , , ) , where x , ~ranges over the discontinuities of g ( x ) , then
I
I
1,
+
+
and the right-hand side of the inequality approaches zero with n. Since
it follows further that
so that
+ 0) g(xn - 0 ) ) = C f ( d ( g ( x + 0) - g ( x - 0 ) ) .
J,"fdgh= C f ( X ? O( g k ,
-
n
2
This result is not entirely reversible, since for the norm integral f ( x ) must be continuous where g ( x ) is discontinuous, and for the u-integral f and g cannot have discontinuities on the same side of any point. It does suggest that if g ( x ) is of bounded variation and f (x) is bounded and if J: fdg, exists, then we might define ffdg
=
J b f d g c+ f ( a ) ( g ( a
+ 0)
-
g(a))
and similarly for any closed interval in [a, b]. When g ( x ) is of bounded variation, it is possible to obtain the a-integral by a type of convergence which involves the norm of 0.
62
11. RIEMANNIAN TYPE OF INTEGRATION
In u-convergence it turns out that the points of discontinuity of g ( x ) play an important role. Arranging these points in any convenient order as a single sequence (for instance, in decreasing order of the magnitude of the breaks I g ( x J - g(x,, - 0 ) 1 g ( x , 0 ) - g f x , ) I, we introduce an order in the subdivisions by the condition that alRa, if 1 al 1 < 1 o2 I and o1 contains more points of the sequence x,, ..., x , ... of points of discontinuity of g than a2 does. Then it is easy to see that the order R makes a directed set of the subdivisions a. We then have:
I+
+
13.9. THEOREM. If g ( x ) is of bounded variation on [a, b], a necessary and sufficient condition that aJ:f d g exists is that Z,,f(x,') (g(x,) - g(x,-J)
converge in the R sense just defined. The sufficiency is immediate. For the necessity suppose that a J:.fdg exists. We can then assume that f is bounded on [a, b]. Let g n ( x ) = g,(x) 2: = P(x, Zm),where P(X, 5,J is the simple break function determined by g at the point of discontinuity X,. Then a J f f d g , , exists both as a a-limit and as an R-limit, and converges to o JB.fdg. Select no so that simultaneously
+
(a> and (b)
I c7 ["fclg
- 0
1;'fdgn0I < e ,
xaI ( g n o ( x z )
- gll0(xt-J)-
for all subdivisions
o
(g(xJ
-
d X t - J ) I < e,
of [a, b]. The latter is possible since for all a :
and the right hand side approaches zero with n. Further select d , so that if 1 a I < d , and a contains X1, ..., x , , ~then: ; (c>
I C,f(X,O
fg,lo(x,)- g ? ? p L - J-J;fdgn0 ) I < e.
Then from inequality (b) we obtain : (dl
I C,f(X,') -
f g n o ( x J- g ? , p - J )
C,f(X,'l f g f x , ) - g(x,-J)
I -=ck f e
13.
FUNCTIONS OF BOUNDED VARIATION
63
where M is the 1.u.b. of I f ( x ) 1 on [a, b]. By combining inequalities (a), (c) and (d), we have:
Ip
g
-
C , f ( X , / ) (g(x,l
-
g ( x , - , ) ) I < (2
+M) e ,
-
provided la I < d , and a contains X,,..., xn0. It is obvious that the R-ordering of subdivisions a makes of them a directed set of sequential character, the required cofinal sequence aptbeing for instance such that I a?,1 < l / n and apLcontains the points x , , ..., x , of points of discontinuity of g ( x ) . As a consequence of the theorem on limits of functions on a directed set of sequential character (1.5.1) we have: 13.10. T H E O R E M . If g ( x ) is of bounded variation on [a, b], then a necessary and sufficient condition that a J: fdg exist is that lim ,rZo,.f(xi’)
such ( g ( x , ) - g ( x , - J ) exist for every sequence of subdivisions that I al( I i 0, and the ultimately contain all of the points of discontinuity of g ( x ) . Note that limon,if it exists for all {a,}described in the theorem, will be independent of {a,}, since two such sequences {a,} and {olz’} can be combined into a single sequence {an”} of the same character by alternating on and o n r . For the norm integral it is possible to replace in the condition N J: o ( J ; I ) v ( I ) = 0, the integral of an interval function by an integral of a point function. We have: 13.11. T H E O R E M . A necessary and sufficient condition that N Jffdg exist is that N J: w ( f , x ) d v ( x ) = O.> Here o(f; x ) = g.1.b. [oj(’ Z) for all I containing X I , the point oscil-
lation function off. We prove equivalence of the condition of the theorem with N J: ( o f f ; I)v(I) = 0. Suppose then N J: w(f; I ) v ( I ) = 0. Then (e, d,, I a 1 < d , ) : C , w ( f ; I)v(I) < e. Let I a I < d J 3 . Then if xP1= xo= a, and x,,, = x,, = b,
since x,‘ is interior to [ x t t l , x t P 2 ]We . can rearrange the points x , into three subdivisions each with I (T 1 < d , as follows:
64
11. RIEMANNAIN TYPE OF INTEGRATION
= (a, x,, xq, x,, ...), g2 = (a, x2, x,, x8, ...) and g3 = (a, xB,x,, ...), so that Zu = zIul Zuz Xc,.Now Z u t w ( f ) I)v(I) < e, i = 1, 2, 3. Then Z u w ( f , xi’) (v(xi) - v(xi-,)) < 3 e provided I u I < de/3, and consequently N w(’ x)dv(x) = 0. On the other hand, assume N J: w ( f , x)dv(x) = 0, or (e, d,, I u 1 < d,) : .Xmw(f, xi’) (v(xi) - v(xiJ) < e. Now for each point x,, of [a, b] there exists an interval I., containing x,, as an interior point such < l.u.b.[w(f, x), that w ( f ; Ix)< w ( f , xo) e. Consequently, w’( for x in IxJ e. By using the Bore1 theorem we find (e, de‘,l(I) < de’): w ( 8 I) < l.u.b.[w(f, x) for x in I] e. If I u I < d, and de‘,then 0,
+ +
Jl
+
+
Ix.
+
Consequently, N J: w(f; I)V(I) = 0. We finally give the extension to Stietjes integrals of the existence theorem for Riemann integrals involving the measure of the points of discontinuity of f(x). As we shall see later, this theorem follows easily from the Lebesgue integral theory, but it can be proved without invoking all of the ponderous measure theory which underlies the Lebesgue integral. Let g(x) be of bounded variation on [a, b] and as usual v(x) = J,” dg 1. For I = [c, d], let v(I) = v(d) - v(c), and for I open, let v(I) = v(d - 0) - v(c 0). Then we define:
I
+
13.12. Definition. A set E is of v-measure zero, if for every e > 0, there exists a finite or denumerable set of intervals {I,} covering E, in the sense that every point of E is an interior point of some I,, such that L‘,v(I,) < e. It is immaterial whether the intervals I , are open or not in this definition. The only property of the class of sets of v-measure zero we shall use is: 13.13. THEOREM. If En are sets of v-measure zero, then the union Z,E, which contains all of the points in any En is also of v-measure zero. For let I,, be a set of intervals covering En such that Z , V ( I , , ) < ~ / ~ ~ . Then the intervals I,, will be a denumerable set of intervals covering ZnE,with Zmnv(Inm) < e. If xois a point of continuity of g(x) and so of v(x), then the v-measure of the set consisting only of x,, is of v-measure zero. Consequently:
13.
FUNCTIONS OF BOUNDED VARIATION
65
13.14. Corollary 1. A denumerable set of points at each of which
g ( x ) is continuous is of v-measure zero. 13.15. Corollary II. A set of v-measure zero cannot contain a point of discontinuity of g . In terms of set of v-measure zero, we now have: 13.16. THEOREM. If g ( x ) is of bounded variation, then necessary and sufficient conditions that N J: fdg exist are: (1) f be bounded on the complements of afinite number of intervals on each of which g ( x ) is constant and (2) the points of discontinuity o f f form a set of v-measure zero. Observe that the theorem is restricted to the norm integral. The necessity of condition (1) has been proved in 11.10.10 for the case of any g ( x ) . To derive condition (2) we note that if ~ ( f x,) is the oscillation function for f(x), then f ( x ) is discontinuous at x , if and only if w ( f , xo) > 0. Let Ekbe the set of points of [a, b] for which w(S, x ) I k > 0. Then Ek is a closed set for all k. Since N f f d g exists, we have (e, d,, I CT 1 < d,) : Z5w(f,I)v(I) < e. Take any g o such that I (T, I < d,, and let 11,..., I, be the intervals of g o containing points of Ek as interior points. Then k Ztv(I,) 5 Z:bow(f) I)v(I) < e. It might happen that some of the end points of I,, i = 1, ..., m are points of Ek. We can then find a subdivision g1 with I g1 I < d,, whose intervals contain the end points of g o as interior points. If 11’, ..., I,’ are the intervals of 0, containing points of Ek as interior points, then k Ziv(Ii’) < e. Then all of the points of Ek will be interior to the intervals I,, ..., I,,, I,’, ..., I?,’and Zzv(I,) Z7v(If’)< 2 elk. Since k is fixed and e is any, it follows that Ek is of zero v-measure. If now we let k = l/n, then all of the points of discontinuity of f ( x ) will belong to the union of E,,97,and consequently this set of points is of v-measure zero. For the sufficiency, assume first that f ( x ) is bounded on [a, b] with 1 f ( x ) < M . Let E be the set of discontinuities of f ( x ) and enclose E in a set of open intervals {I,} such that ZrLv(In) < e. Since the I??are open intervals, their union is an open set G consisting of the disjoint open intervals J,. Consequently, J , covers E and Zmv(Jm) < Znv(In)< e. For any closed subinterval of an interval J,, can by the Bore1 theorem be covered by a finite number of the intervals I,L. Let F be the closed set complementary to G relative to [a, b]. Then f ( x ) is continuous for each point xo of F and consequently uniformly continuous, so that (e, d,, xo in F, I x - x , I < d,) : [ f ( x ) -
+
I
66
11. RIEMANNIAN TYPE OF INTEGRATION
+
f(x,) I < e, and so w ( f ; [x, - d,, xo d,]) < 2e. Consider now any subdivision a of [a, b] such that a < d,. Let 11',... I?' be the intervals of (T containing at least one point of F, and 11",... Isr' be the complementary intervals, none of which contains a point of F. Then < 2Me. On the Ziv(Ii") < ZYnv(J,)< e, and so Ziu(f; Ii")v(Ii") other hand, Z i w ( f ; Ii')v(Ii') < e Ziv(Ii') < e(v(b) - v(a)). Consequently, if a I < d,, then -',w(f; I)v(I) < e(2M v(b) - v(a)). Then by 11.13.2: N fdg exists. In case f(x) is unbounded, but bounded on the (closed) complements I' of a finite number of intervals of constancy of g(x), then J I fdg exists for all such I f . For if f(x) is discontinuous at any end point x, of an I f , then the v-measure of x, must be zero, so that g(x) is continuous at x,, and f and g have no common discontinuities at the end points of the intervals I ' . Consequently, by II.4.4N: N J: f d g exists and is equal to ZInJ I #fdg. The parallel existence theorem for J: fdg offers some difficulty in elegant statement since the continuity conditions for the a-integral on f are onesided at each point of discontinuity. Some concept of zero measure involving the fact that the points of x are two faced in this situation is needed. It is possible to bypass this difficulty as follows :
I I
1
Jl
+
13.17. T H E O R E M . A necessary and sufficient condition that a
J: fdg
exist, f bounded and g of bounded variation on [a, b] is that f and g have no common discontinuities on the same side of any points and J: fdg, exist, where g , is the continuous part of g. The previous theorem can then be applied to J: fdg,. Summarizing, we have the following necessary and sufficient conditions for the existence of the N J: f d g and a J:fdg, if g is of bounded variation and f(x) is bounded on [a, b] : (a) the Cauchy condition of convergence on the approximating sums Z, f d g for both;
(b) J: w ( f ; I ) * v ( I ) = 0 in each case; g.l.b.,,Zow(f; I)v(I) = 0 is necessary for N-integral, necessary and sufficient for a-integral ; (c) J: f(x)dv(x) exists for each case, where v(x) = (d) for the norm integral: N J: w(f, x)dv(x) = 0;
Jz I
dg I;
(e) for the norm integral: the set of discontinuities of f(x) have v-measure zero.
14.
PROPERTIES OF
J: fdg
67
EXERClSES 1. For each interior point x of [a, b] define w(f, x+) = g.l.b.d,o o(f; [x, x d]) and w(f; x-) = g.l.b.d>o~ ( f [x ; - d, XI). Show that a necessary and sufficient condition that a dg exist is that (T J: w'( x * ) dv(x) = 0, where in the approximating sum S,w(f, x,I) (v(x,) - v(x,-J), the
+
Ji
value x,' may be x+ or x- if x , - ~< x < xL,or x + , - ~or x - ~ . 2. Modify the definition of a set of v-measure zero to a v*-measure zero, so that a fdg exists for f bounded if and only if the discontinuities of f
s:
form a set of v*-measure zero. 14. Properties of
J: fdg
with g of Bounded Variation on (a, b)
The linearity and interval properties derived for the general Stieltjes integral, obviously apply also when g(x) is of bounded variation. We consider properties when the assumption that g(x) is of bounded variation is pertinent.
[a, b] andf,(x), are bounded functions on [a, b ] such that the integrals J:f,dg, i = 1, ... n exist; if h(y,, ... J I J is a continuous function in (yl, ... y J ' on an n-dimensional rectangle containing the value set of ff, fx) * * * f , (XI 1 for x on [a, b I , then J: h ff, (x) * * * f I / (x) )dg(x) exists also. For the continuity points of h(f,(x), ...A,(x)) will be those for which all of the functions f , , ...A, are continuous. Consequently, for the case of the norm integral, h(f,(x), ...A,(x)) will be continuous excepting for a set of v-measure zero, the union of the sets for which one of the functionsf,, ...A2is discontinuous. It follows that h(f,(x), ...A,(x)) is integrable with respect to g on [a, b]. For the case of the a-integral, we note in addition that all of the functionsf,, ...f,, will be continuous on theside of a point whereg(x) is discontinuous, so that the same applies to h ( f , ( x ) , ...f7 t('x)). So the theorem is valid for a-integrals also. Immediate consequences are : 14.1. TH E O R EM . If g(x) is of bounded variation on
..., f,,(x) 7
7
14.2. THE ORE M. If fl(x) and f2(x) are integrable on [a, b ] with respect to the function of bounded variation g(x), so is f,(x) - f,(x). 14.3. THEOREM. Iff(x) is integrable on [a, b ] with respect to the function of bounded variation g(x), then If(x) is also. Moreover, J: dv exists and J: fdg I 5 J: I dv.
If
I
I
If
I
68
11. RIEMANNIAN TYPE OF INTEGRATION
The second part of this statement follows from the existence theorem 11.13.4 and from a comparison of the corresponding approximating sums. More generally, we have :
f f,dg and Jfl f,dg exist; and if f l ( x ) I I I f 2 ( x ) 1 for x on [a,b], then I Jf:fldg I 5 I fi I dv. In particular then: 14.4. T H E O R E M . If g is of bounded variation and
I
s:
14.5. If g ( x ) is monotonic nondecreasing on [a, b] and M = 1.u.b. of f ( x ) on [a, b], and m = g.1.b. of f ( x ) on [a, b], and if S:fdg exists,
then m(g(b)
- g(a))
5 J)dg 5 M f g ( b ) - g(a)).
From this we derive in the usual way the: 14.6. Mean Value Theorem. If f ( x ) on [a, b] takes. on all values between two values [in particular, if f ( x ) is continuous], if g ( x ) is monotonic nondecreasing on [a, b] and if S:.fdg exists, then there exists an x, on [a, b] such that Jf:fdg = f ( x , ) (g(b) - g(a)). 'This mean value theorem leads to: 14.7. Second Mean Value Theorem of the Integral Calculus. If
f ( x ) is Riemann integrable, g ( x ) monotonic nondecreasing on [a, b], then there exists a point x, on [a, b] such that J:f(x)g(x)dx
=
g(a) Jh".f(x)dx
+ gfb)
J b2 0
f(x)dx.
For any monotonic nondecreasing function is Riemann integrable so that J: f ( x ) g ( x ) d x exists. If we set h ( x ) = J:.f(x)dx, then by the substitution and integration by parts theorems we have: j:fogfx)dx=
J; g(x)dh(x) = hfb)g(b) - ,; h ( x ) d g f x ) .
Applying the mean value theorem gives an x, of [a, b] such that J : f f x ) g f x ) d x = g(b) f f f x l d x - p ( x ) d x ( d b ) - d a ) ) = g(a)
,:"fX)dX
+gfb)
f(x)dx.
Jb XO
Another application of the integration by parts and substitution theorems leads to:
15.
69
CONVERGENCE THEOREMS
14.8. T H E O R E M . If g ( x ) is of bounded variation on [a, b] and
1:fdg
vanishes for every continuous function f ( x ) , then g ( x ) = g ( b ) except for a denumerable set of points on a < x < b. For set f ( x ) = J: (g(b) - g ( x ) ) d x . Then f ( x ) is continuous and 0=
=f(b)g(b)
-
ffdg
=f ( b ) g ( b )
1;g ( x ) ( d b )
-
-
f(a)g(a) -
g ( x ) ) dx
=
1:gdf
1: ( d b )
- g(x))''dx.
Now if a function h ( x ) is Riemann integrable and positive or zero on [a, b], and J: h(x)dx = 0, then h ( x ) vanishes at all points of continuity of h(x). Consequently, g ( x ) = g ( b ) , excepting at the points of discontinuity of g ( x ) which are denumerable in number. The fact that g ( a ) = g ( b ) follows from J: f ( x ) d g ( x ) = 0 by setting f ( x ) = 1 on [a, b ] . This theorem is closely related to the fundamental lemma of the calculus of variations, which asserts that if g ( x ) is continuous and J: g ( x )f ' ( x ) d x = 0 for every continuous function f (x) which vanishes at a and b, and for which f ' ( x ) is Riemann integrable, then g ( x ) is a constant. An alternative proof of 11.14.8 is as follows: Set f ( x ) = x for a 5 x 5 .yo; f ( x ) = x, for x , 5 x 5 b. Then integration by parts leads to
Taking derivatives with respect to x, yields g(x,) = g ( b ) except for the points of discontinuity of g, with a < x, 5 b. That g ( a ) = g ( b ) follows as above. Obviously if g ( x ) = g ( a ) excepting for a denumerable set of points interior to [a, b ] , then fdg = 0 for every continuous function. For the points of subdivision in the approximating sums can be chosen so as to avoid the points of discontinuity of g .
J':
15. Convergence Theorems
As in the case of Riemann integrals we have: 15.1. T H E O R E M . I f g ( x ) is of bounded variation on [a, b ] , f , ( x ) are
such that J:f,dg exists for all n and lim,f,(x) = f ( x ) uniformly on [a, b], then J: fdg exists and lim, J: f,dg = J: fdg.
70
11. RIEMANNIAN TYPE OF INTEGRATION
This follows from the iterated limits theorem 1.7.4. For
If limnf,(x) =f(x) uniformly on [a, b], then it follows from this inequality that lim,<2,fJg = 2JOg uniformly as to 0. Consequently, limJimoCofndg = ~ i m ~ i m 7 ~ ~or u fEm, 7 z ~J:,f,,dg g = Jifdg. 15.2. T H E O R E M . If g,, and g are of bounded variation on [a, b] and such that lim. J: I d(g,, - g) I = 0; if f(x) is bounded, and if J:fdg, exists for all n, then lim,! Jffdg, and Jifdg exist and are equal. This also follows from the iterated limits theorem if we note that
A slightly more sophisticated theorem is:
f(x) is continuous, if g,,(x) are uniformly of bounded variation on [a, b] with J: 1 dg, I 5 M for all n, and if limngn(x) = g(x) for all x, then limn J: fdg,, = J:fdg. 15.3. T H E O R E M . If
Since the g, are uniformly of bounded variation, g is also of bounded variation so that J:fdg exists. Now
I
Jlfdgn -
C,f%, I 5 C,W(f;I )
*
JI
1 dg, I s max,uff;
1)
j;I dg, I
5 max,w(f; I ) M . Consequently, because of the uniform continuity of f(x) on [a, b] it follows that lim , o , Z0fogn = J: fdg,, uniformly as to n. The iterated limits theorem then gives the interchange of limits with integration. It is, however, not necessary to assume that g,(x) converges to g(x) for all x. It is sufficient to assume that the convergence holds on a denumerable dense set D on [a, b], including a and b, and that g(x) is of bounded variation on [a, b].In the proof one can then limit the subdivisions to a sequence consisting of points of D such that lim, I urn1 = 0. It might be observed in this theorem that the continuity of f(x) assures us of the existence of the limit integral J:fdg. This suggests:
-
15.
CONVERGENCE THEOREMS
71
If g,,(x) are monotonic nondecreasing and limr,gy,(x)= g(x) for all x of [a, b], and if further J:fdg, and cfdg exist, then lim, J:,fdg. = J:,fdg. We prove the theorem for the a-integral. It will then also hold for the norm integral. Since J: fdg,, and J:,fdg exist, it follows from 11.13.3 that limUZuw(fi I) -g,(I) = 0 and limUZOw(f;I ) *g(I) = 0. Now lim,lCuw(f; I ) * g,,(I) = I) . g(I). Moreover, Zuw(.f; I) - gTl(I) is monotone in a for every M. Since limfllimoCBW(f;I) . g,(I) = limo limflEUco(.f; I) . gTl(I), it follows from 1.7.8 that lim~EOco(f;I) g,(I) =O uniformly in n. Hence J:Jllg,, exists uniformly in n, the iterated limits theorem 1.7.4 applies and lim,l Jefdg,, exists and is equal to J: fdg. This theorem can be extended to functions of bounded variation in the following form: 15.4. T H E O R E M .
Eo(li(f;
15.5. If g,,(x) and g(x) are of bounded variation such that limrlg,l(x)
= g(x) for every x, and lim7[J: [ dg,, ] = J: I dg I ; if further J:fdg,, and J:fdg exist, then limTlJ:.fdg,, exists and is equal to J:fdg. For the hypotheses of the theorem imply that lim, J: I dg,, I = J: I dg I for all x [see 11.8.7 (3)]. Consequently, if P and N stand for positive and negative variations, then lim,,P,,(x) =P(x) and lim,,NT,(x) = N(x). The rest is obvious. A theorem for Riemann integrals not requiring the uniform convergence of the functionsf,,(x) is due to Arzela and Osgood, and is sometimes called Osgood’s theorem. In the Stieltjes integral setting we have :
15.6. T H E O R E M . If the sequence of functions f , , ( x ) is uniformly bounded on [a,b];if lim f,, (x) = f(x) for every x ; if g (x) is of bounded variation on [a, b ] ; and if ff,dg and J: fdg exist for every n ; then lim, J;.frldg = J: fdg. We prove this theorem more or less in reverse. We have 1,
where v(x) = J,”I dg I. As a consequence, it is sufficient to prove that under the hypotheses of the theorem, the right-hand side of this inequality converges to zero. This will be accomplished by proving the following : 15.7. Lemma. If f,,(x) 2 0 on [a, b] for all n ; if f,(x)
5 M for all x
and n ; if lim,f,,(x) = 0 for all x ; and if g(x) is monotonic nondecreasing then lim, Jb f,dg = 0. -0
72
11. RIEMANNIAN TYPE OF INTEGRATION
Jl
Note that the existence of fndg is not assumed here. We give a contrapositive proof. The assumption that J3fndg does -a not converge to zero gives rise to an e > 0, and a subsequence n, such that for every m : -a Jbfnmdg > e. Since f ,m would satisfy the same conditions as f,, we can replace n, by n. Then for each n there exists a subdivision un such that Zo m , ( I ) g ( l ) > e, where n mn(I) = g.1.b. f n ( x ) . xonI
For 7 > 0 let I , ] , ... Ink, be the intervals of cr7Lfor which m,(I) > 7 and Irnl, ..., I',,kn,be the remaining intervals of uPL.Then ML',g(I,,) + 17 Z j g ( T n j )> e. Or since .Z9g(Irn9) 5 g ( b ) - g ( a ) : M Z a g ( I n , )> e T(g(b) - g ( a ) ) . If we select 7 small enough we can make the right-hand side of this inequality positive; for instance, if 7 ( g ( b ) g ( a ) ) < e / 2 , then for each n : Zczg(I,,)> e / 2 M . Then for each n, we have a finite number of intervals In,, ..., Ink, on each of which .fn(x) > 7, and such that Zczg(In,) > e / 2 M , a fixed number C independent of n. Now the function g ( x ) maps the interval [a, b] on the interval [ g ( a ) , g ( b ) ] on the Y-axis. We shall assume the map is such that the closed interval [ g ( c ) , g ( d ) ] corresponds to the closed subinterval [c, d]. In case x, is a point of discontinuity of g ( x ) this means that both [ g ( x , - O ) , g(x,)l and [g(x,), g ( x , 0)l correspond to x, depending on the direction from which x, is approached. To any point y , between g ( a ) and g ( b ) there will be at least one x such that g ( x ) = y . As a result of this mapping, the intervals [x,, ,--I, x,] are mapped into J,, = [ g ( x n ,,-1, g ( x n , ) ] . Then Z,g(Ina)> C becomes L'J(J,J > C, where we can discard any J of length zero. If now we knew that there exists a sequence nmim such that the intervals J , r n m have a point y o in common, there would be a corresponding point x, in Inm ,r n for which fn m (x,) > 7 for all m. Then lim,f, rn (x,) f 0 contrary to the hypothesis of the lemma. Consequently our lemma and the theorem is proved if we prove a lemma due to Arzela:
+
15.8. Arzela's Lemma. If -3, is a sequence of finite sets of intervals Inl, ... Ink,, and there exists a constant C such that Zil(Ini)> C for all n, then there exists at least one point x belonging to a subsequence of the -3,. In terms of concepts used in the theory sets, this lemma asserts that
15.
CONVERGENCE THEOREMS
73
lim.3, is not vacuous, where for a sequence of sets E,, the set E = lim,E,, is the totality of all elements appearing in an infinite number of the sets En. This set E can be obtained by taking the union Urn of the sets E , for n 2 m, and then taking the intersection of the U,. For the proof of the Arzela lemma assume first that the intervals I,, are closed, and that the set of points 3, is contained in that of -3%-,. We then have a nonincreasing sequence of closed sets on [a, b] none of which is vacuous. It is well known that there is a point common to these sets, so that the lemma holds for this case. The general case is reduced to this one. For the general case, assume that the I,,, are open intervals, which = C:='=3,,, , that is, will not affect the validity of the lemma. Let the union of the sets 3,, for n 2 m. Then anL is an open set. If the intersection of the ( s f m is not vacuous, i.e., if the (sfjm have a point in common, then the lemma is proved. Note that (sf?, is contained in am being an open set it consists of a denumerable set of open intervals JwL,. Since @, contains 3 ,, we have Z J ( J m 7 , > ) C for all m. Let el, ..., em, ... be a decreasing sequence of positive numbers such that Zme, < C/2. We then discard from (sf,, a set of intervals J,,, n = n, 1, ..., the sum of whose lengths is less than el/2. From the remaining intervals Jll, ..., Jl,,l,we lop off both ends totaling in length at most e,/2 leaving the closed intervals JIll, ..., J',,,. Set ZaJ',, = g1. Then for the intervals in the intersection of (sf, with 5,, we have diminished the total length by at most e,. Treat the common part of (sf2 and 5 , the same as B1 using e,, resulting in closed intervals J',,, ..., J r Z nmaking z up the closed set 5 , contained in 5 , with Z(5J = L')(JJ2J > C - e, - e,. This process can be repeated and gives rise to a sequence of sets of closed intervals Z,,S2, ..., each contained in the preceding with Z(8,) = Z:lI(J'ma) > C/2. Then each 5 , is not vacuous, and the intersection of the 5 , is not vacuous. The same holds for (sf, and there exists a point x, common to a sequence of intervals
+
Lrnlm. It might be noted that the intersection set 5 of 5,, which is a closed set, is not of measure zero. For suppose {K,} is a set of open intervals enclosing 5. Then by the Bore1 theorem a finite subset K , , , ... K, r, which we will denote by 92, will cover 5. We assert that there exists an m, such that for m 2 m, the sets 5 , will be covered by 91. If not, there exists for each m a point x, in 5 , not in 92. The sequence x, will have a limiting point x, which because of the clo-
74
11. RIEMANNIAN TYPE OF INTEGRATION
sure and monotoneity of the sequence iYm will belong to 5. But since x, are not in 91, and 91 is open, xo is not in 91 and so not in 5 . It follows that Z(91) = Z’)(K,, z ) > C, so that the greatest lower bound of the sums of the lengths of any set of open intervals covering 5 is positive, and not zero. This completes the proof of 11.15.6 which we shall call Osgood’s theorem. We can generalize our theorem as follows: 15.9. THEOREM. Ifg(x) is ofbounded variation with v(x) =
J: Idgl:
if the sequence of functions f,(x) is uniformly bounded on [a,b ] ;and if limnfn(x) = f(x) excepting possibly at a set of v-measure zero; and if f,dg and J:fig exist for all n, then lim. J: f,,dg= fdg, even in the sense that lim. - f dv = 0. We note that the condition that cfdg exist cannot be dropped in this theorem. For suppose g(x) = x on [0, 1 1 and set f,,(x) = 1 for x = p / q , p and q integers with 0 5 p 5 q 5 n, and f,(x) = 0 elsewhere. Set f(x) = 1 for x rational, and = 0 for x irrational. Then lirn?,f,,(x) = f(x) for all x, J’: f,,(x)dx = 0 for all n, but J,’f(x)dx does not exist. We note, however, that lim. J,’f,,(x)dx exists, which as we shall show below always happens under the hypotheses of the theorem. The convergence theorems developed are stated for sequences of functions. For some of them, the sequence of functions can be replaced by a directed set of functions and the proofs are still valid. We have:
Jl
JB If,
Jl
I
15.10. If Q is a directed set; fJx) a set of functions on
[a,b] such that
lim,.f,(x) = f(x) uniformly on [a,b];if g(x) is a function of bounded variation; and if J: fq(x)dg(x) exists for all q, then lim, J;,f,(x)dg(x) and J: f(x)dg(x) exist and are equal. 15.11. If Q is a directed set; g,(x) and g(x) are of bounded variation on [a, b] and such that lim, J: I d(g, - g) I = 0 ;iff’(x) is bounded on [a, b ] ; and if fdg, exists for all q, then lim, fig, and J:fdg exist and are equal.
JB
s%
15.12. If Q is a directed set; if g,(x) and g(x) are of bounded variation such that lim,g,(x) = g(x) on [a,b], lim, I dgq I = J: 1 dg 1; and if J: fdg exists and fdg, exists for all q, then lim, fdg, exists and is equal to J: fdg.
Jg
JB
Jt
15.
CONVERGENCE THEOREMS
75
The proof of this last theorem for sequences requires some slight modifications for the directed set case. The proof of the extension of Osgood's theorem depends on the fact that we are dealing with sequences of functionsf,(x). To obtain a generalization to directed sets, we must limit our directed sets to those which have sequential character (see 1.5) that is there exists a sequence {q,} of Q which is cofinal with Q. The resulting theorem reads: 15.13. If Q is a directed set of sequential character; ifg(x) is of bound-
ed variation on [a, b ] ;if &(x) is a uniformly bounded set of functions on [a,b] such that lim,f,(x) = f(x) except at a set of v-measure zero, where v(x) = J,"I dg I ; and if cfdg exists and J: f,dg exists for all q, then lim, J:.f,dg exists and is equal to Jlfdg, even in the sense that lim, J: If, - f ] dv = 0. For then for every sequence { q T l ' }cofinal with Q we can apply the sequential form to show that limn J:f,,,dg = fdg. It then follows by 1.5.1 that lim, J;f,dg = Ji,fdg. This extended Osgood theorem gives us: 15.14. THEOREM. If g(x) is of bounded variation on [a, b];iff,(x) are uniformly bounded on [a, b] and such that limJ,(x) = f(x) except for a set of v-measure zero; and if Jif,,dg exists for every n, then lim. Jlf,dg exists even in the stronger sense that
If we assume the double sequence (m, n ) ordered by the condition that (ml, n,) 2 (in2, n,) if and only if m, 2 m, and n , 2 n,, then Q = (m,n) is a directed set of sequential character, (n, n ) being cofinal with Q.If we set h,,[(x) = I fm(x) - f,,(x) 1, then hm,(x) form a directed set to which the preceding theorem applies so that limm,lzJ: If, - f,, I dv = 0, so that limn J:@g exists. The extension to directed sets reads: 15.15. If Q is a directed set of sequential character; if g(x) is of bounded variation on [a,b ] ;with v(x) = 1 dg ] ;if.f,(x) is a set of functions uniformly bounded in q and x, such that lim,f,(x) exists except at most a set of v-measure zero, and if J;f,dg exists for all q, then lim, J:.f,dg exists and limqlq2J: I f q l - f g 2 I dv = 0. For if Q is sequential character, then Q x Q when ordered as above for sequences, is also of sequential character.
J'r
76
11. RIEMANNIAN TYPE OF INTEGRATION
EXERCISES 1. Suppose g ( x ) is monotonic nondecreasing on [a, b] and . f ( x ) 2 0 on [a, b], and so that J: fdg = 0. Show that f ( x ) vanishes excepting for a
set of g-measure zero.
Sl
2 . Suppose that g ( x ) is of bounded variation and fdg = 0 for all continuous functions vanishing at both a and b. Show that g ( x ) is a constant excepting for a denumerable set of points on [a, b]. 3. Show that if g ( x ) is not of bounded variation on [a, b], the uniform convergence of .f, to f on [a, b] and the existence of J':f,dg for all n are not sufficient to ensure that limn f,dg = J: fdg.
Jl
4. If g , ( x ) are monotonic nondecreasing and lim,g,(x) = g ( x ) on a
dense set of points of [a, b] including a and 6 ; if is lim, J: f d g , = J: fdg?
s: ,fdg,, and J: f d g exist,
5. Prove that if Q is a directed set; if g,(x) and g ( x ) are monotonic nondecreasing and such that lim,g,(x) = g ( x ) for x on [a, b ] ; and if J l f d g , exist for all q, and f d g exists, then lim, J: f d g , = J: fdg.
Jl
6. Show that if f ( x ) 2 0 for x on [a, b] and only if x is a point of discontinuity of J
S: f ( x ) d x = 0, then f ( x ) > 0
7. Construct an example of a directed set Q of elements q, and a set of uniformly bounded functions f , ( x ) on [0, I ] such that lim, f , ( x ) = . f ( x ) exists for each x and ,f,(x)dx and J: f ( x ) d x exist for all q, but lim, J,' f , f x ) d x # J: f ( x ) d x .
Jt
16. The Integral as a Function of the Upper Limit
We have already shown in 11.11.3 that the continuity properties of the indefinite integral J,Z fdg reflect those of g ( x ) . If g ( x ) is of bounded variation on [a, b], then g ( x 0) and g(x - 0) exist at all points. Then h(x 0) and h(x - 0) exist for all x and h(x 0) - h ( x ) =
+
+
f(x) (g(x+O) - g ( x ) ) ; h ( x ) - h ( x - O )
+
=f(x) ( g ( x ) - g ( x - O ) ) .
In addition, we have: 16.1. THEOREM. If g ( x ) is of bounded variation on [a, b], then h ( x ) = JIfdg is also of bounded variation and J: I dh I 5 M J: I dg I, where M is the least upper bounded of I f ( x ) I on the intervals complementary to those for which g ( x ) is constant.
16.
77
INTEGRAL AS A FUNCTION OF THE UPPER LIMIT
This follows at once from the inequality I J: fdg I 5 M ( A [c, d ] ) I dg 1, for any interval [c, d ] on which f is bounded. We cannot expect h(x) to have a derivative with respect to x . However when g ( x ) is monotone nondecreasing we have a kind of parallel theorem : 16.2. T H E O R E M . If g ( x ) is monotonic nondecreasing on [a, b ] , and h ( x ) = fdg, then limAwo (h(x Ax) - h ( x ) ) / ( g ( x Ax) g ( x ) ) =f ( x ) for all points where f ( x ) is continuous which are not interior to an interval where g ( x ) is constant. For if Ax > 0, then
I h f x , + Ax) 5 1.u.b. x , < x S x,+Ax
+
+
JI
-
W,) - f f x , )
fgfx,
+ Ax)
-sfx,))
I
I f f x ) -f f x , ) I f g f x , + AX) - g(X0)) .
+
Division of this inequality by the positive valued g ( x , Ax) - g ( x , ) yields the theorem for the limit on the right of x,,. A similar procedure works on the left. The following theorem embodies a generalization of the fundamental theorem of the integral calculus: 16.3. T H E O R E M . If g ( x ) is strictly monotone increasing ( x , < x , implies g ( x , ) < g ( x , ) ) on [a, b ] ; if the function h(x) is such that (h(x Ax) - h ( x ) ) / ( g ( x Ax) - g ( x ) ) = f ( x ) exists for lim,,,, all x on [a, b] (with onesided limits at a and b) ; and if N J: fdg exists; then h(b) - h(a) = J: fdg. For the case of Riemann integrals With g ( x ) = x , the proof of this theorem is usually based on the mean value theorem of the differential calculus. Since we have derived no parallel to this theorem here (is there one?) we proceed differently. We have for every x of [a, b] fe, d,,, I Ax I < de,) :
+
I h(x+Ax)-hfx)-ffx)
+
( g f x + ~ x ) - g ( x ) ) 15 e f g f x + A x ) - g f x ) ) .
+
To the set of intervals 1, = [ x - Ax, x ] and r, = [x, x Ax] with 0 < Ax < d,,, attached to each x of [a, b], we can either apply the Bore1 theorem or proceed directly to the right of x = a, and prove that there exists a finite number of intervals r,, or/and 1, laid end to end reaching from a to b inclusive. These intervals can be chosen so that
78
11. RIEMANNIAN TYPE OF INTEGRATION
their maximum length is less than a predetermined d,. The end points of this selection of intervals can then form a subdivision a for [a, b] with I a I < d, such that for given e,
I Cf(XJ M X , )
- g ( x t - l ) )-
J : f d g I < e.
In f( x,’ ) (g(x,) - g(x,-,) if [x,-,, X J is an r, take x,’ = x % - ~if, [ x ~ -xz] ~ ,is an 1, take x,’ = x,. Then
I Cf(xt‘, (g(x,) a
-
g(xt-J) -
a
(hb,)-W t - J )
I
17. The Integral as a Function of a Parameter 17.1. Continuity. The convergence theorems proved in 11.15 can serve as a basis for obtaining properties of the integral, where f ( x ) and/or g ( x ) are functions of a parameter. As a matter of fact, we can think of the integers n as such a parameter. In the theorems below, we shall consider y as a parameter on the range c 5 y 5 d. The following continuity theorems are immediate consequences of Theorems 11.15.6, 11.15.3, and 11.15.4 because of the sequential character of limits as to y . 17.1.1. T H E O R E M . If g ( x ) is of bounded variation on [a, b ] ;if f ( x , y ) is continuous in y for each x of [a, b];if I f ( x , y ) I 5 M for a 5 x 5 b, c 5 y 5 d ; and if J: f ( x , y)dg(x) exists for y on [c, d ] ; then h ( y ) = J: f ( x , y)dg(x) is continuous in y on [c, d]. 17.1.2. T H E O R E M . If g ( x , y ) is of bounded variation in x on [a, b]
uniformly in y on [c, d ] ; if g(x, y ) is continuous at y = y o on a dense set Duoof [a, b] including a and b ; and if f ( x ) is continuous in x ; then Jff(x)d,g(x, y ) is continuous in y at y = yo.
17.
79
THE INTEGRAL AS A FUNCTION OF A PARAMETER
17.1.3. T H E O R E M . If g ( x , y ) is monotone nondecreasing in x for
each y , and bounded on a S x 5 b, c 5 y I d ; if g ( x , y ) is continuous in y for each x of [a, b ] ;and if J:f(x)d,g(x, y ) exists for each y , then J: f ( x ) d , g ( x , y ) is continuous in y . 17.2. Bounded Variation. 17.2.1. T H E O R E M . If g ( x ) is of bounded variation on [a, b ] ; if f ( x , y ) is defined on [a, b] x [c, d ] with J: I d, f ( x , y ) I 5 F ( x ) on [a, b ] ; if f ( x , y ) d g ( x ) exists for each y ; and if F ( x ) d v ( x ) < co, where v ( x ) = I dg 1, then h ( y ) = f ( x , y ) d g ( x ) is of bounded variation on [c, d ] and J: I dh I 5 F ( x ) d v ( x ) . For
Jl
I
Jz
WY,)
7%
-
WY,)
I = I J:
7:
Jg
(f(X,Y,) - f ( X , Y , ) ) d d X )
I
s I f ( & v,) - f ( x , Y J I dv(x).. Consequently, if
CT
is a subdivision of [c, d ] ,
leading to the conclusion of the theorem. In particular, the theorem applies if F ( x ) is bounded in x , that is,f(x, y ) is of bounded variation in y uniformly in x. Another theorem guaranteeing that J: , f ( x ) d g ( x , y ) is of bounded variation in y is to be found below in connection with two-dimensional variation (see 111.7.13). 17.3. Integrability. We have the following elegant theorem on integrability and interchange of order of iterated integrals : 17.3.1. T H E O R E M . If g ( x ) is of bounded variation on [a, b] and h ( y ) on [c, d ] ;iff(x, y ) is bounded on [a, b] x [c, d ] ;if J: f ( x , y ) d h ( y ) exists for each x and J : f ( x , y ) d g ( x ) exists for each y ; then the iterated integrals d g ( x ) Y)dh(Y) and d g ( x ) f ( x , Y))dh(Y) both exist and are equal. We prove this for the case when the integrals are all norm integrals. Set G ( y ) = J: d g ( x ) f ( x , y ) and H ( x ) = J: f ( x , y ) d h ( y ) . Then the theorem states that f G(y)dh(y) and H ( x ) d g ( x ) both exist and are equal. Let u be any subdivision of [a, b] and consider the approximating sum for J: H ( x ) d g ( x ) :
s:
s;m,
s; cs:
Jt
80
11. RIEMANNIAN
TYPE OF INTEGRATION
for each y . Also for each a: J’: Fu(y)dh(y) = ZuH(xi’) ( g ( x , ) g ( x i T 1 ) )exists. Since the u directed via I u I are of sequential character, it follows by 11.15.15 that
J’t
exists or, in other words, H(x)dg(x) exists. By parity it follows that J: G(y)dh(y) exists also, -so that the Osgood theorem 11.15.13 applies and we have
Since g ( x ) and h(y) are of bounded variation, the a-integrals exist if a are ordered via R,i.e., as 1 a I 0 and u ultimately includes the points of discontinuity of g and h, respectively, and this ordering is of sequential character (see 11.13.10). Consequently, the theorem also holds if a-integrals are used instead of norm integrals. A different type of iterated integral theorem is the following: --f
17.3.2. THEOREM. If g ( x , y ) is defined on [a, b] x [c, d ] and of bounded variation in x uniformly in y , or J: I d,g(x, y ) I 2 M for y on [c, d ] ; if f ( x ) is continuous on [a, b ] ; if h ( y ) is of bounded variation on [c, d ] ; if J: g ( x , y)dh(y) exists for each x of [a, b ] ; then
J:f(x)d*
r
g f x , Y)dhfY) =
c
( j ba f ( x ) d x g f x ,Y))dhfY),
where all the integrals involved exist and are of the same type (norm or a). We prove the theorem for norm integrals, the proof for a-integrals follows the same procedure. By Theorem 11.17.2.1, J’: g ( x , y)dh(y) is of bounded variation in x , so that since f ( x ) is continuous J : f ( x ) d , g ( x , y)dh(y) exists.
18.
81
LINEAR CONTINUOUS FUNCTIONALS
On the other side, consider the approximating sums
2 ffCX)d&, a
YJ MY,)
- h(Yi-1)) = /)-(X)dFU(X),
a
where we have set F J x ) = ZUg(x,y,‘) (h(y,) - h ( y i - J ) . The functions Fu(x) are of bounded variation in x uniformly as to a, because
Cj I Fg(xj) - Fa(xj-1) I
Xi I g ( x , y , ’ ) 5 l.u.b.i f 1 d,g(x,
5
a
-
I I MY,) - h(Yi-1) I I dh I 5 M I dh I.
g(xj-pYi’)
y,’) I
id c
C
By hypothesis lim,ol+o Fu(x) = Jf g(x, y)dh(y) exists for all x . Consequently, since f ( x ) is continuous, we have by 11.15.3:
I” c
(Sbf(xJd,g(x, a Y)dh(Y)
18. Linear Continuous Functionals on the Space of Continuous Functions
We have previously defined a linear normed space (1.9.2) and noticed that the space of continouus functions on [a, b] is such a space if 11 f 11 = 1.u.b. (I.f(x) I; x on [a, b ] ) . Then limn 11 f , - f 11 = 0 is equivalent to uniform convergence off, to f on [a, b]. 18.1. Definition. A linear form or functional on a linear space is a real valued function L ( f ) on the space such that L(c,f, c&) = c,L(f,) c,L(f,), for all constants c, and c, and elements f , and f , of the space. A linear form on a linear normed space is continuous if limnlIf , - f l l = 0 implies limnL(fn) = L(f). For a linear form L(0) = 0. A linear form on a linear normed space is obviously continuous if and only it is continuous at f = 0.
+
+
18.2. THEOREM. A linear form on a linear normed space is continuous if and only if it is bounded or limited, that is, if there exists a positive constant M such that for all f : I L ( f ) I S M 11 f 11. The fact that boundedness implies continuity follows from I L ( f n ) - L ( f ) I = I L ( f , - f ) I 5 MI1 f , - f 11. On the other hand,
82
11. RIEMANNIAN TYPE OF INTEGRATION
assume L ( f ) continuous and if possible not bounded. Then there exists a sequence { f , } with 11 f, 11 = 1, such that 1 L(f,) 1 > n. But 11 f,/d/n11 = 1/2/E, converges to zero in n while I L ( f n / v ' Z ) I > 2 / Z , which contradicts the continuity at f = 0. The l.u.b.[l L ( f ) I for I I f I I = 11 is usually denoted by 1 I L I 1, pointing the way to the fact that the space of all linear continuous forms on a linear normed space is again a linear normed space. The problem of the most general linear continuous form on the normed space of continuous functions concerned analysts for some time. It was observed that J: f(x)g(x)dx for any Riemann integrable function g(x) is such a form. Also that Zic,.f(x,), where 2% I ci I < co and xi are any points of [a, bj was a possibility. Bore1 proved that if L ( f ) is such a form, then there exists a sequence of Riemann integrable functions g,,(x) such that L ( f ) = limn J:,f(x)g,(x)dx. The elegant solution of the problem by the use of Stieltjes integrals is due to F. Riesz who proved [see Sur les operations fonctionnelles lineaire, Compt. rend. acad. sci. Paris 149 (1909) 974-9771: 18.3. THEOREM. If L ( f ) is a linear continuous form on the space of continuous functions on [a, b] normed via 11 f 11 = 1.u.b. (I f(x) I; x on [a, b]), then there exists a function g(x) of bounded variation on [a, b] such that L ( f ) = J: f(x)dg(x). We shall prove the theorem for the interval [0, I], the proof for the interval [a, b] can be deduced by indulging in the linear transformation x = (b - a)y a. Our proof is based on the use of Bernstein polynomials in the proof of the Weierstrass polynomial approximation theorem for continuous functions: " for any continuous function f(x) on the closed finite interval [a, b], there exists a sequence of polynomials P,(x) such that lim?lP,,(x) = f(x) uniform1,y on [a, b]." Since the proof is rather simple and elegant we repeat it here, limiting ourselves to the interval [0, 13, the necessary changes to any interval [a, b] being simple.
+
18.4. Definition. The Bernstein polynomials for any function f ( x )
on [0, 11 are defined B,(f; x)
=
C f ( m / n ) .Cmxm(l - x),-~,
m=O
where ,Cm are the binomial coefficients n!/(m!(n - m ) !). The expression for B,(f; x) is reminiscent of the definition of an integral.
18. LINEAR
83
CONTINUOUS FUNCTIONALS
If we take f ( x ) = 1 for all x, then
c ,lC,nXm(l n
BJl; x)
=
-
1.
X)n-m=
m=O
If we take the derivative of this identity and multiply by x ( l we obtain ,,cm(m- nx)xm(l- x)n-m = 0.
-
x)
2
m=O
Taking a second derivative and multiplying by x ( l
xnllCm(m
- nx)'
x m ( ~- x)n-m
- x)
we find
= nx(1 - x ) .
m=O
If we remember that the maximum value of x(1 - x ) on [0, 11 is 114, and divide this expression by n', we conclude that
2 n C m ( ~ m/n)'x"(l -
-
x)"-"5 1/4n.
m=O
Consequently, if m" denotes values of m for which for a given x and d we have 1 m/n - x 1 > d, then
Consider now for fixed x
=
C
( f ( x ) - f ( m / n ) ) nCmxm(l- x)n-m
m=O
I
Then because of the continuity of f ( x ) we have: (e, d,, I x' - x < d,) : 1 f ( x ' ) - f ( x ) I < e. If m' are the rn for which I x - m/n I < d, and m'' those for whicki I x - m/n I 2 d,, then
+ 2 I f( x ) - f(m/n) I nCmxm(l 5 e + M/2nd:, m"
-
X ) n-m
where M = 1.u.b. (I f ( x ) I ; x on [0, 11). Since, because of uniform continuity, d, is independent of x , we will have 1 f ( x ) - B(', x ) I < 2e provided n > n, and n, > M / 2 ed:. This is the desired uniform con-
84
11. RIEMANNIAN TYPE OF INTEGRATION
vergence. The demonstration shows incidentally that for any function f(x) bounded on [0, 11, Bn(f; x) converges to f(x) at every point where f is continuous. Returning to the proof of the Riesz theorem and to our linear continuous form L(f), the continuity and linearity of L give: n
L(f) = lim, L(Bn(f; x))
= lim,
C f(m/n)
L(,Cmxm(l - x ) ~ - ~ ) .
m=O
We define a function g,(x) as follows: gn(0) = 0; gn(x) = L';=o L (,Cmxm(l - x ) ~ - ~for ) p l n < x 5 ( p l)/n, 0 5 p < n ; gn(l) = L(1). Then
+
Whether limngn(x) exists is uncertain. But the gn(x) are uniformly of bounded variation, since
where
I
E~
c
=
sgn L(,Cmxm(l - x ) " - ~ ) and obviously n
n
Em
*
I 5 2 , c m x y l -x>,-,
ncmxm(l- xy-,
= 1.
m=O
m=O
Using the Helly theorem 11.8.9, there exists a subsequence g,Jx) and a g(x) of bounded variation such that lim,g,,(x) = g(x) for all x. Consequently, by 11.15.3, L(f) = lim, J,' f(x)dgnk(x) = J,'f(x)dg(x). This completes the proof of the Riesz theorem. Suppose a different subsequence of gn(x) leads to a different g,(x) . Then J,'fdg = J,'fdg, for all continuous functions f(x), and SO g(x) = g,(x), excepting for a denumerable set of points on [0, I]. It is customary to assume that g(x) is chosen in such a way that for all x, g(x) lies between g(x - 0) and g(x 0), which does not affect the value of the integral. Such a function of bounded variation is said to be regular. Now if g(x) is regular in this way, then the interval function P( I ) = g(d) - g(c) is pseudoadditive at every point since I gfx 0) - gfx - 0 ) = I gfx 0 ) - gfx) I I gfx) - gfx - 0)l. As a consequence J'i I dg I exists as a norm integral. Moreover, we have:
+
+
I
I
I
+
+
18.
85
LINEAR CONTINUOUS FUNCTIONALS
18.5. THEOREM. If g ( x ) is a regular function of bounded variation on [a, b] and if L ( f ) = J: f d g on the space of continuous functions,
then 11 L I( = J: I dg I, where 11 L 11 = g.1.b. [ M such that I L ( f ) I 5 M for (1 f 1) = 11. Since J: f d g 5 11 f 11 J: I dg 1, it follows that 11 L 11 5 J: I dg I. To show that 11 L (1 2 J: 1 dg 1, we find a a. such that for a 2 go, we have J: I dg I 2 ZU1 d i g I - e, and since J: I dg 1 exists as a norm integral, we can assume that g ( x ) is continuous at the points of subdivision x i . Then for i = 1 , ... n - 1, we can select points xif,xiff such that x;Ll < x i f < x i < x i f f< and ZiI g ( x i f f )- g ( x i f ) I < e. We then 5 x 5 xi', i = 1 , ... define f ( x ) = sgn ( g ( x i f ) - g(xiLl)) for n - 1 ; f ( x ) = sgn ( g ( x l f ) - g ( a ) ) on [a, x,'I; f ( x ) = sgn ( g @ ) gfx;:,)) on [x,;;,, b ] ; and f ( x ) linear on [ x i f ,x i f f ]Then .
Since
llfll
S 1 on [xif, x i f f ]we have
Consequently if a consists of a, xlf, ..., xif, xi, x i f ' , ..., x " ~ - ~b,, then
I rfdgI2
xuI 4 I
-
2
Xi I g ( - W - gfxJ
I4
coI & I
-
2e
2 s ; IdgI - 3 3 .
Since
llfll
=
1 , it follows that
11 L 11 2 J: I d g l .
EXERCISES 1. Show that for all n, the Bernstein polynomials of f ( x ) = x are BJx, x ) = x , while for f ( x ) = x2, Bn(x2;x ) = x2 x(l - x ) / n .
+
2. In general show that if P ( x ) is any polynomial in x , then B,(P(x) ; x ) = P(x) Q,t(x), where Q J x ) is a polynomial in x whose coefficients are polynomials in l/n, having l / n as a factor.
+
3. Show that any function f ( x ) on [a, b] which has only discontinuities of the first kind (breaks), can be uniformly approximated by functions of the form co Z:=l c,,B(x, x,,,; dmf,d,") where as usual ,B(x, xo; 4 , d,) is the simple break function with discontinuity at x,.
+
86
11. RIEMANNIAN TYPE OF INTEGRATION
19. Additional Definitions of Riemann Integrals of Stieltjes Type
The existence of the norm Stieltjes integral N J” f d g we have considered imposes, among other things, the restriction that f and g have no common discontinuity, that of the a-integral requires that f and g have no common discontinuity on the same side of a point, Modifications have been proposed which to some extent bypass these restrictions. Usually such modifications sacrifice some other desirable property of the Stieltjes integral. One of the simplest modifications is that in which the basic function of intervals is F[c, d ] = Q ( f ( c ) + f ( d ) ) (g(d) - g ( c ) ) . We define 19.1. The Mean Stieltjes Integral.
Mean Stieltjes Integral A4 J” fdg of j with respect to g is the limit of Z J f ( x , ) + f ( x , - , ) ) (g(x,) - g ( x t - J ) / 2 , where we obtain the norm integral N M J f d g if the limit exists as 1 a 1 0 and the a M J” fdg if the limit exists as a spreads. If the ordinary integrals exist, so that both C,,f(x,) ( g ( x , ) g ( x t - l ) ) and Zof ( x , - , ) (g(x,) - g(x,-l)) have the same limit, then obviously the corresponding mean integrals exist and give the same value for the integral. Many of the usual properties of integrals, especially those based on the integrals of interval functions are valid, such as the bilinearity of M J” fdg as a function of f and g, the existence of J: fdg with a 5 c < d 5 b if J: fdg exists, and the additive property of the resulting interval function. The integration by parts theorem is almost trivial, since 19.1.1. Definition. The
--f
c 9 ( f ( x , ) + f ( x , - , ) ) ( d X J g(x,-,)) + c 9 (g(x,) + g ( x , - J ) ( f ( x , ) -f(Xt-1)) -
z
a
=
c (f(x,) d x , ) z
-
f(x,-,l g ( x , - J ) = f
-
f (a)g(a)
for all subdivisions, so that we have: 19.1.2. THEOREM. If M
J”: exists, then M
fdg f g d f exists also and s:fdg M S:&f = f ( b ) g ( b ) - f ( a ) g ( a ) Further, since q f ( x , )+ f ( x , - , ) ) ( f ( x , ) - f ( x t - l ) = , w f ( x , ) ) ‘ -f(x,-,)‘) = (f(b))’ - (f(a))‘ for all a, we have: if f ( x ) is any finite valued function on [a, b],then M fdf exists and is equal to ((f(b))‘
+
J”:
4
19.
87
RIEMANN INTEGRALS OF STIELTJES TYPE
- (f(a))2). This shows that f and g may have common points of discontinuity for this definition of integral, even under the norm limit. On the other hand, the substitution theorem: “ if f d g exists and h ( x ) is bounded on [a, b], then J%hfdg exists if and only if J: h ( x ) d J:fdg exists and are equal,” does not always hold. If we take for f ( x ) = g ( x ) the break function P(x, x o ;d,, d,) on [a, b],and for h ( x ) the break function p ( x , x0; d,‘, d2‘), then M J: fdg = M s” f d f = 3 (f(x))‘. Then N M J: hfdg exists if and only if (dl‘ - d,’)dli2 = 0, while N M 1: h ( x ) d , s” f d f exists if and only if dl’di - d2‘d,2= 0. If d,’ = d,‘ but d: then N M J: hfdf and N M J: h ( x ) d , f d f do not exist simultaneously. If d , = d, and d,’ = d.’ they do exist 2.’ simultaneously, but the values might be different with M J : hfdf = 9 d:d,‘ and M J: h ( x ) d , s” f d f = d:d,‘. If g f x ) is of bounded variation and f ( x ) is bounded on [a, b] then the inequality I J: fdg I 5 K J: I dg 1, where K is the 1.u.b. of I f ( x ) 1 on [a, b], leads to the uniform convergence theorems:
Jl
+2,
19.1.3. limn M
J: f,dg
=
Jl
M J: fdg for f, approaching f uniformly on
[a, b] and 19.1.4. Iff is bounded and lim
liml, M
J: 1 d ( g ,
-
g)
1
= 0,
then
J’”fdg,, = M j’fdg.
From the second of these two convergence theorems and the fact that U M Jb f ( x ) d P ( x , xo; d,, d,) exists, if f ( x o 0) and f ( x o - 0) exist at xo,nwhichcan be easily verified, it follows that OM J: f d g exists if f ( x ) has only discontinuities of the first kind. For additional properties of these integrals the reader is referred to H. L. Smith: Trans. Am. Math. Soc. 27 (1925) 491-515; H.S. Kaltenborn: Tokohu Math. J. 44 (1938) 1-11; R.E. Lane: Proc. Am. Math. Soc. 5 (1954) 59-66; P. Porcelli: Illinois J. Math. 2 (1958)
+
124-128. 19.2. Cauchy Left and Right Integrals. Conceptually, the right and left Cauchy integrals based on the interval functions F[c, d ] = f ( d ) (g(d) - g ( c ) ) and F[c, dl = f ( c ) ) ( g ( d ) - g ( c ) ) , respectively, are similar to the mean integrals. Obviously if both the right and left Cauchy integrals exist, then the mean integral exists and is the average of them. For the case where g(x) = x, D.C. Gillespie [Ann. Math. (2) 17 (1915) 61-63] has shown that the existence of the right
88
11. RIEMANNIAN TYPE OF INTEGRATION
or left Cauchy integral implies the existence of the Riemann integral and so equivalence. Modifications of this result if g(x) is of bounded variation have been developed by G.B. Price. The reader is referred to the article by Price: Bull. Am. Math. SOC.49 (1943) 625-630, for details. See also R.F. Deniston, Koninkl. Ned. Akad. Wetenschap. Proc. 52 (1949) 1111-1128 or Indagationes Math. 11 (1949) 385-402. In case g(x) in the discontinuities of the first kind, that is g(x finitely for every point, we can take account the discontinuities of g in the basic function that for any interval I = [c, d ] 19.3. The Y-Integral.
F(I)
= F k , dl
=f(c) (g(c+O)
integral J: fdg has only 0 ) and g(x - 0) exist of this fact by including of intervals. We assume
+
-
g ( c ) ) +f(x') (g(d - 0)
-d C + O ) )
+f ( d l (g(d) - g(d - 0))
with c < x' < d. Such a function of intervals was suggested by W.H. Young [Proc. London Math. SOC.(2) 13 (1914) 109-1501but the resulting integrals he considers are in the norm sense and it turns out that for the case when g ( x ) is of bounded variation, this mode of convergence adds very little in integrability. We shall call Stieltjes integrals based on the above function of intervals Y-integrals. Obviously, the existence and approximation theorems for integrals of functions of intervals apply. In particular, those involving w(SF; I) prove useful, where it should be noted that for the interval function of the Y-integral, the oscillation function w(SF; I ) does not depend on the value of f or g at the end points of the interval I: The reasoning used in 11.10.10 shows that: 19.3.1. THEOREM. If Y J:fdg exists, then f ( x ) is bounded on a finite number of closed intervals, which are complementary to a finite number of open intervals on the interior of which g(x) is constant. For the pseudoadditive condition on F(I) which links the norm and a-integrals, substitution and rearrangement of terms shows that for c < x, < d, we have
F k , dl
-
(Fk, xol + F[x,, 4) = (f(x/)-f(x'')) k ( x 0 - 0)
+ 0)) - (f(x,) -f(x')) (g(x0 + 0) - g(x0 - 0)) + ( f ( x 0 -f(x'/')) (g(d - 0) - g(x, + 0))
- g(c
where c < x' < d, c < x" < xoand x, < x"' < d. For pseudoadditivity this expression must have limit zero as c+ x, - 0 and d-+ x, 0.
+
19.
Now lim g ( c c-z,-o
89
RIEMANN INTEGRALS OF STIELTJES TYPE
+ 0)
= g(x,
-
0 ) and lim g ( d
- 0) = g ( x o
d-xo+O
+ 0).
Consequently, if f ( x ) is bounded, which we assume, we must have fim f f f x , ) - f f x O ) ( g f x , Z'--rX0
+ 0) - g f x o - 0))
= 0,
+
or f ( x ) is continuous if g ( x , 0) # g ( x , - 0 ) . This condition is also obviously sufficient for pseudoadditivity of F(I) . We therefore have : 19.3.2. T H E O R E M . The interval function F(I) for the Y-integral
is pseudoadditive at xo if and only if f is continuous at xo when g ( x , 0) f g ( x , - 0 ) . If g ( x ) is regular at xo in the sense that g(x,) lies between g ( x , 0) and g ( x , - 0), then F(I) is pseudoadditive at x, if and only if f and g have no common discontinuities at x,. The Y J f d g is an extension of the ordinary Stieltjes integral J fdg, for :
+
+
19.3.3. T H E O R E M . If f fdg exists in the ordinary sense, then Y J: f d g exists and the two integrals are equal. For consider the function of intervals F(I) for the Y-integral:
FfI) = f f c ) ( g f c
+ 0)
+
g(c)) f f x ' ) ( g f d - 0) + f f d ) f g f d ) - g f d - 0)) -
-gfc
+ 0))
with I f ( x ) I 5 M. Then for e , > 0 we can find points c < c, < x' < d , < d such that I g ( c 0 ) - g ( c , ) I < e0/3M, and I g ( d - 0 ) g ( d l ) I < e0/3M. Then for
+
F, (I) =f f c) f g ( c , ) -g ( c ) ) +f fx0 ( g f d , )- g f C J ) +f f d ) f g f 4-8 f 4 ) )
I
we have I F(I) - F,(I) < e,, and moreover, Fl(I) is a function of intervals for the ordinary integral. The validity of the theorem then follows from a comparison of F(I) and F,(I) or rather ZnfAg, where ul includes (T and points like c1 and d , for each x i of a. In the reverse direction we can assert: 19.3.4. T H E O R E M . If Y
f f d g exists, and if g ( x ) is continuous, then
$: f d g exists in the ordinary sense.
If g ( x ) is continuous, then F(I) = F[c, d ] = f ( x ' ) ( g ( d ) - g ( c ) ) x' is restricted to c < x' < d for the Y-integral, while for the ordinary
90
11. RIEMANNIAN TYPE OF INTEGRATION
integral c 5 x’ 5 d is permitted. Suppose (e, d,, 1 (T I < d,) : I ,ZUf2lg - Y fdg I < e. Select an arbitrary (T such that I (T I < d,/2 and conf ( x i ’ )(g(xi) - g(xi-l)), with n points of sider the ordinary sum L’,, subdivision and xi-l 5 xi‘ 5 xi. To compare with a sum for a Y-integral we need worry only about the points xi’ which are either ximlor xi. If for a given i: xi’ = xi = x’% + I , then f f x i ’ ) ( g f x , )- gfxi-,)) +ffX‘i,J k f X i + J - g f x , ) ) = f W k f X i + J - g f X i p l ) where X i - 1 < xi < xi+l and I xi-l - xiPl < d,. If for a given i, we have xi’ < xi - x ‘ ~ +the ~ , continuity of g ( x ) at x i gives us an xi” such that xi‘ < xi” < xi < xi+l so that g(x”%)- g(xi) I < e/2nM, and so
Ji
I
I
I -
+fW k f X i + J
[ f f X i O M X i ) - gfXi-1))
[ffXi’)
fgfx,”) - g f x i - J )
+ffxJ
kfXi+J
-
gfx,))I
- gfx,”))l
I < eln.
By considering each point of subdivision of (T in succession we arrive at a new subdivision (T’such that I u’ I < d, and such that
I
c .ffXi’lkfX,) a’
-g6i-J) -
c ffx,’)fgfx,)
I
-gfXi-J)
<e
(I
where Xi-1 < Xi‘ < Xi. From this it follows that
I
2 f f x i ’ ) f g f x i )- g f x i - J ) - Y
/ * f d g 1 < e, for
1 0 I < d,/2
U
and xi-1 5 xi’ 5 xi, for all i. If g ( x ) is the break function ,8(x, x,; d,, d2), then (T Y Jfdg exists for any function f bounded in the vicinity of x , and (T Y fdg = f(x,)(g(x, 0 ) - g(x, - 0 ) ) , where x , - 0 is replaced by a if x , = a and x , 0 is replaced by b if x , = b. On the other hand, for d, # 0, NY J: fdg exists if and only if f ( x ) is continuous at x,.
+
Ji
+
19.3.5. The Y-integrals have the usual linearity properties relative to f and g. Also if Y fdg exists, then Y J: fdg exists for a I c < d 5 b, and the resulting interval function is additive. Note that Y J: does not depend on the behavior of f or g for x < c and for x > d.
Ji
19.3.6. If Y J: fdg exists, and f is bounded, then the function h ( x ) =
+
1
fdg reflects the continuity properties of g ( x ) in that h(x 0) h f x ) = f f x ) f g f x + O ) - g f x ) ) and h f x ) - h f x - O ) = f f x ) f g f x )g(x - 0 ) ) for all x. For by 11.4.7 lim z-z,+o
I
I’ xo
F(I)
- F[x, x,]
I
= 0.
19.
91
RIEMANN INTEGRALS OF STIELTJES TYPE
Now for our F(I) we have
ma,
XI
=f(xo) k ( X ,
+ 0)
-
g(x0)) + f ( x ’ ) ( g ( x
+f(x)(g(x
- 0) -g(x)).
Also lim g ( x ) = lim g ( x l + Z
+o
+ 0 ) - g(x,+OU
-
0)
= g(xo
z-l,+o
+ 0),
so that h(x0
+ 0) - h(xJ
= z-
lim F[x,, X I = f ( X o ) k ( x 0 Z0+O
+ 0)
-
g(x,)).
Similarly for xo - 0. The substitution theorem is valid in its usual form: If Y J: fdg exists, and h ( x ) is bounded on [a, b], then Y J: h ( x ) d , J: fdg exists if and only if Y h ( x ) f ( x ) d g ( x ) exists, and then the two integrals are equal. This follows in the usual way. For for any interval [c, d ] as a result of the discontinuity properties of J l f d g , we have 19.3.7.
JR
I [h(c) j.:+h? + h ( x ‘ ) .fd-Ofdg c+o + h(d) -
[h(c)f(c)(g(c
-
g(c
=
+ 0)
-
g(c))
+ 0)) + h(d)ffd)(g(d) I h(x’,J (ja-of%-.N-x‘)fg(d c+o
a-0
fdgl
+ h(x’)f(x’)(g(d - 0 )
-
g(d
-
0)
- 0))l -
g(c
I
+ 0)) I
5 M w ( S f i l g ; [c, d ] ) where I h ( x ) I 5 M for all x. The rest of the proof follows that given in 11.11.6. If g ( x ) is of bounded variation on [a, b] and f ( x ) is bounded and if Y x f d g exists, then obviously I Y J: f d g I 5 M J: I dg 1, where I f ( x ) I I M on [a, b]. This inequality gives the usual convergence theorems based on uniform convergence : 19.3.8. If lim/,fJL(x)= f ( x ) uniformly on [a, b], g ( x ) is of bounded
variation on [a, b] and if Y J: f,!dg exists for all n, then Y Jgfdg and lirn,( Y J:f,dg exist and are equal. 19.3.9. If f ( x ) is bounded on
[a, b], g,(x) and g ( x ) are of bounded
variation with limn J: I d(g,, - g ) I = 0, and if Y J:fdg,, exist for all J: f d g and Iim, Y J: f d g , exist and are equal.
n, then Y
92
11. RIEMANNIAN TYPE OF INTEGRATION
From this last theorem we deduce in the usual way that if g ( x ) is a pure break function of bounded variation on [a, b], that is, g ( x ) = ZnB(x, x,; d,', d,,") with 2, (1 d,' I I d," - d,' I) < co; and if f( x ) is bounded on [a, b],then (T Y J: fdg exists and has as value
+
Cnf(Xn)dn"
Consequently:
+ 0) - g(xn - 0)) = C 2 f f x ) ( g f x+ 0) - g ( x - 0)).
= C n f ( x n ) (g(x,
19.3.10. THEOREM. If (T Y Slfdg exists, then (T Y fdg, exists and so (T Y s",fdg, exists, where g, and g , are the continuous and break parts o f g. It follows then that s",fdg, exists, so that
Jl
+
where a - 0 = a, and b 0 = b. From this theorem, we can conclude that if f ( x ) is bounded, and g ( x ) is of bounded variation on [a, b], and if J: fdg, exists, then Y J: fdg can be defined in two ways, viz.: limo
[(ffXi-JkfXi4 - g(Xi-1 0))
+
+ 0)
- gfXi-1))
+f(x,') k ( X , - 0)
+mi)M X , ) - g ( x , - 0))l or as s:fdg, + ~ 2 f f x ) ( g f+x 0) - g ( x - 0)).
Since a function which is bounded and has only discontinuities of the first kind has at most a denumerable number of discontinuities, we have : 19.3.11. If f ( x ) is bounded and has only discontinuities of the first kind, and if g ( x ) is of bounded variation, then (T Y s",fdg exists. In particular, (T Y J: fdg exists if f and g are of bounded variation on [a, b]. For the v,-measure corresponding to g,, the continuous part of g for a denumerable set of points is zero, so that J: fdg, exists. If N Y J: fdg exists, then f ( x ) is continuous at all points where g ( x 0) f g ( x - 0 ) . If then g ( x ) is regular, i.e., g ( x ) lies between g ( x - 0) and g ( x 0) for all x, then f ( x ) is continuous at all points of discontinuity of g ( x ) . It follows that if gb(x) is the break function for the function g ( x ) of bounded variation which is regular, then N Y J: fdg, and N J: fdg, exist or do not exist simultaneously, so that we have:
+
+
19.
93
RIEMANN INTEGRALS OF STIELTJES TYPE
19.3.12. If g ( x ) is a regular function of bounded variation then NY J: fdg exists if and only if N J: fdg exists. This holds in partiuclar if g ( x ) is monotone on [a, b]. The NY J: fdg, with g ( x ) of bounded variation is then essentially equal to the ordinary N J: fdg. However, the u Y J: fdg contributes something new in that it allows for common discontinuities of f and g, and gives a formal expression which is closely related to that of the Lebesgue-Stieltjes integral considered later. 19.3.13. Integration by Parts Theorem. Iff and g are both of bounded variation on [a, b] so that both u Y J: fdg and u Y Jlgdf exist, then
+ Y/"df f f b k f b )- f f a ) g f a ) - c [ f f f x+ 0) - f f x ) ) f g f x + 0) - g f x ) ) YJ"dg
0
=
0
2
-
fffxl - f f x
- 0))
f g f x ) - g f x - 0))l.
For the u Y-integral then, the integration by parts formula involves corrective terms due to the common discontinuities of f and g , when f and g are both of bounded variation on [a, b]. Suppose f and g have a finite number of discontinuities at the points x1< x , < ... x , ~Then . on the open intervals (xi-l 0, x i - 0 ) , f and g are both continuous, so that the Y integrals reduce to ordinary integrals, and the usual integration by' parts formula applies. Then
+
where xo - 0 = xo = a, and x , + ~+ 0 = x,, = b. To the first term on the right hand side we apply the ordinary integration by parts theorem, while for the second term we note that
+ gdf) = f f x ) f g f x + 0) g f x + g f 4 f f f x+ 0) - f ( x 0)).
Y/x'o ( f i g 2-0
-
- 0))
-
Then
Yf f d g + Y f gdf = n&l a
+f(x,)(g(x,
+ 0)
i =O -
[f(xi-O)g(xi-O)
g(xi
-
0))
+g(xJ
-f(x,-,+O)gfx,-,+O) fffXi
+ 0) -ffxi-o)A
94
11. RIEMANNIAN TYPE OF INTEGRATION
The expression under the summation sign can be arranged in various ways. It can be written for x, = x as ( f ( x ) - f ( x - O))(g(x) g ( x - 0 ) ) - ( f ( x 0 ) - f ( x ) ) ( g ( x 0 ) - g ( x ) ) , or as the determinant f(X+O) g ( x - 0 ) 1
+
+
f (x) f(x
-
g (x)
0) g f x
+ 0)
1
1
We have then shown that the formula of the theorem holds if f and g have a finite number of discontinuities. Since if f and g have an infinite number of discontinuities, they can be approximated by function f,, and g , having only a finite number of discontinuities in such a way that J: I d(f?,- f ) [ and J: I d(g,, - g ) I converge to zero, it follows easily that the formula is valid for any f and g o f bounded variation on [a, b]. If the ordinary integration by parts theorem is to hold for [a, b], then the corrective term involving the discontinuities must vanish. If this is true for all intervals [c, d ] and so by continuity for all ( X - 0, x 0 ) for all x , then ( f ( x 0) - f ( x ) ) ( g ( x 0 ) g ( x ) ) - ( f ( x ) -.f(x - O ))(g(x) - g ( x - 0 ) ) = 0 for all x. This happens for instance when f and g have no common discontinuities on the same side of a point. It also happens when there exist functions A ( x ) and B ( x ) such that f(x) = A ( x ) f ( x - 0 ) B ( x ) f ( x 0 ) and g ( x ) = B ( x ) g ( x - 0 ) A ( x ) g ( x 0 ) for all x, for instance, A ( x ) = B ( x ) = 4 for all x. The Osgood convergence theorem for integrals is valid for the Y-integral in the form:
+
+
+
+
+
+
+
19.3.14. If j ; , ( x ) are uniformly bounded on [a,b],if g ( x ) is of bounded variation on [a, b], if lim,,f , ( x ) =f(x) except for a set of v-measure zero ( v ( x ) = I dg I), and if Y J: f,,dg and Y J: f i g exist then limn Y J: f,dg = Y J: fdg. From the corresponding theorem for ordinary integrals it follows that Iim,, Y J: f,dg, = Y J: fdg,, since because of continuity of g,
Jz
19.
95
RIEMANN INTEGRALS OF STIELTJES TYPE
these are ordinary integrals and a set of v-measure zero will also be of vc-measure zero. For the convergence of Y J: f,,dg, to Y f d g , we use the following theorem from the space 1' of absolutely convergent series :
Ji
19.3.15. Lemma. If the double sequence a,,, is bounded and 1im,,amn= a, for all m, and if C,,, b,, < co, then lim,/ C,a,,,b, = Z,a,b,.
1
I
Obviously, under the hypothesis of the theorem the sequence a, is bounded so that C,a,b, is absolutely convergent. We then write
Since 2, I b, I < co, and a,,, and all/ are uniformly bounded, for a given e > 0, we can find a p such that
Since further
there exists n, such that for n > n,, we have
Consequently, (e, n,, n > n,) :I 2',,lam,Lb,,l - Z,a,b,,
+
I < e.
Now y s:f),dg, = q J ( x , , J ( g ( x , n 0) - g ( x , - O)), where x , are the points of discontinuity of g . Since lim,{f , , ( x ) = f ( x ) except for a set of v-measure zero, it follows that lirnTL f , ( x , ) = f ( x , ) if g(x, 0) - g ( x , - 0 ) f 0, since then x , will not be of v-measure zero. If we set am,l= f n ( x , ) , a, = f ( x , ) and b, = g ( x , 0) g ( x , - 0 ) , the lemma applies. The Osgood theorem for Y-integrals then follows from the decomposition Y J: f d g = J: fdg, Y J:,fdg,. Whether a theorem holds, relating Y J: to Y J: fdg after the manner ofII.15.4, we do not know. The fact that if lim,,g,,(x) = g(x) for all x , the discontinuities o f g ( s ) are not related to those of g , , ( x ) is a source of trouble in deriving such a theorem.
+
+
+
96
11. RIEMANNIAN TYPE OF INTEGRATION
EXERCISES 1. Prove the theorem that if f ( x ) is bounded on [a, b], and g ( x ) is of fdgc exists by bounded variation, then a Y s", fdg exists if and only if applying the condition limuZu~(Sfdg,I) = 0.
s:
2. Show that for the Y-integral, the a-convergence can be replaced by the R-convergence of 11.13.10, where a,Ra, if al < I a2 1 and a1 contains more points of discontinuity of g ( x ) than a2.
I I
3. Show that if f ( x , y) is bounded on [a, b] x [c, d ] ; g ( x ) is of bounded variation on [a, b] and h(y) on [c, d ] ; a Y dg(x) f ( x , y) exists for each y of [c, d ] and a Y J : f ( x , y)dh(y) exists for each x of [a, b ] ;then Ys", dg(x) f ( x , y)dh(y) and ( s l d g ( x ) f ( x , y ) ) d h ( y ) both exist and are equal, the integrals being a Y-integrals throughout.
J'!
x
4. Let g ( x ) be of bounded variation on [a, b] and f ( x ) bounded and suppose aY s", fdg exists. Set F(x) = f ( x ) for U SX Sxo and F(x) = 0 for xo < x l ; b. Show that UYf
Ffx)dg(x) = q ' " d g
+ffxo)(gfxo
+ 0) - gfxo)).
5. Under the hypotheses of Ex. 3 above, with [c, d ] = [a, b], show that if the integrals are s Y-integrals, then
+ C2f(XY x ) [ ( g f x ) - g ( x - 0)) ( h f x ) - hfx - 0)) - ( g ( x + 0) - g f x ) ) (h(x + 0) - h f x ) ) l . 6. A modified Stieltjes integral defined by B. Dushnik (University of Michigan, Ann Arbor, Michigan, Dissertation, 1931) is based on the interval function F(I) = F[c, d] = f ( x ) (g(d) - g ( c ) ) with c < x < d. Determine which of the usual properties hold for the resulting norm and s-integral and in what way it is related to other definitions of Stieltjes integrals. In particular show that if g ( a 0) and g(b - 0) exist and if the modified integral exists, then the corresponding mean integral exists.
+
7. Let f ( x ) be of bounded variation, and g ( x ) be bounded and have only discontinuities of the first kind on [a, b]. Does Y f fdg exist?
8. Let g,(x) and g ( x ) be such that s", I dg, I l;M for all n and lim,,gn(x) = g ( x ) for all x . Let f ( x ) be also of bounded variation. Does it follow that lim,sY s", fdg, = OY s", fdg?
20.
97
STIELTJES INTEGRALS ON INFINITE INTERVALS
20. Stieltjes Integrals on Infinite Intervals
For Stieltjes integrals on (- co, cedure and define:
+ co), we can follow the usual pro-
20.1. Definition. If f ( x ) and g ( x ) are defined for --.a <x
< + co,
and J:fdg exists for all a and b, then JIzf d g is defined as lima+-m,b++oo Jffdg provided this limit exists. fdg involves an iterated limit, We note that the definition of viz., lima,, limuL'lJAg. One might then consider the inverse order. In particular, if a norm integral is involved we start with a subdivision of (- co, co) of finite norm and set up Za.f(x,')(g(x,) - g ( x t w 1 ) ) . On the infinite interval this is an infinite series. It is then natural to assume that for all (T of finite norrp and all choices of x,' with x , - ~5 x,' 5 x,, the expression Z,,f(x,') ( g ( x , ) - g(x,-,) is convergent, order being prescribed by x,-~< x,, i ranging over the positive and negative integers. Then JTz f d g could be defined as
Jzz
+
Sufficient conditions that the two iterated limits yield the same value can be obtained by applying the iterated limits theorem 1.7.4. For instance, it is sufficient to have limlu l + o L' JAg = J: f d g piformly relative to a and b. Or to have the infinite series ZufAg converge uniformly in cr in the sense that for e > 0, there exist a, and be such that if a < a , and b > b e then I zlu,fAg I < e for all cr' of finite norm of (- co, a] and [b, co). The second of these uniformities occurs if f is bounded and g is of bounded variation on (- co, co) in the sense that J: I dg 1 is bounded in a and b.
+
+
20.2. Definition. If g is of bounded variation on every finite interval then f is said to be absolutely integrable on (- co, co) if Jffdg exists for all finite intervals [a, b] and JTz 1 f l dv exists, where v ( x ) = I dg 1. Obviously if f is absolutely integrable on (--a,co), then f is integrable, and I f d g I 5 J?: I f l dv. Stieltjes integrals with infinite limits play an important role in the theory of integral transforms, such as the La Place transform: JLz e-"dg(x) ;the Fourier transform: JLz eiz"dg(x);the Mellin transform: J,"z-"dg(x) ; and the Stieltjes transform: JZz [dg(x) / ( z x ) ] . The use of the Stieltjes integral permits simultaneous consideration of distributions such as JLz e-"g(x)dx and infinite series such as 2n a ne-'"n = Z,,anbnz with b,, = ex*.
+
Jz
Jzz
+
+
98
11. RIEMANNIAN TYPE OF INTEGRATION
Theorems concerning Stieltjes integrals with infinite limits can be deduced from the corresponding cases for finite limits by adding additional assumptions. We give two samples: 20.3. Osgood Theorem Generalized. If g ( x ) is of bounded variation for all finite intervals [a, b ] ; if f , , ( x ) are uniformly bounded on every [a, b ] ; if lim,f,(x) = f ( x ) for all x ; if JLZ f,,dg exist for all n in such a way that lim,,,Jh+53 f,,dg = 0 and lima---m JYm f , d g = 0 both uniformly as to n ; and if J: f d g exists for all a and 6 , then limn J22 f,,dg and JI: f d g exist and are equal. For by the Osgood theorem for finite intervals limn J: f,,dg = f ’ f d g for all n. Then by the iterated limits theorem 1.7.4, limnlima,bJ: f n i g = lima,Jimn J:f,,dg, or limTZ J?gf,,dg = J Z . f d g . In a similar way we prove:
20.4. If gn(x) are uniformly of bounded variation on every [a, b ] ; if lim,g,(x) = g ( x ) for x dense on - co < x < co, and g ( x ) is of bounded variation on every [a, b ] ; if f ( x ) is continuous on - co < x < co; and if J?: fdg,, exist uniformly as to n, then limn JLz f n d g and :.J! fdg exist and are equal. Other theok-ems such as an iterated integral theorem can be proved from the case for finite limits by adding sufficient conditions for interchange of limits and integrals and iterated limits in general. The integration by parts theorem in its usual form need not hold. However, the existence of JLz fdg will carry with it the existence of
lim a--co,b++
[f(b)g(b) -ffa)g(a) 00
-
,;
gdf
I.
Consequently, we can state: 20.5. THEOREM. If JLg f d g exists, and if lim,+-,f(a)g(a) and limb++a, f ( b ) g ( b ) exist, then gdf exists and J?: (fdg gdf) = firnb++a,f ( b ) g ( b ) - %+-m ffa)g(a).
J?z
+
21. A Linear Form on the Infinite Interval
In recent years considerable attention has been paid to linear forms on the class of continuous functions C , each of which vanishes outside of some closed bounded interval. This class of functions is obviously linear, and can be normed by defining llfll = l.u.b.,.f(x) I, since obviously f ( x ) is bounded. However, the space is not complete un-
21.
A LINEAR FORM ON THE INFINITE INTERVAL
99
+
der this norm. For if you set f,,(x) = (sin n x)/n for n 5 x 5 n 1 and - n - 1 5 x I - n, and zero elsewhere, then for the sequence we will have limn,,811F,(x) - Fm(x) 11 = 0. But Fm(x) = -y= .= : f,(x) which is not in C,. Fm(x) converges in the norm to Z 21.1. Definition. A linear form L ( f ) on C, will be said to be continuous if for every sequencef,(x) such that there exists a finite interval [a, b] independent of n such that f , , ( x ) vanishes outside of [a, b], and lim,f,(x) = f(x) uniformly on [a, b], then lim,L(f,) = L(f). For such a continuous linear form, the Riesz theorem reads:
21.2. THEOREM. There exists a function g(x) of bounded variation
on every finite interval, such that for every f of C, we have L ( f ) = Jzg fdg, where L ( f ) is a linear continuous form on C,. By the Riesz theorem for a finite interval, there will exist a function of bounded variation g,(x) on [- n, n] such that if f(x) is continuous on [- n, n] then L ( f ) = J?,fdg,,. This expression will be effective also if f(x) vanishes for x 5 - n and for x 2 n. Now for every f(x) vanishing for x S - n 1 and for x 2 n - 1, we will have J-Y;E:lfdg,-, = Jz:lfdg,. Consequently, g,(x) = g,&) except for 1 < x < n - 1. If we alter a denumerable set of points on - n 1I x 5 n - 1, for all n, and g,(x) to agree with g,-,(x) on - n set g(x) = g,(x) for - n < x < n, we obtain a function of bounded variation on every finite interval such that L ( f ) = fdg for every f vanishing outside of a finite interval. This integral gives then the most general linear continuous form on C,. It should be mentioned in passing that if C is the class of bounded continuous functions on - 00 < x < 00, with I [ f I I = 1.u.b.J f ( x ) 1, then the most general linear continuous form is not usually expressible as a simple Stieltjes integral J22fdg.
+
+ +
flz
+
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CHAPTER Ill
INTEGRALS OF RIEMANN TYPE OF FUNCTIONS OF INTERVALS I N T W O OR HIGHER DIMENSION
1. Interval Functions
We consider briefly integrals of interval functions on spaces of dimension greater than one. Their properties parallel in some respects those for the integral in the one-dimensional case. For convenience we shall limit ourselves to the two-dimensional case from which one can usually deduce the situation for the n-dimensional one. Let J denote the basic rectangle or interval: a, 5 x, 5 b,, a, 2 x, d b, or [a,, b,; a,, b,]. We shall denote by I = [c,, 4; c,, d J , with a, 5 c, < d, 5 b,, a, 5 c:,< d, 5 b, any subrectangle or subinterval of J with sides parallel to those of J. A function of intervals F(I) assigns one or more finite real numbers to every interval I of J. We shall not distinguish in this chapter between open and closed intervals, but in general the values of F(I) for any I will depend only on the points interior to or on the boundary of I. We give some instances of interval functions which to some extent parallel those of the one-dimensional case. (1) F(I)
=
(d, - c,) (d, - c,), that is, the area of I.
(2) F(I) = fffd,) -ffc,)) (gfd,) - gfc,)), whereffx,) andgfx,) are functions on [a,, b,] and [a,, b,], respectively. (3) F(I) = f ( d , , d,) -f(c,, c,), where f(x,, x,) is a function on J. (4) F(I)
=m,,d,)
d,) - f f d , , c,) + f ( C , Y CJ x,), where f(x,, x,) is defined on J. ( 5 ) F(I) = 1.u.b. [I, f(x;', x,") - f ( x l ' , x i ) I for all (x,', x i ) and (x,", x,") of I], that is, the oscillation w ( f ; I ) of the point function f on I. = A::
-ffC,,
4:f b , ,
101
102
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
( 6 ) If F(I) is any interval function on J , then I F(I) I and 1.u.b. [F(I,) for all I, in I ] , or g.1.b. [F(I,) for all I,, in I ] define additional interval functions. The last two are finite valued only if F(I) is bounded on J. (7) If g(x,, xz) is a point function on J and F(I) an interval function, then g(xl’, xz‘)F(I) defines a many-valued interval function, if (XI’, xz’) ranges over I. 1.l.Definition. A subdivision a of an interval Iconsists of a finite number of intervals having at most edges or vertices in common, whose sum or union is I A subdivision of the fundamental interval J obviously induces a subdivision on any subinterval I. A subdivision will be called a net if it is determined by lines completely across 1 parallel to the coordinate axes, so that a net is determined by subdivision a and ax2of the projections of I on the coordinate axes. We shall 21 denote a net by a1xo9. 1.2. Definition. A function of intervals F(I) is additive if it is singlevalued, and if for any subdivision a of an interval I consisting of the intervals I,, ..., I,, we have F(I) = ZzF(Ii). To show that a given interval function F(I) is additive it is obviously sufficient to verify additivity on subdivisions of net type since we can superimpose on any subdivision a net by extending the sides of the intervals of a to the boundary of I, and then reassemble the net divided intervals 11,..., I , constituting a. If f ( x l , x2) is a point function on J, then F(I) = At; Af: f = f ( d l , d,) - f(c,, d2) - f ( d , , c2) f(c,, d,) is additive. Conversely, if F(I) is additive, then the point function f(x,, x2) = F[a,, xl; a*, xq]determines F(Z) in the sense that F(I) = A:: 4: fb,,X J .
+
1.3. Definition. A single-valued function F(I) is upper semiadditive, if for any subdivision (T = I,, ..., I , of any interval I we have F(I) 5 ZiF(Ii). The definition for lower semiadditivity reverses the inequality. If F ( I ) is additive, than I F(I) I is upper semiadditive.
2. Subdivisions
In dealing with Riemann type of integrals on a rectangle J, we note that we have available two types of subdivision: (a) a general subdivision a, and (b) a net type of subdivisions a,xa,. Subdivisions can be ordered by the condition that a’ 2 a“ provided 0’consists of sub-
3.
103
INTEGRALS
divisions of intervals of a”, or if every interval of a’ is contained in some interval of a”. Since any subdivision of J induces a subdivision of any interval I of J, it follows that the product of two subdivisions g ‘ . a’’is again a subdivision of J such that a‘a“2 a’ and a‘a“2 a“. So that subdivisions a of J form a directed set under this ordering. If we define partial order on nets by the condition that a1’xa2’ 2 and a’2z2 a” then nets form a dia,” x a,”, provided a 21 2 rected subset of subdivisions which is cofinal with the set of all subdivisions, as any subdivision contains the net obtained by extending the edges of all its intervals to the boundary of J . We can also order subdivisions by norm. At least two norms are available (a) if I I I A is the area of I, then I a is the maximum I I I A for any I of a ; (b) if I I = maximum side length of I, then I a is the maximum of I I for all I of a. A limit taken as I a 0 is weaker than a limit as I a I A + 0, since any statement true for all a such that I a I A < d will be true for all a such that I a < 4 2 . For nets we can take 1 0 l A = I ax, * I axzI and l a the greater of I az11, I o x z ( . $2’
Is Is
Is+
I
Is
Is
Is
3. Integrals
The interval function F ( I ) will have a Riemann integral in case the expression G(a) = .ZtF(Ii), where a = I,, ..., I,, is a subdivision (or net) on J, has a limit in a according to an ordering of a of the type discussed in the preceding paragraph. In particular, we have the a-integral, the net a-integral, the area normed, and the side normed integrals, and the normed net integrals. If F ( I ) is an additive function of intervals, then all of these integrals exist and J I F ( I ) = F ( J ) . If F (I ) is upper semiadditive (lower semi additive) and if G(a) is bounded above (below) in a, then o J j F ( I ) exists and agrees with the least upper bound (greatest lower bound) of G(a) as to a. In this case it is sufficient to limit one’s self to subdivisions of the net type. Since the net subdivisions are cofinal with the general subdivisions, if an integral exists for general subdivisions, it also exists in the same sense if one limits one’s self to nets, and yields the same value. However, the net integral a,x a, JjF(I) can exist without having the a JJF(I) exist. For example if J = [0, 1 ; - 1, 11 and for I = [c,, d,; c,, d,] we define F ( I ) = d2 - c2if 0 5 c2 < d,; = d, c2if c, < 0 < d, and = c, - d, if c2 < d, I 0. Then a x a, J j F ( I ) exists and has value
+
104
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
0. But a J j F ( I ) does not exist, since by proper choice of (T any value can be approached by the approximating sums. Unless otherwise indicated, we shall limit ourselves to integrals based on general sudivisions, and for the norm integral take as norm I a Is the maximum side length of intervals of the subdivision. Since any of the definitions of integrals depend on directed sets, the corresponding Cauchy conditions of convergence are applicable. For the integrals we are considering, the following theorem can be proved along the lines of 11.3.6 for the space of one dimension: 3.1. THEOREM. A necessary and sufficient condition that the (T or norm integral exist is that Jjw(SF; I ) = 0 for the corresponding integral. Here w(SF; I ) = 1.u.b. [I ZulF-Zo2FI for all al and a2of I ] . This theorem is not valid for net-convergence, as the example cited above shows. For if I is any interval entirely in [0, 1; 0, 11 or entirely in [0, 1; - 1, 01, then "(SF; I ) = 00. If the norm integral exists, so does the a-integral. In one dimension, the condition to be added to the existence of the a-integral in order to obtain the existence of the norm-integral was the pseudoadditive property at each point of J. Since in two dimensions, the boundaries of intervals I are lines and points, pseudoadditivity at points is not sufficient. We suggest the following more complicated notion :
3.2. Definition. F ( I ) will be said to be pseudoadditive along a line 1 in J parallel to a coordinate axis, if for every e > 0, there exists a d , such that if a finite number of nonoverlapping intervals I,, ..., I , with I I, Is < d, are cut by the line 1 into intervals Ill, I1", ..., In', In1', respectively, then I Z,F(I,) - Z,(F(I,') F(I,")) I < e. It is obvious that if N J j F ( I ) exists, then F ( I ) is pseudoadditive for all lines 1 parallel to the axes. We need only apply the Cauchy condition of convergence to a comparison of the subdivision a' obtained from a by replacing I , by I,' and I," for i = 1, ..., n. Conversely, if a JjF(I) exists and F ( I ) is pseudoadditive along all lines 1 of J parallel to an axis, then the norm integral N J,F(I) exists and is equal to a JJF(I). The proof follows the procedure in one dimension, in that we determine a ae such that a 2 a, implies I ZuF(I) - a JjF(I) I < e. We then determine d, so that I a I < d, implies that I ZoF(I) ZaaeF ( I ) I < e, by using the pseudoadditive condition along the finite number of sides of intervals of ae, Combining statements gives: if I a I < d,, then I Z J ( I ) - a JjF(1) I < 2e.
+
3.
105
INTEGRALS
3.3. If JjF(Z) exists and Z is contained in J then JIF(I) exists in the
same sense, so that G ( I ) = JIF(Z) defines an interval function on J, which turns out to be additive. This follows in the usual way from the Cauchy condition of convergence by comparing the sums for subdivisions which vary only on Z and remain the same on the complement of Z relative to J. The theorem does not hold if convergence is only on nets as the above example shows. Here JjF(Z) = 0. However if I = [0, 1; 0, 11, then JrF(Z) = co. The difficulty is traceable to the fact that in net subdivisions an alteration of the subdivision on I affects also the division of the complement of Z. The additivity of the function JIF(Z) follows in the usual way. The converse to additivity reads : 3.4. If Zl,
..., I , is a subdivision of J and
JIiF(Z) exist for i = 1, ..., n, then u JJF(Z) exists and (T JjF(Z) = Zia JIiF(Z). For norm integrals, the condition that F(Z) be pseudoadditive along the sides of all Zi must be added. In two dimensions we have: (T
3.5. Approximation Theorem. If u = I,, ..., Z , is any subdivision of J and JJF(I) exists, exists, then 1 JJF(Z) - ZoF(Z) I 5 Zmlw(SF;I ) . This leads at once to:
I = 0. In particular, if Z converges to the point P = ( x ] , x 2 ) , then the convergence is uniform in ( x l , x2) on J . If (T J,F(Z) exists then 3.6. If N JjF(1) exists, then lim,I,-ol N JIF(Z) - F(Z)
if P is the same oriented vertex (lower right hand, upper right hand, etc.) for the intervals involved. Because of the validity of the approximation theorem the following substitution theorem is available : 3.7. T H E O R E M . If F(Z) is an interval function such that JjF(I) exists giving rise to the interval function G ( I ) = JIF(Z); if h(x,, x 2 ) is a bounded point function on J and H ( I ) = h(xl', x2')G(I), (xl', x2') any point in I, then JjH(I) = JJh(x,, x,)G(Z) exists if and only if JJh(x1, x2)F(Z) exists and the two integrals are equal.
106
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
It is understood that in this theorem, the same convergence process using general subdivisions is used throughout. For h(x,, x,) = I on J , this theorem asserts in effect the differential equivalence of the additive interval function G ( I ) with that of F(I). Linearity and convergence properties of integrals of interval functions in two dimensions carry over without difficulty from the onedimensional case. 4. Functions of Bounded Variation in Two Dimensions 4.1. Definition. In passing from functions of one variable to func-
tions of two or more variables, that is, f ( x , , x,) defined on the rectangle or interval J = [a,, 6 , ; a,, b,,],the choice of the interval function and its use in defining a function of bounded variation has taken different forms, depending on the objective of the investigator. If I is the rectangle [c,, d,; c2 d,], then a possible F ( I ) is I f ( d , , d,) f(c,, c,) I = I A,, f 1, a first order difference. Closely associated with this is the oscillation function of f on the interval I : (0
('I)
= 1.u.b. [If(x,',
x z ' ) -f(x,", x,")
1
for (xl', x,')
and (xl", x,") on I ] . A second possibility is the double difference which involves all four vertices of I, namely, F ( I ) = I f(d,, d,) - f(c,, d,) - f(d,, c,) f(c,, c,) I = I A,A, f ( I ) I = I A,A, f ( I ) I. Since this second expression is more closely related to integration in two variables, we shall limit our discussion to bounded variation based on this interval or difference function. Bounded variation based on the interval function d,A, f ( I ) is due to Vitali, whose definition was later modified by Hardy and Krause. A full discussion of the interrelations between a variety of definitions of functions of bounded variation in two variables together with references will be found in the paper by J.A. Clarkson and C.R. Adams: On the definitions of bounded variation for functions of two variables, Trans. Am. Math. SOC.35 (1933) 824-854.
+
4.2. Definition. The function f ( x , , x e ) on the rectangle J:[a,, b +' a,, b,] is of bounded variation on J if E0F(I) = L o [ A,A,f(I) I 1s bounded as to subdivisions CJ of J, where if I = [c,, d,; c2,d,], then a
F(I)
=
I A,A,f(I) I = If @ ,
d,) -ffcl, d,) -f(d,, c,) +f(c,,
c2)
I.
4.
BOUNDED VARIATION IN TWO DIMENSIONS
107
Since the interval function d,d,f(Z) is additive, the interval function F ( I ) = I d d f ( I ) I is upper semiadditive on J. Then f(x,, x2) l ? is of bounded variation on J if and only if (T SjF(I) = (T SJ I dld2f I exists. Moreover, since the net subdivisions ( T x~ 8, are cofinal with (T, it is sufficient to know that SoF(I) is bounded on nets (T,X(T,. The total variation of f on J is expressible as lim u102
c I o,o,fV)I
= a,xa,
JJ I dld2fl -
ula2
We shall write V ( J ) = JJ I d,d,fl. From the properties of integrals we deduce that if SJ 1 d,d,fI exists, then for any I contained in J SI I d,d,,fI = V ( I ) exists. Further, V ( I ) is an additive function of intervals, positive or zero for all I so that V ( I l )5 V(I,) for I , in I,. As a consequence, V ( I ) generates and is generated by the point function v(xl, XJ = V[a,, xl; a,, x2]. If f(x1, xz) = g(x1) f h(x,), then SJ I d l d ~ f l= S I I d ~ d z f l== 0 for all I . Conversely, if I d,d,fl = 0, then F(1) = I d,d,f(Z) I 5 JI dld,fl = 0 for all 1, so that f(x,, x,) =f(xl, a,) + f ( a , , xz) f(a,, a,), or f(xl, x,) is the sum of a function of x, alone and a function of x, alone. This combination of functions replaces the constant function for functions of bounded variation in one variable. Because of the linearity of the operator d,d,, the addition of such a function to any function of two variables does not affect the bounded variation property of the function. If f(x,, x,) =g(x,)h(x,), then J l d 2 f = A,gd,h, so that f is of bounded variation on J for nonconstant functions f and g if and only if g(x,) is of bounded variation in x1 and h(x,) is of bounded variation in x,, and then
SJ
I
If d,d,f(I) L 0 for all I, then F(I) = I d,d,fl is additive so that SIF(I) = F ( I ) . The functions f(x,, x,) such that f(x,, a,) = f(a,, x,) = 0 and for which old,f ( I ) >= 0 for every I have the additional property thatf(x,, x,) is monotone in x, for each x, and monotone in x, for each xl. For
4::' 4: f = f(xl'" x,) =
fh,'"x,)
-
ffxl',x,)
- f(X,
' 9
0,)
+f(xl"
-f(x,', xzl
which is 1 0 for x,' < x,". In the same way it can be shown that
108
111. RIEMANN INTEGRALS
IN TWO DIMENSIONS
f(x,, x,") - f(x,, x,') is positive or zero if x,' < x;', and so f(x,, x,) is monotone in x, for each x,. The same monotone properties can be shown to hold if the vanishing of f for x, = a, and for x, = a, are replaced by the conditions that f ( a x,) and f' f(x,, a,) are monotone nondecreasing in x, and x,, respectively. 4.3. Definition. We shall call a function f(xl, x,) positively monotonely monotone i f d , A , f ( I ) 2 0 for all I, and f ( x , , a,) and f(a,, x,) are monotonic nondecreasing in x, and x,, respectively. The function v ( x l , x,) is positively monotonely monotone. More-
over, since V ( I ) f A,+ f ( I ) = J I I d,d,fl & Old2f ( I ) 2 0 for all Z and are additive functions of intervals, it follows that the interval functions P ( I ) = ( V ( I ) A,PZf ( I ) ) and N ( I ) = ( V ( I ) d,d, f ( I ) ) are additive and positive for all I, and consequently give rise to the two positively monotonely monotone functions:
+
4
4
P(X,, xz) = [v(x,,X,)+(f(XI, n(x,, x,)
=
4 [v(x,, x,)-
4
x,) --f(a,, xz) -f(x,, a,) +f(a,, a,))]
(f(x,, x,) -f(a,, xz) -f(x,,
a,)
+f(a,, a,))].
4.4. THEOREM. The functionsp(x,, x,) and n(x,, x,) effect a Jordan decomposition of the function f(x,, x,) - f ( x , , a,) - f ( a , , x,) f(a,, a,), as do P ( I ) and N ( I ) for G ( I ) = d , d , f ( I ) .
+
We can duplicate some of the statements for functions of bounded variation in one variable; in particular, 4.5. P ( J ) is equal to the 1.u.b. as to
0 of .Z Ald2f ( I ) , the summation being extended over all intervals of CT for which d,dzf ( I ) 2 0, while N ( J ) is equal to the 1.u.b. as to (T of - .Z d,d, f ( I ) , the summation being extended over all intervals of CT for which d,d, f ( I ) 5 0.
P ( J ) is equal to the 1.u.b. as to 7t of Cnd,dzf ( I ) for all n of J, where 7t = I,, ..., I , is any finite set of nonoverlapping intervals of J, while N ( J ) is the negative of the g.1.b. of Cnd,d, f ( I ) for all 7t of J. 4.6.
a,) -f(a,, x,) +f@,, a,) = P'fX,, x,) n'(x,, x,) where p ' ( x l , x,) and n'(x,, x,) are positively monotonely monotone, vanishing on x, = a, and x, = a2,then d,A2(p' - p) I 0 and d,d,(n' - n ) 2 0, that is p(x,, x2) and n(x,, x,) are minimal for functions having the properties of p' (x,, x,) and n'(x,, x,) . The Jordan decomposition and the monotone properties of p(x,, x,) and n(x,, x,) give us:
4.7- If f h , , x,) -f(xl,
5.
109
CONTINUITY PROPERTIES
Iff(x,, x2) is of bounded variation on Jand f(x,, a,) and f(a,, x,) are of bounded variation in x, and x,, respectively, then f(x,, x 2 ) is of bounded variation in x, for each x, and in xp for each xl. 4.8. T H E O R E M .
5. Continuity Properties
We consider first, the continuity properties of the interval functions F(I) defined on J, additive on intervals, with F(I) 2 0 for all I. Then if I, contains I,, we have F(I,) 2 F(12).Consequently if we have a monotone sequence of intervals I,zsuch that I,,contains ZVL+,, with lim7,1*z = I, then lim?,F(I,J exists. In particular, if I reduces to a single point, lim,LF(In)exists. As a matter of fact, if we direct the intervals containing a fixed point (XI, x,) by inclusion, then limIF(I) exists and is equal to the g.1.b. of F(I) for all I containing (xl, x,). Because of the additive and positive property of F(I), for any fixed e > 0, the number of points (x,, x,) such that limI-.(Tl,z,)F(I) > e, is finite. If we take e = l/n, we can conclude: 5.1. T H E O R E M . If F(I) is a positive valued additive function of
F(I) > 0 is intervals on J, then the set of points such that limI+(z,,T,) denumerable. The conclusion of this theorem can also be stated that F(I) is continuous as a function of I except at a denumerable set of points provided we define continuity of F(1) at a point (x,, x,) by the condition limI-,(zl,xs) F(I) = 0, intervals being directed by inclusion. In the preceding considerations, we can assume that the point (x,, x2) is on the boundary of intervals I, in particular, it can be fixed as the lower left-hand corner (or any of the other three corners) of all I involved. If we consider the point function f ( x l , x2) = F[a,, x,; a,, x2] generated by such an F(I) and if I is the interval [x,, xl‘; x,, x,’] with x,‘ > x, and x,‘ > xz, then =f ( x 1 ‘ 3
xz’) - f ( x 1 9
x2’)
- ffxl‘>
x2)
f ffxl,
x2)7
and l i m( z ~ , ~; ) ~ ( ~ , + o ,T*+o) FfI) will exist. Since f ( x , , x,) is monotonely monotone, it follows that f ( x , , x , 0) andf(x, 0, x,) exist for each x1 and x,, respectively. Consequently, l i m ( ~ ~ , z ; ) ~ ( ~ ~ +f(x1” o , z , +xi) o)
+
+
110
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
will exist also. Similarly the other three quadrantal limits at (xl - 0, x, 0), (xl 0, x2 - 0), and (x, - 0, x, - 0) exist also. We can consequently state:
+
+
5.2. T H E O R E M . If f(x,, x2) is positively monotonely monotone on J , then for each (xl, x,,) of J, the four quadrantal limits: f(x, 0, x, O), f(x, - 0, -x, O), f(x, 0, xz - O), and f(X, - 0, x, - 0) exist. For any positively monotonely monotone function gives rise to a positive additive interval function. This result can easily be extended to functions of bounded variation in the form:
+
+
+
+
5.3. T H E O R E M . If f(x,, x2) is of bounded variation on J and f(xl, a2) and f ( a , , x2) are of bounded variation in x,and x,, respectively, then the four quadrantal limits o f f at (x, 0, x2 0), (xl - 0, x, 0), (xl 0, x, - 0) and (x, - 0, xz - 0) exist at each point-(x,, XJ of J. It is to be noted that we assume that (xl', x2') '--> (x, 0, x, 0) includes the conditions xl' > x, and x2' > x, and similarly for the other three limits. By way of proof, we need only apply the preceding theorem to the Jordan decomposition off (x,, x2),which gives us two monotonely monotone functions p(x,, x,) and n(x,, x2), such that
+
+
+
+
fb,,XJ
-f(X,,
u2) -f(a,, x*) + f ( a , , a,)
+
+
= P(X,7 X2) - 4x1, X2)'
A consequence of this theorem is: 5.4. T H E O R E M . If f(x,, x,) is of bounded variation on J,f(x,,a,)
and f(a x,) of bounded variation in x, and x,, respectively, then the 1: set of discontinuities of f(x,, x2) lie on a denumerable set of lines parallel to the coordinate axes. Since f(XI, x,) has double limits at every point of J if approach is limited to quadrants, it follows from the Cauchy condition of convergence that for each (x], xz) of J and for a given e > 0, there exists a circle with ( x l , x2) as center, such that if (x,', xz') is interior to this circle but not on the lines through (x,, x,) parallel to the coordinate axes, then ~ ( f(xl', ; x,')) < e, where w(J. (x,', x,')) is the usual oscillation of f at (x,', x2'). By the Bore1 theorem, a finite number of these circles cover J . If the centers of these covering circles are the points (xi') , xp) ), k = 1, ..., 12, it follows that if (xl', x,') is not on the
5.
111
CONTINUITY PROPERTIES
lines x, = xik)or x, = xf) for any k, then w(f; (x,’,x,‘) < e, so that the points of J for which w(f; (x,,x,)) 2 e lie on the finite number of lines x, = xik), x, = xf), k = 1, ..., N. By setting e = l/m, we obtain a method for counting the lines on which the discontinuities of f(xl, x,) lie. 5.5. THEOREM. If f(x,, x,) is of bounded variation on J, and f(x,, a,) and f(a,, x2) are of bounded variation in x, and x,, re-
spectively, and if f(~,, x,) is continuous in x, for each x,, then f(x, 0, x,) and f(x, - 0, x,) are continuous in x, for each x,. Under the hypothesis of our theorem, f(x,, x,) is of bounded variation in x, for each x, and in x, for each x,,so that f(x, 0, x,) and f(x,-0, XJ exist for each xi. Further, by Theorem 111.5.3
+
+
lim ( T I ’ ,7 ’ z ) - ( l 1 + 0 ,
ffx1’,XZ’) l d o )
exists as a double limit. Because of the bounded variation properties
offfx,,X J , f(xl‘,x,’) and
lim x;
+
x,+o
1%‘
-lim
,f(xl’, x2‘)
x1 +o
exist for all x,‘ and x,‘,respectively. Then by 1.7.2 the iterated limits lim .r,l-x,‘
’0
lim
f(xI’,x,‘) and
1;+0
.);+
lim 7
2 . 1
lim
-0
f(XI”
x,’)
x2‘+ T , + O
both exist and are equal to the double limit. Because of the continuity of f(x,, x,) in x,, this tells us that lim SI’+
r,+o
ffx,
+ 0, XJ
= f(x,
+ 0, x,),
+
that is, f(x, 0, x,) is continuous on the right at x,. The left-hand continuity at x,, as well as the continuity of f(x, - 0, x,) in x, follow in a similar way. 5.6. T H E O R E M . If f(x,, x,) is monotonely monotone with f(a,, x,) = .f(x,, a,) = 0 and A 1 A 2 f ( I ) I 0 for all I, and if f(x,, x,) is
continuous in x, for each x, and in x, for each xl,then f(xl, x,) is continuous on J. For then lim z’,-zl+o
lirn z’,+r,+O
lirn
f(x,‘,x2’)=
z‘ -* z 2
%
+0
lim zfI- zl+0
f ( x l ‘ , xr‘) = f(x,,x2).
112
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
Since f(xl, x,) is monotonic nondecreasing in x, for each x,, it follows from 1.7.8 that the double limit lim
f(x1’, xz’)
(5‘,,~’8)-r(z,+o.z,+o)
exists and is also equal to f(xl, x,). The same procedure gives the same value for the other quadrantal limits. Because f ( x , , x,) is continuous along lines parallel to the coordinate axes, we can conclude from this that for the double limit we have
or f is continuous at (x,, x,). For functions of bounded variation in one variable, we have a decomposition of the function into a continuous and a purely discontinuous part, the latter a sum of an absolutely and uniformly convergent sequence of simple break functions. For two variables, we consider first the case where y(~,,x,) is positively monotonely monotone with f(a,, x,) = f ( x , , a,) = 0. Then for each x, and fked x,, f ( x , , x, - 0) and f ( x , , x2 0) exist so that we have the breaks f(x,, xz) -f(xl, x2 - 0 ) f(x,, x, 0 ) - f f x , , x, - 0 ) . These will be monotonic nondecreasing functions in x,. These suggest the simple break function
+
fh,,xzl
- Nx,, x,;
f(x,, x,’
+
x , ’ ; f ( x , , x,‘) -f(x,, x,’
- 0)
-
fh,, x,’ + 0 ) )
- O),
will be continuous at x,’, for every x,. If x, = x , ( ~ )are the lines of discontinuity of f parallel to the x,-axis, it follows as in the case of one variable that
5.
113
CONTINUITY PROPERTIES
is continuous in x2 for each x,. The infinite series can be replaced by the single function
f*f.,,
x,) = 52
c
( f f x , ,Y ,
5 fI8 < 5%
+f f x , , xzl -
+ 0) -f(x,, Y ,
f(X,Y
x,
-
- 0))
0).
+
Then if f ( x , , x,) = f,(x,, x,) f,(x,, x,), f,(x,, x,) is continuous in x,for each x,. It is possible to treat f , ( x , , x,) relative to x, as we have f f x , , x,) and write f,fx,, x,) =fec(xl, x,) +fb,fx,, x,) where
f+,,
x2)
= 51
c
S ~1
< ~1
+f,fX,Y
ff,(Y,
+ 0, xJ
--f,fY,
- 0,
x21)
x,) - f , ( x , - 0,x,)
so that fcc(xl, x,) is continuous in x1 for each x,. As a matter of fact, fce(xl, x,) is also continuous in x, for each x,. For as shown above in 111.5.5, since f , ( x , , x,) is continuous in x2for each x,, f , ( x , 0, x,) --f,(x, - 0, x,) and f , ( x 1: x2) - f ( x , - 0, x,) are continuous in x,. Moreover, the series of positive monotone (in x,) functions Cyl[f , ( y , 0,x,) - f , ( y , - 0, x,)] is term by term less than the absolutely convergent series Z,,[f,(y, 0, b,) - - f , ( y , - 0, b,)] and is consequently uniformly convergent in x,. Thenf,,(x,, x,) is continuous in x, for each x,, and the same thing holds forfcc(x,, x J as the difference of two continuous functions. Nowf,,(x,, x,) is obviously a monotonely monotone function withf,,(x,, a,) = fcc(al,x,) = 0, and is continuous in x, for each x, and in x, for each x,. Consequently, by 111.5.6, fee(x1,x2) is continuous in (x,, x,) on J . We can then state:
+
+
+
5.7. T H E O R E M . If f ( x , , x2) is positively monotonely monotone on J with f(a,, x,) =f f x , , a,) = 0, then f f x , , x,) = frc(xlyx2) fbZl(x1, x,) +fbz,(xl, x,), where f,,(x,, x,) is continuous in (xlyxJY fbZl(xl,x,) is the sum of a uniformly convergent sequence of break , x,) is functions having breaks along the lines x, = x , ( ~ )andfbZ2(x,, a similar function with breaks along the lines x, = x , ( ~ ) ,the lines of
+
discontinuity off. From the Jordan decomposition of a function of bounded variation it now follows: 5.8. T H E O R E M . If.f(x,, x2) is of bounded variation on J, f(a,, x,) andffx,, a,) are of bounded variation in x, and x,, respectively, then
114
111. RIEMANN INTEGRALS I N TWO DIMENSIONS
f(x,, xz) can be decomposed into the sum of a continuous function and break functions of the kind described in Theorem 111.5.7. With respect to the continuity of the total variation function v(x,, x,) of a continuous function of bounded variation f(xl, x,) we have : 5.9. THEOREM. Iff(x,, x,) is of bounded variation and continuous in (x,, x,) on J, then the variation function v(x,, x,) = V[a,, x,; a,, xz] is also continuous. The continuity of f(x,, x,) yields that limII l - o V ( I ) = 0, where 1 I I is the maximum side length of I, since lim,I,+o [V(I) I O,O,f(I) I] = 0. This does not seem to be sufficient to prove that v(x,, x,) is continuous. We use the Jordan decomposition off(x,, x2) :
fb,,X p )
-f(x,, a,) -f(% x2) +f(a,, a,) = P(X,, xz) v(x,, xz) = Ph,,x,) n b 1 ,XJ.
+
-
d x , , X,)’
If v(x,, x,) is discontinuous, then since the right-hand side of the first equality is continuous, p(x,, xp) and n(x,, XJ are discontinuous at the same points and the discontinuities would cancel out in the difference p(x,, x,) - n(x,, x,). But by virtue of the decomposition of p and n into their continuous and discontinuous portions, this would permit diminishing both p and n by a positively monotonely monotone function, which would contradict the minimal character of p and n. Then p(x,, x,) and n(x,, x,) and consequently v(x,, xz) are continuous iff(x,, xz) is. 6. The Space of Functions of Bounded Variation in Two Variables: BV2
The following statements are obvious or can be proved as in the one-dimensional case : 6.1. The space BV2 is linear, with
JJ I cd,d,f(
SJ I d , W , +fz) I 5 J-J I d,d,f, I + SJ I
d,d,f,
1 I
= c
JJ
I d,d,f(
and
I.
6.2. The space BV2 is a lattice if partial order fl(2)fz is defined by
the condition d,d,(f, - f,)( I ) 2 0 for all I, or iff, d , d , p ( l ) 2 0 for all I .
=
f,+ p, where
6#3.The subspace of functions of BV2 for whichf(a,, xz) and f(x,,a,) are of bounded variation in x2and x,, respectively, is also a linear space and can be normed by setting
6.
BOUNDED VARIATION IN TWO VARIABLES
115
Limn 11 f,, 1 I = 0 implies that lim,,f(x,, x,) = 0 uniformly on J , lim, I d , f , ( x , , a,) I = 0 uniformly in x,,lim, I d,f,(a,, x,) I = 0 uniformly in x,, and lim. J::;:; 1 d,d,f,, I = 0 uniformly in (x,, x,) on J. But these conditions are not sufficient to yield lim, I( f, ( 1 = 0.
Jz:
Jz1
6.4. If f , ( x , ,
xz) are in BV2 and lim,f,(x,, x,) =f(xl, xz/ on J ,
m.
SJ I d,d,f, I. then JJ I d,dzf I 5 This follows from the upper semiadditive property of the interval function F(I) = I dlAzf(I) I and an extension of 11.5.4. In particular, if the functions f, are uniformly of bounded variation in the sense that JJ I d,d, f, I 5 M for all n, then f is also in BV2.
f,(x,, x,) is a sequence of functions in BV2, such that for some M , JJ I d,d,f, I 5 M , J:i I d,f,(x,, a,) I 5 M , I d,f,(a,, x,) I 5 M and I f,(a,, a,) I 5 M , ‘for all n, then we can a find a subsequence f,m off, such that lim,nf, m (x,, x,) exists for all (x,,x ).’ and the variations of the limit function will satisfy the same inequalities as those for the variations off,. The proof of this follows in the footsteps of the theorem for functions of bounded variation in one variable. It is proved first for a bounded sequence of monotonely monotone functions f,(XI,x,) with f,,(xl, a,) =fTl(al,x,) = 0 and A,d, f n ( I ) 2 0 for all n and I. By selecting a sequence of points Pk dense in J , one determines a subsequence f ,m which converges at the points Pk.This determines a function f at points Pk subject to the condition f(x,, a,) = f(a,, x,) = 0 and d,d, f 2 0, the latter inequality limited to intervals all of whose vertices are points of Pk.The four quadrantal limts (xl‘, x,’) approaching (XI 0, x, O), (x, - 0, x2 O), (x, 0, x, - O), (x,- 0, x, - 0) where (x,’,x,’) are limited to points of Pk,will exist at all points of J , and be equal except on a denumerable set of lines parallel to the coordinate axes defining a function f ( x , , xz) at such points. It turns out that lim,f, m (x,,x,) =f(xl, x,) at all such points. The application of the Helly theorem in one variable to the lines of discontinuity of f(x,, x,) produces a subsequence of which converges for all points of J . The Jordan decomposition fnm theorem together with the Helly theorem in one variable then yield the Helly theorem in two variables as stated. 6.5. Helly Theorem. If J:p
+
+
+
+
116
111. RlEMANN INTEGRALS IN TWO DIMENSIONS
EXER U E S 1. Show that the function f(xl, x2) = 0 for OIS'x,5 x 2 5 1 and for 0 5 x2 < x1 5 1, is not of bounded variation on [0, 1; 0, 11.
=1
2. What theorems for functions of bounded variations in two variables (suitably modified) carry over to functions of bounded variation in three variables based on the interval function F(Z) = 1 A , A 2 d , f ~f, being defined on a three-dimensional interval J. 7. Functions of Bounded Variation According to Frhchet
We devote a few pages to the discussion of a definition of bounded variation of functions in two variables, which is due to M. FrCchet [see M. FrCchet : Sur les fonctionelles bilineaires, Trans. Am. Math. SOC.16 (1915) 215-2341. FrCchet found this form of bounded variation useful in extending the Stieltjes integral representation of the most general linear continuous functional on the normed space C of continuous functions on a finite interval to the determination of the most general bounded (and so continuous) bilinear functional on the product space C, x C,. [ B ( f , g ) is bilinear if for functions f , , f , of C, and g,, g, of C, and constants a,, a,, b,, b, we have
B(a,f,
+ a2f29 b,g, + b,gJ = a,b,B(f,, g,) + a,b,Bff,, 8,) + a,b,B(f,,g,) + a,b,Bff,,g,) ;
B(f, g) is bounded if there exists a constant M such that I B ( f , g ) 1 5 MI1 f 11 11 g 11 for all f of C, and g of C,].
For notational convenience, we shall replace the variables x1 and x, of the preceding sections by x and y , and assume that J is the interval [a,a'; b, b'] defined by a 5 x 5 a', b 5 y 5 b'. The FrCchet bounded variation is based on net subdivisions of J . Let as = a, = (a = x, < x, < ... < x, < ... x , ~= a') and uy = a2 = ( b = yo < y , < ... < yi < ... < y, = b') be subdivisions of [a, a'] and [b, b'], respectively; let Izi = [x,-,, x t ; yip,, y , ] ; further let E~ = f 1 and E ~ '= & 1 for all i and j . Then: 7.1. Definition. f ( x , y ) on J is said to be of Frichet bounded var-
iation if the expression
7.
BOUNDED VARIATION ACCORDING TO FRECHET
117
is bounded in uz and av for all choices of E % = & 1, E ~ ’= & 1. If f ( x , y ) is of FrCchet bounded variation on J, then the FrCchet total variation FV(J) is the 1.u.b. ZuX x uY E ~ E ~ ’ A , A ~ ~for ( I ,all~ ) = f 1, E ~ = ‘ i-1 and a z x a v of J. We shall denote the class of functions of FrCchet bounded variation by FBV. We note that P V ( J ) is also - g.1.b. [ Z a z x o Y ~ z ~ j ’f(Iti)] A , A , since a change in the signs of all E % gives a negative value corresponding to any positive value of the sum. h ( y ) , then A I A 2 f ( I ) = 0 for all I so that If f(x, y ) = g ( x ) FV(J) = 0. If f ( x , y ) = g ( x ) h ( y ) , then
+
It follows that f(x, y ) = g ( x ) h ( y ) is of FBV if and only if g(x) and h ( y ) are of bounded variation on [a,a’] and [b, b‘], respectively, provided f and g are not constant functions. Since
f ( x , y ) will be of FBV iff is o f bounded variation on J, and FV(I) I V ( J ) . However, f(x, y ) can be of FVB without being of bounded variation on J. For an example of such a function, the reader is referred to C. R. Adams and J. Clarkson: On definitions of bounded variation for functions of two variables, Trans. Am. Math. SOC.35 (1933) 837-841. The total FrCchet variation FV(J) is not expressible as an integral of an interval function, so theorems developed for such integrals do not apply. If we define the functions of nets F(az x av) = max [ L ’ i i ~ i ~ i ‘ A I A z . f(I ij)for all ci = f 1 and E ~ = ‘ f 11, then F(az x aw) is monotone nondecreasing in 0%x crv ordered by az’>= az and av’ 2 uy. For if we add to crz the point xo with xi-l < x,, < xi and take E 21. = E t2. = E . on [xi-l,xi] for the new subdivision a;, then the sums for a=’x av and for az x u y will be the same. It follows then that t
FV(J)
= 1.u.b. aXX%
F(uz x aw) = lim F(aZ x
aJ.
a,Xay
7.2. THEOREM. If f(x, y ) is of FBV on J, and I $ J, thenfis also of FBV on I and FV(I) 5 FV(J) or, more generally, if I , 5 Iz,then
118
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
FV(Z,) 5 FV(12), so that FV(Z) is monotone nondecreasing in Z, the intervals I being directed by inclusion. Consider the case when Z = [a, c; b, b ' ] , with a < c < a', and let oz be a subdivision of [a, a ' ] containing c. If uz' is the part of oz in [a, c ] and uz" the part in [c, a ' ] , then
Since subdivisions of [a, c ] and [c, a ' ] are independent of each other, we can select oz" and the corresponding E , so that Za31,,xov ~ ~ ~f ( Z~l j ) ' is positive or zero so that
for all oz' of [a,c ] and ov of [b, b']. The extension of this result to any subinterval of J and to I , 5 I?, is a simple matter. 7.3. T H E O R E M . If
f(x, y ) is of FBV on J, then the interval function FV(Z) is upper semiadditive, i.e., if u = ( I , ... Zt,) is any subdivision of Z, then FV(Z) 5 ZkFV(IJ. For any e > 0, uz and ay can be selected so that the sides of I , ... Z), are on lines determined by oz and uu, and so that
5 2 FV(ZJ. k
Then for all e > 0, FV(Z) semiadditivity of FV(Z) .
-
e 2 ZkFV(Ik),which gives the upper
7.4. T H E O R E M . If FV(I) is additive, then f(x, y ) is of bounded variation on J. For then the function of intervals FV(I) f A,d,f ( I ) will be additive and positive or zero valued, since F V ( I ) 2 A I d 2 - f ( I )for all I. If we set FV(I) d,d, f ( I ) = P ( I ) and FV(I) - d , A , , f ( Z ) = N(Z), then .l,d, f ( I ) = 4 [ P ( I ) - N ( I ) ] and so
+
It follows that F V ( I ) is not always additive; further, that a Jordan decomposition for a function in FBV may not exist. As for the case of ordinary bounded variation we have:
d
~
7.
BOUNDED VARIATION ACCORDING TO
FRBCHET
119
f(x, y ) is of FBVon J , and f(x, b ) is of bounded variation in x on [a, a ' ] , then f(x, y ) is of bounded variation in x, uniformly for y on [b, b']. Similarly if f(x, y) is of bounded variation in y on [b, b'] then f(x, y ) is of bounded variation in y, uniformly for x on [a, a ' ]. 7.5. THEOREM. If
For
C,E?[f(X,,Y)-ffx,,b ) -ffX,-,LY) + f f X , - , ' b ) l 5 FV[a, a ' ; b, y ] 5 F V ( J ) , for all &?
E?
= f 1.
Then
[f(x,,v) - f(Xz--l,Y l 1 5 F V f J )
r'
f(X1-1, b )
1
I d,f fx,b ) I. 1 d, f(x, y ) 1 5 F V ( J ) + Jl' I d, f(x, b ) 1, 5 FVfJ!
Jr
+
+ C?&? [f(x,,b ) -
Consequently for all y . Since for a function f(x, y) in FBV, FV(I) is monotonic in I, it follows that limI+(z,y) FV(I) exists when the intervals Icontain (x,y ) and are ordered by inclusion. If further we knew that limI+ ,,( y) [FV(I) - d , d , f ( I ) ] exists, which would be true if FV(I) - O,O,f(I) were monotone nondecreasing in I, then we could conclude that lim (S',Y')P(l
f(x',Y ' )
+O,?/+O)
+
+
exists, provided f ( x 0, y) and f(x, y 0) exist for all (x,y). This latter condition is fulfilled if f(x, b ) and f ( a , y ) are of bounded variation in x and y, respectively. Even though there seems to be no simple way of proving that limI+(7,il)[FV(I) - d,d, f ( I ) ] exists, we do have:
f(x, y) is of FBV on J , and if f(x, b ) and f ( a , y ) are of bounded variation in x and y , respectively, then the four quadrantal limits of f(x', y') exist at every point of J.
7.6. THEOREM. If
Since the proof which depends on showing that if, for instance, the (.c'
9
v')+
lim
f(x', Y ' )
(Z+O, Y f O )
does not exist, then f(x, y ) is not of FBV on J , is rather complicated, we refer the reader to the original demonstration by M. Morse and W. Transue in Functionals of bounded variation, Can. Jour. of Math. 1 (1949) 153-165. See also A Calculus for FrCchet Variations, Jour. Ind. Math. SOC.14 (1950) 65-117 by the same authors.
120
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
As in the case of functions of bounded variation on J , it follows that : 7.7. THEOREM. The discontinuities of a function f(x, y ) of FBV
on J, where f ( x , b) and f ( b , y ) are of bounded variation in x and y , respectively, lie on a denumerable set of lines parallel to the coordinate axes. The following statements are obvious: 7.8. If f(x, y ) is of FBV on J , then cf(x, y ) is also of FBV on J and
FV(cf; J )
=
1 c I FV(f; J).
7.9. Iff,(x, y ) and fi(x, y ) are of FBV on J , then so is f,(x, y )
f,(x,y) and F V ( f , FBV is linear.
+
+ fi; J ) 5 F V ( f , ; J ) + V(f,; J ) . Then the space
7.10. The subspace of FBV for which f ( x , b ) and f ( a , y ) are of bounded variation in x and y , respectively, is also linear and can be normed
7.11. If f,(x, y ) is a directed set of functions of FBV on J and
lim, f,(x, y ) = f ( x , y ) for all (x, y ) of J , and if there exists an M such that FV(&; J ) 5 M for all q, thenfis also FBV and FV(’ J)SM. For if F V ( f q ;J ) 5 A4 for all q, then for all q, uz x u,, and ei = 31 1, E ~ ‘= f. 1, we have
From the linearity properties of limit it follows that ~ , ~ ~ f (’Z 5 ij) ; 5 5M ~ axxaY
also. We can replace 111.7.1 1 by the more precise results: 7.12. If f,(x, y ) is a directed set of functions of FBV on J , and lim, f , ( x , y ) = ,f(x, y ) for all (x, y ) of J , then F V ( f ; J ) 5 lim$V(f,; J ) . For if lim F V ( f q ;J ) = a,there is nothing to prove. If lim,$‘V(f,; J) = c < coTlen for e > 0 and q, there exists q,, such thxq,,Rq and FV(fqep; J ) < c e. The q,, are cofinal with Q so that lim, f q e g ( xy, ) =
+
7.
BOUNDED VARIATION ACCORDING TO F ~ C H E T
f ( x , y ) . Then by 111.7.11, we have FV(’ J ) < c so that FV(A J ) 5 c. As an application of 111.7.11, we show:
+e
121
for all e > 0,
7.13. THEOREM. If g ( x , y ) is of FBI/ on J = [a, a ‘ ; b, b ’ ] ;if f ( x ) is bounded on [a, a ’ ] and such that h(y) = J:’f(x)d,g(x,y ) exists
for all y of [b, b’], then h(y) is of bounded variation on [b, b’]. For consider
122
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
A consequence of this theorem is the following iterated integrals theorem : 7.14. THEOREM. If f ( x ) is continuous on [a,a ' ] ;if g ( y ) is bounded on [b, b ' ] ;if h(x, y ) is of FBV on [a, a'; b, b'] and of bounded variation
in x for y = b (and so for ally); and if J:' g(y)d,h(x, y ) exists for all x , then s y c x ) d , s:'gfY)d,h(x, Y ) and s:'gfY)d, S;ffx)d,h(x, Y ) both exist and are equal. The existence of the first two of these integrals is a consequence of the continuity of f ( x ) and the bounded variation of J:' g(y)d,h(x, y ) proved in the preceding theorem. Also, f(x)d,h(x, y ) exists for each y , since f ( x ) is continuous and h(x, y ) is of bounded variation in x for each y. To prove the existence of the second integral, we consider :
JT
c g(YJ
p ( X ) d , [ h f x ,Y i )
- W x , Yi-1)
1
UY
where yt- l 5 y,' 5 y,. By the proof of 111.7.13, the functions Gug(x)= Zuvg(yt') [h(x, y,) - h(x, Y % - ~ )are ] uniformly of bounded variation in x relative to 0,. Further, by hypothesis lim G , ( x )
=
%
C g(y,')
[ h f x ,Y , )
-
h(x, Y,-,)I
1,g(Y)d,hfx, b,
=
Y)
UY
for every x . Consequently, since f ( x ) is continuous, the convergence theorems 11.15.3 and 11.15.12 give us that J : ' g ( y ) d , Sz'f(x)d,h(x, y ) exists and is equal to J:'f(x)d, J:'g(y)d,h(x, y ) .
EXERCISES 1. Show that if f(x, y) on J is such that FV(f,; J ) = 0, then .f(x,y ) = f(x, b) f(a, Y ) - f(a, b). 2. If f(x, y) is of bounded variation on J, is FV(I) an additive function of intervals?
+
3. Is it possible to weaken the hypotheses of the last theorem by replacing the condition that f(x) be continuous on [a, a'] by the conditions that f(x) be bounded on [a, a']and f(x)d,h(x, y) exist for every y of [b, b']?
s:'
4. Is there a theorem of Helly's theorem type which is valid for functions
in FBV on J?
8.
R-S INTEGRALS IN TWO VARIABLES
123
8. Riemann-Stieltjes Integrals in Two Variables
Riemann-Stieltjes integrals in two variables are integrals of the interval functions based on two point functions f ( x , y ) and g ( x , y ) defined on an interval J= [a, a ' ; b, b ' ] ,where F ( I ) = f ( x ' , y')d,d,g(Z). Here ( x ' , y ' ) is any point of the interval [x,,x,; y l , y,] and d,d,g(Z) = d x , , Y,) - d x , , Y J - g h , , Y , ) g b , , Y 2 ) . Such an F ( I ) is of the formf(x', y ' ) G ( I ) where G(Z) is an additive function of intervals. The value of d,d,g(I) is obviously unchanged if we add to g ( x , y ) a function of the form h,(x) h,(y). If we replace g ( x , y ) by g(x, y ) = g ( x , y ) - g ( x , b ) - g(a, y ) g ( a , b ) , then for all intervals Z f ( x ' , y')d,d.,g(I) = f ( x ' , y')d,d,g(Z), so that for the properties of integrals we can usually assume that g ( x , b) = g(a, y ) = 0 for all x and y . We recall that there there is some variety in the definitions of integrals of interval functions which are based on the limits of functions of subdivisions a of J: G ( a ) = ZaF(Z). We have a choice of subdivisions a of the interval J, either a is general = [I, ... I n ] ,where the I t have at most parts of a side in common and Z J / ( = J; or ~7 is a net, where a is determined by subdivisions ax of [a, a ' ] and ay of [b, b ' ] by drawing lines parallel to the coordinate axes through the points of ax and au. The limits of the approximating sums G ( a ) = Z a F ( I ) depend upon the partial order. We have either: the a are oidered by inclusion u1 2 a2, if every interval of a , is a subinterval of some interval of a2, or a, is obtained from a2 by redividing the intervals of a2. Similarly for nets ax'x our2 ax" x au" if a%'2 a%'' and ay'2 ay". Or: the a can be ordered by some metric or norm. The usual norm I a I is the maximum of the length of the sides of the intervals constituting a. For nets this norm becomes the larger of I ax I and 1 ag I. In addition, conditions may be made on the choice of the points (x', y ' ) in I. We shall usually assume that ( x ' , y ' ) is any point of I (interior or boundary), depending only on I. For net subdivisions one obtains a weak type of integral by assuming that the points ( x ' , y ' ) in the intervals I are chosen so that they form a net also, that is, in Iti = [ x ~ -x~t ;,yj-,, y i ] we assume for ( x ' ~ y~' ,J that x ' , ~= x,' for all j and y J t i= y: for all i. Corresponding to these choices we have a-integrals (a J) and norm integrals ( N J) based on general subdivisions, a-net integrals (0% X ag J) and norm net integrals ( N az x a y J), based on net sub-
+
+
+
124
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
divisions, and weak 0-integrals (weak 0-J) and weak norm integrals y',J form(weak N J) based on net convergence with the points (xttj, ing a net. We shall usually denote integrals by JJfdld,g, omitting the variables (x, y ) and sometimes the interval J where no confusion results. Also, we shall be concerned largely with the 0 and norm integrals based on general subdivisions. If f(x, y ) = 1 on J , then obviously all of these integrals exist with value d,d,g(J). Ifg(x, y ) = xy, then d,d,g(I) = (x, - xl) (y2- yl), leading to double Riemann integrals. If g ( x , y ) = h , ( x ) h , ( y ) , then A,A,gfI) = ( h , f x , ) - h l f X 1 ) N h 2 f Y 2) h2fYl)), leading to double S tie1tjes integrals. In the matter of existence theorems, Cauchy conditions of convergence formulated for the type of limit involved are necessary and sufficient for the existence of the corresponding integrals. The finiteness and equality of upper and lower integrals, defined as greatest and least of limits, is also available. When it comes to more specialized conditions the corresponding theorem for integrals of interval functions 111.3.1 gives us': 8.1. A necessary and sufficient condition that the
JJfd,d,g (or N JJ fdld,g) exist when based on general subdivisions is that u Jjw(Sfdg; I ) = O (or N s j ~ ( s f d gI;) = 0). For the case of net integrals there is no difficulty in showing that JJw(sfdg;I) = 0 (where the S extends over nets of I ) has the corresponding Cauchy condition of convergence as a consequence, so that it gives a sufficient condition for the existence of net JJfd,d,g. The proof of the necessity of this condition for unrestricted g ( x , y ) runs into difficulty, since a net subdivision of an interval I t j induces subdivisions of for i' = 1 ... TI and I T j ,for j ' = 1 ... rn. A simple necessary condition for the existence of the integral JJfd,d2gemerges frqm the observation that for any given subdivision u of J and for all e > 0, there exist points ( x ' , y ' ) and(x", y") in I such that f f f x ' ,Y w , d , g f I ) - f f x / / , Y " ) d , d , g f I ) ) 1 0
I xu
2
xu
w f f ; I ) I d,d,gfI)
I
-
e-
For if ZuI d,d2g(I) I = M , we need only determine (x', y ' ) and (x", y") in each I so that sgn(f(x', y ' ) - f ( x " , y " ) ) = sgn d,d,g(I) and If(x', y ' ) -f(x", y") I > w ( f : I ) - e / M . Using the Cauchy condition of convergence we obtain :
8. R-s
8.2. A necessary condition that J,fd,d,g
J 4 f ; I) I
d,d,g(I)
I
125
INTEGRALS IN TWO VARIABLES
exist is that correspondingly
= 0.
This result is valid for integrals based on general and net subdivisions but not for weak integrals. It follows that if NJ,fd,d,g exists then limlIl+o~(’; Z) I d,d,g(Z) I = 0 uniformly on J, where I Z I = the maximum side length of I. Consequently, limI+(z,v)m(f;Z)dldzg(Z) = 0 for all (x, y ) in J . If f(x, y ) is discontinuous at (x, y ) so that w ( f ; x, y ) > 0, then = 07 if l i m I + ( z , v ) ~ l ~ f z g0,~ thenffx, q y ) must lim,+,z,y)dl~zg(I) be continuous at (x, y ) . In an extended sense then, if N fd,d,g exists, thenfand g have no common discontinuities. For the a-integral, similar necessary conditions can be developed involving quadrantal convergence at the points (x, y ) of J . It is obvious that if N JJfd,d,g exists, then the corresponding a JJ fd,d,g exists also. No simple condition added to the existence of a J, fd,d,g which yields the existence of the norm integral seems to be available. The pseudoadditive condition defined for interval functions when applied to the function f(x’, y’) d,d,g(I) does not seem to produce a simple relation between the behavior of f ( ~y,) and d,d,g(Z).
sJ
exists, then for any I i n J, the corresponding fd,d,g exists, provided the integrals are based on general subdivisions. Further, the resulting function of intervals is additive. This follows from 111.3.3. This does not seem to be true for net integrals unless g is restricted. It is not always true for weak integrals based on net convergence. For let J = [0, 1 ; - 1, I]: let f ( x , y ) be independent of y , equal to f(x), and set g(x, y ) = x I y I. Then 8.3. THEOREM. If J,fd,d,g
s,
for all az x ay of J since zlny(J y j I - I y,-, 1) = 0 for all subdivisions aa/of [- 1,1]. Consequently weak SJf(x)d,d&(x, y ) = 0. If I = [O, 1 ; 0, 13 then
so that J,f(x)d,d,g(x, y ) will exist only if f(x) is Riemann integrable on [0, I]. There is the usual converse theorem :
126
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
8.4. THEOREM. If I ,
... I , are a subdivision of J and
fd,d,g exists for each i, then a sJ,fd,d,g exists and is equal to the sum of these integrals. For norm integrals this theorem does not hold without additional conditions on the behavior of f ( x , y ) and g ( x , y ) on the boundaries of the I,. On the boundedness of the function f ( x , y ) , when the integrals exist we have: a
JI
1
sJ
8.5. THEOREM. If either N lJfd,d,g or a fd,d,g exists, then there exists a subdivision a, of J such that the subintervals of a, fall into
two categories: I’, I” such thatfis bounded on each I’,f is unbounded on each I “ , but d , d , g ( I ) = 0 for every I in any I”. Then J,fd,d,g = C,, SIPfd,d,g, that is the integral JJ fd,d,g depends only on the values off and g on the intervals I ’ where f is bounded. Since the integrals exist, there exists a subdivision a, such that if 0 2 a,, then C b f ( x ’ , y ’ ) d , d , g ( I ) is bounded for all choices of ( x ‘ , y ‘ ) in the corresponding intervals I. If thenf(x, y ) is unbounded on the interval I = [x,, x , ; y,, y,] of a,, then d,dzg(I) = 0. Divide I into intervals I , = [x,, x ; yl, y,] and I , = [ x , x,; y,, y,]. Then f ( x , y ) is unbounded on either I , or I, and consequently d , d , g ( I J = 0 or d,d,g(Zz) = 0. Since 0 = d , d , g ( I ) = d , d , g ( I J d,d,g(I,), it follows that both d , d , g ( I , ) = 0 and d,d,g(I,) = 0. Similarly, if I is divided into the intervals I,’ = [x,, x,; y I , y ] and I,‘ = [ x , , x , ; y , y,], then d,d,g(I,’) = d,d,g(I,’) = 0. If I is divided into four rectangles by lines through the point ( x , y ) parallel to the coordinate axes, it follows that f(x, y ) is unbounded on one of these rectangles, say I,, and d,d,g(I,) = 0. From the additive character of d,d,g, and the preceding considerations, we conclude that d,d,g(I‘) = 0 for all four I‘ having ( x , y ) as common vertex, so that in particular d , d , g ( I ’ ) vanishes for I’ = [ x l , x ; y,, y ] . Consequently, g ( x , y ) - g ( x , y,) g ( x , , y ) g ( x l , y,) = 0 for all ( x , y ) in I and d , d , g ( I ” ) = 0 for any interval I” in I. This theorem is valid for the integrals based on general subdivisions and the net integrals, but does not need to hold for weak integrals as the example in the preceding paragraph shows. The proof of the following substitution theorem follows the usual lines :
+
+
8.6. THEOREM. If f ( x , y ) is bounded on J, and g ( x , y ) and h(x, y )
are such that JJg(x, y)d,d,h(x, y ) exists and if k ( x , y )
=
J:;: gd,d,h,
8.
127
R-S INTEGRALS IN TWO VARIABLES
then J J f ( x , y ) g ( x , y)d,d,h(x, y ) exists if and only if s J , f ( x ,y)d,d, k ( x , y ) exists and the two integrals are equal. Here the same integral (norm or u) based on general subdivisions is used throughout. The theorem is limited to these since we do not have assurance that k ( x , y ) exists for net type integrals. 8.7. Integration by Parts. There is difficulty in proving any integration by parts theorem for the general subdivision or net integrals since in the interval function f ( x ' , y ' ) d , d , g ( I ) , the points ( x ' , y ' ) do not determine a subdivision of J into rectangles. The proof of the following integration by parts theorem is valid only for weak net integrals, where both subdivisions and the points ( x ' , y ' ) form nets. 8.8. THEOREM. If f ( x , y ) and g ( x , y ) on J = [a, a ' ; b, b ' ] are such that weak sJ[f(X,Y ) - f ( X , b ) - f ( a , Y ) + f ( a , b) Id,d,g(X, Y ) exists, then weak SJ[g(& Y ) - g(X, b') - g ( a ' , Y ) -k g(a', b')]d,d,f(x, Y ) exists and the two integrals are equal. This could be expressed :
It is sufficient to prove this theorem for the case when f ( x , y ) vanishes on x = a, and y = b, and g ( x , y ) vanishes on x = a', and y = b'. To obtain the theorem for any two functions , f ( x , y ) and d x , Y ) we set f ( x , Y ) = .Nx,Y ) - f ( x , b) - f ( a , Y ) f f a , b ) and i ( x , Y ) = g ( x , Y ) - g ( x , b') - g ( a ' , Y ) g(a', b'). As in the case of one variable we rearrange the approximating sum
+
+
cc 11
71
i=l i=1
f(XI',
Y J ( g ( x l ,Y J -g(xl+13 Y J
-
.Xi-,)
+g(x,-,, Y + J )
with a = x , 5 x,' 5 x , ... 5 x ~ ~5- x,,' , 5 x,* = a' and b = y o < =Y, ' < = Y l 5 ... 5 y,t-l 5 y y L 5 ' y , = b ' , so as to collect the multipliers of g ( x i , y J . In general, these will be of the form f ( x ' i + l , yli+,) . N X a ' , Y'j+J - f f X r i + l , Yi') +f(Xi" Yi')= d,d,,f (1'1, where I' = [xi', xli+,;yi', Y',,~]. The exceptions are the terms g(a, y J , g(a', y j ) , g(x,, b ) , and g(x,, b'). For instance, the coefficient of g(a, y i ) with 0 <j < n will be f ( x l ' , yli+,) - f ( x I ' , yi'). But since ,f(a, y ) = 0 for ally, we can write this term: g(a, y J [f ( x l ' , yli+J -f ( x l ' , y,') f ( a , Y ' ~ + ~ f)( a , y j ' ) ] . The term involving g ( x i , b ) can be treated in the same way. Since g(a', y ) = 0, the term involving g(a', y i )
+
128
111. RIEMANN INTEGRALS I N TWO DIMENSIONS
+
can be written gfa', Y j ) [ f f a ' , Y'j+l) - f f a ' , Y J - f f x m ' , Y f j + J f(x,', y,')] and similarly for g(xi', b ) . In the same way the values of g at the vertices of J, for instance, g(a, b ) , can be provided with a factor of the form A,d,f, since g(a, b)f(xl', yl') can be written g(a, b) f f f x l ' ,v,') - f f x l ' , b) - f ( a , y , ' ) + f ( a , b ) ) . It follows that
-m i + , ,
YJ
+
ffXi',
rJ1
where x,,' = a, yo' = b, km+] = a' and yr,,+l = b'. The same type of formula holds if f and g are interchanged. The remainder of the proof proceeds as in the case of one variable. If N JJ fd,d,g exists, then (e, d,, I axI 5 d,, I oy I 5 d , ) : I ,Ziif(xi', yi') AIA,g(Iti) - J,fd,d,g I < e. Select uz = (a = xo < x , < ... < xm=a') and ay = (b = y o < y1< . . . < y,, = b') such that 1 uz I < dJ2 and 1 ay I < d,/2. Then
If we set ax'= ( a = x,,' 5 x,' < ... 5 x ' , + ~= a') and uy' = (b = y,,' 5 yl' < ... 5 y'n+l = b'), then 1 az' I < d , and I oy'I < d,, so that (e, d,, I az I m
I
C i-1
I < d,/2) :
c g f x A Y ; ~ + L t - f I i j ) jJfd,d& I < n
j=l
-
e,
from which the theorem follows at once. The proof for the corresponding a-integral situation for the one variable case can also be adapted here, by using the trick of dividing the rectangle [xi',x ' ~ +y ~j ' ,;y'j+l] into four rectangles having the point ( x i , y J as common vertex and applying the additivity of the interval function A , A , f ( I ) . 9. Stieltjes Integrals in Two Dimensions with Respect to Functions of Bounded Variation
In the theorems of the preceding paragraphs, the function g ( x , y ) with respect to which the integrals are taken does not belong to any special class of functions. We obtain additional properties of the
9.
129
STIELTJES INTEGRALS IN TWO DIMENSIONS
integrals when we assume g ( x , y ) to be of bounded variation on J. Since the value for g ( x , y ) for x = a, or for y = b do not affect the existence nor the values of the integrals, we shall usually assume that g ( x , y ) is normalized so that g ( a , y ) = g ( x , b) = 0 for all x and y . We denote by v(Z) the function of intervals determined by I d,dzg I and by v(x, y ) the corresponding point function J:::,”: I d,d,g I. We have at once:
SI
9.1. THEOREM. If g ( x , y ) is of bounded variation on J , then a necessary condition that JJfdld2gexist is that J J w ( f , ;I ) I d,d,g(I) I = 0 and a sufficient condition that J J w ( f ; Z)v(Z) = 0. Here the integrals involved are assumed to be of the same type throughout, norm, a, norm-net, or a-net integrals. The theorem is not valid for weak net integrals. The necessity part of the theorem has been demonstrated in 111.8.2. For the sufficiency, we note that if a and al are such that a, 2 a, then I ZUfdld2g- Zu,fd,dzgI 5 ZOw (f;Z)v(l). For if Z is any interval of u, then
5 w(f; 1)
c I 4 d 2 gI 5
w(f; Z)V(Z).
0l.I
If, then, SJw(f; Z)v(Z) = 0, and for e > 0, o1 and g2 are such that Z , 51w ( f ; Z)v(Z) < e, for i = 1,2, then for a’ 2 a, and az we have
a2
a‘
Then J J o ( f ; Z) v(Z) = 0 implies the corresponding Cauchy condition of convergence and the existence of the corresponding integral. As an immediate corollary we have : 9.2. THEOREM. If g ( x , y ) is monotone on J in the sense that for each Z of J : d,d,g(Z) 2 0, then JJfdld2gexists if and only if for the corresponding integral J J w ( f ; I)d,d,g(Z) = 0. Further : 9.3. Iff(x, y ) is continuous on J and g ( x , y ) is of bounded variation,
then JJfd,d,g exists for all definitions of integral considered.
130
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
As in the case of one variable we can show by a contrapositive procedure that if $J fd,d,g exists for every continuous function f(x, y) on J, then g(x, y) is necessarily of bounded variation on J . The demonstration can be made on the assumption that the integral is of the a-net type, and will then hold also for the three other types of integrals which are stronger. As a matter of fact, one can even limit f(x, y) to be of the form fl(x)f2(y) where fi(x) and&(y) are any continuous functions on [a, a ' ] and [b, b ' ] , respectively. On the matter of existence of integrals, we can prove the stronger theorem : 9.4. If f(x, y) is bounded on J, and g(x, y) is of bounded variation then a necessary and sufficient condition that a sJ.fd,d2gexist is that 0 J J w ( f ; I ) v ( I ) = 0. The sufficiency has already been shown. For the necessity, we note that if a $ fd,d2g exists, then a J" w ( f ; I ) I d,d,g(I) I = 0, so that (e, ae, a 2 ae) : zlaw ( f ; I) 1 d,d,g(Z) I < e. If g is of bounded variation, then (e, ae', a 2 ae') : I v(J) - L'"1 d,d2g(Z) ( 1 < e, or X0(v(Z) - I d,d,g ( I ) I) < e, since v ( I ) is additive and v(I) 2 I d,d2g(Z)I for all I . Consequently, if a 2 a e and at', then
=
C"W(f;
I ) fvfl)
-
I 45,gfU I) < 2 Me,
where 1 f(x, y ) 1 5 M on J. Consequently, a s J o ( f ; I)v(Z) = 0. Since the variation function v(x, y) is monotone, we can assert:
J and g(x, y) of bounded variation, then a necessary and sufficient condition that a $J fd,d,g exist is that u fd,d2v exist. Consequently, if p(x, y) and n ( x , y) are the positive and negative variation functions of g(x, y ) , then a fd,d2g exists if and only if u $J fd,d2p and a $J fd,d,n both exist. It is probable that the same theorem holds for norm integrals but simple proofs of the necessity condition, in particular: if N $J fd,d2g exists, then N $J fd,d,v exists also, are lacking. The method of proof given for the a-integral extends to the norm integral if we know that the total variation v(J) = $J I dldzgI exists as a norm integral. The boundedness conditions on f in the above theorems can be dropped if we add the convention that w(f; I ) v(I) = 0,when v(I) = 0. Also although the proof has been made for the a-integrals based on 9.5. THEOREM. If f ( x , y ) is bounded on
sJ
sJ
9.
STIELTJES INTEGRALS IN TWO DIMENSIONS
131
general subdivisions, it is also valid for the a-integrals based on net subdivisions, since the total variation exists as a net integral. Since for given f and g, the function Loco(fiI ) v ( I ) is a monotone nonincreasing function of a, and the net subdivisions are cofinal with general subdivisions, we conclude : 9.6. T H E O R E M . A necessary and sufficient condition that o JJ fd,d,,g exist is that net a J J w ( f ; Z)v(I) = 0, or that net o JJfd,dzv exist.
We also note: 9.7. T H E O R E M . If g,(x,y) and g,(x, y ) are of bounded variation on J and such that JI I d,d,g, I 5 JI I d,d,g, I for all I of J, and if o JJ fd,d,g,, exists, then a JJfd,d,g, exists also. The same statement holds for norm integrals if g, and g, are monotone. Further we conclude that if net-a J,.fd,d,g exists and if Z is any interval of J, then net-a J,fd,d,g exists for all I of J, and the resulting interval function is additive. The corresponding result for norm net integrals is valid if g ( x , y ) is monotone in the sense that d,d,g(Z) L 0 for all I, but it is an open question whether it holds when g is any function of bounded variation. The following theorems are proved in the usual way:
9.8. Approximation Theorem. If g(x, y ) is of bounded variation on J, and f(x, y ) is such that JJfd,d,g exists, then for every subdivision o, we have I JJ fd,d,g - Zo fd,A,g I 5 Zp(f; Z)v(Z). This holds for general or net, norm or a-integrals, the a being correspondingly restricted, but not for weak integrals. The same remark applies also to the following convergence theorems : 9.9. T H E O R E M . If.f,(x, y ) with q on directed Q, converge uniformly
to f(x, y ) on J ; if g(x, y ) is of bounded variation on J,and if JJfgd,d,g exists for all q, then lim, JJf,d,d,g exists and is equal to JJfd,d,g. 9.10. T H E O R E M . If g,(x, y ) with q on directed Q and g(x, y ) are of bounded variation on J ; if lim, J J I d,d,(g, - g ) I = 0 ; iff(x, y ) is bounded on J ; and if J J fd,d,g, exists for all q, then lim, JJ.fd,d,g, exists and is equal to JJ.f(x,y)d,d,g.
Q and g(x, y ) are positively monotonely monotone on J, if lim,g,(x, y ) = g(x, y ) for all (x, y ) of J ; if f(x, y ) is bounded and such that J J fd,d,g, and JJfd,d,g exist for all q of Q,then lim, JJfdldlgp = JJ fd,d,g.
9.11. T H E O R E M . If g,(x, y ) with q on directed
132
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
The last theorem is a generalization to functions of two variables of Ex. 5 at the close of 11.15. 10. Relation between Double and Iterated Integrals
A reduction of double integrals to iterated integrals is possible in the case when the function g(x, y ) of bounded variation with respect to which the integration is taken, is a product of a function of x by a function of y , that is g ( x , y ) = a ( x ) B ( y ) . We shall assume in this section that a ( x ) is of bounded variation in x on [a, a’] and B ( x ) in y on [b, b’]. The basic inequality is contained in: 10.1. THEOREM. If f ( x , y ) is bounded on J = [a, a’; b, b’] and a ( x ) and B(y) are of bounded variation on [a, a’] and [b, b’], respec-
tively, then
Here the upper integrals are the greatest of the corresponding limits and the double integrals are based on general subdivisions. We give the proof for the norm integrals, for the a-integrals the procedure is similar. We have: 9‘ (
I I
’6, a 5 d J :
5
C,f(x’,Y’Idzdya(x)B(y)
fJfk Y ) d , d , ~ f x ) B f Y+ ) e.
For the same e and d,, there exists a with xiPl5 xi’ 5 xi, such that
2
J
a
uz
such that 1 az 1 < d , and xi’
d a f x ) J f f X , Y ) d B f Y ) - e.
If a(xi) - a(xiJ 2 0, we can find uiy such that ylii with yi,f-l5 ylii 5 y i j ,j = 1 ... ni,such that
Ifafxi) we have :
I aiDI < d, and
< 0, there exists a d,‘ 5 d, such that for
I aiy 1 < d,‘
10.
133
DOUBLE AND ITERATED INTEGRALS
As a consequence,
(.(Xi)
-4Xi-J)
+ e c I&)
-4 X i - J
i
I.
Consequently, if 0 consists of the intervals [xi-,, x i ; yii, yi+,1, i = 1 ... m,j = 1 ... ni, then
+- e f l + Va) 5 l J f kY)dUfX)dPfY)+ 4 2 + Va). Since this holds for all e > 0, the inequality of the theorem follows. In a similar way (or by replacing f ( x , y ) by - f ( x , y ) ) , we prove that
As an immediate corollary we have : 10.2. If a ( x ) is monotonic nondecreasing on [a,a'] if P(y) is of bounded variation on [b, b'] and if f ( x , y ) is bounded on J, then
The monotoneity condition is needed for the inner inequalities :
An immediate consequence of these inequalities is : 10.3. If N J J f ( x ,y)da(x)d,L?(y) exists, with a ( x ) monotonic nondecreasing and P(y) of bounded variation, then c ' f ( x , y)dct(y) exists
134
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
except for a set of a-measure zero, and SJ f ( x , y)da(x)da(y) = d a f x ) X ' f f x ,Y)dPfY)* For if JJ f ( x , y)da(x)d,d(y) exists, the end terms of the inequalities of the preceding theorem are equal so that all of the inequalities become equalities. Then
s:
r' r' -a
and
d a f x ) f ' f f x ,Y)dPfY) = -b
d a f x ) J ; ' f f x ,Y)dPfY) =
-a
Sf
d@(X)f ' f f x ,Y)dP(Y)
I:'
d a ( x ) ,:'ffx, Y ) d N Y ) .
-b
This means that Ja'da(x)f ' f ( x , y ) d p ( y ) and Ja' d a ( x y f ' f ( x ,y ) d p ( y ) both exist and are" equal.+hen
is a positive or zero function of x such that N J:' d a ( x ) h ( x ) = 0. But by Ex. 1 at end of 11.15 such a function h ( x ) vanishes excepting possibly at a set of a-measure zero. Then J : ' f ( x , y)dP(y) exists for all x excepting at most a set of a-measure zero and we can write s J f f x , Y)dafx)dpfY)= d a f x ) $i 'f (x,y ) d p f Y ) , where . f : ' f f x ,Y ) dP(y) can be either jb'f ( x , y)d/?(y) or J ; f ( x , y)dB(y), and is defined except for a set bf a-measure zero, By interchanging x and y , we obtain:
sl'
10.4. If a(x) and P(y) are monotone on [a,a'] and [b,b'],respectively,
if f ( x , y ) is bounded on J , and N JJ f ( x , y)da(x)dP(y) exists, then N J : ' f ( x , y ) d a ( x ) exists for all y except for a set of ,&measure zero and N J : ' f ( x , y ) d p ( y ) exists for all x except for a set of a-measure zero, and
provided we interpret the inner integrals properly.
10.
135
DOUBLE AND ITERATED INTEGRALS
The same type of theorem is possible for a-integrals. The modifications necessary are due to the fact that the condition Jl’ h(x)da(x) = = 0 for h(x) 2 0, and a(x) monotone yields only that h(x) vanishes except for a set of a*-measure zero, where the a*-measure takes into account the right and left hand approach to points of [a, a’]. For the points for which h(x) # O will be either those for which h(x) is discontinuous, or those for which h(x) is continuous but which are interior to an interval of constancy of a(x). Our theorem can be extended to the case when a(x) and P(y) are of bounded variation in the following form: 10.5. If a(x) and P(y) are of bounded variation on [a, a’] and [b, b‘],
respectively, if f(x, y) is bounded on J = [a, a‘; b, b’], and if N$Jf(x,y)dVa (x)dVP(y) exists, then the iterated integrals N$:’da(x) $:‘f(x, y)dP(y) and N $:’ d/3(y) Jl’f(x,y)da(x) exist and are equal to SJ f(x,Y)dCY(x)dP (Y)* Here Va(x) and VP(y) are the variation functions corresponding to a and P, respectively, and the inner iterated integrals exist except for sets of Va-measure zero and VP-measure zero, respectively. If p,(x) and n,(x) are the positive and negative variations of “(X) and PJY) and n,(Y) of P(Y), then
fa(x)
- .(a))
(P(Y)
= P,(X)P,(Y)
-
P(b))
+ n,(x)n,(y)
(P+)
= -
- n,fx))
(P,(Xh,(Y)
(P,fY) - n,(y))
+ n,(x)p,(y)).
But we also have
1;::;;I d,d,(a(x)P(y))
I = va(x)VP(Y)
= P,(X)P,(Y)
+ n,(x)n,fv)
+ P,(x)n,(Y) + n1(X)P,(Y)-
+
+
Then the functions p,(x)p,(y) n,(x)n,(y) and p,(x)n,(y) n,(x)p,(y) are the positive and negative two-dimensional variation functions of the function a(x)P(y). Since N fdVadVa exists, it follows that N $,,fdodP also exists. Further, since d,d,p,p,(Z) 5 d,d,VaVP(Z) for every interval 1, it follows that fdp,dp, exists. In the same way $J fdn,dn,, fdp,dn,, and sJfdnldp2exist also. Each $J
sJ
sJ
of these four integrals can be expressed as an iterated integral. Then J:’ .f(x, y)dp,(y) and J:’f(x, y)dn,(y) each exist except for a set of p,-measure zero. Since the sum of two sets of p,-measure zero is also of p,-measure zero, it follows that J:’f(x,y)dP(y) exists excepting
136
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
for a set Elof p,-measure zero. Similarly, the same integral will exist excepting for a set E, of n , measure zero and so x ’ f ( x , y)dP(y) exists excepting perhaps for the points of a set E common to E , and E, which will be ofp, as well as n,-measure zero. Then since for any interval I : Va(1) = p,(Z) n l ( l ) it follows that E is also of Va-measure zero. It follows that J : ’ f ( x ,y)dP(y) exists except for a set of Va-measure zero and by combining parts we can write
+
SJf(X,y)da(x)dP(y)
=
T’
da(x) / : ‘ f ( x , Y l d g ( Y )
9
where J : ’ f ( x ,y ) d P ( y ) is defined except for a set of Va-measure zero. To-ensure integrability it can be taken as the function jb’f (x, y)dp,(y) - f ’ f ( x , y)dn,(y) on the exceptional set. Since a ( x ) ind P(x) enter symmetrically, the reduction to an iterated integral in which the order is reversed is also possible. For the case of the a-integrals, it is sufficient to assume that a J J f ( x y)da(x)dP(y) , exists, since the variation function corresponding to a(x)P(y) is V a ( x ) VP(y), so that by Theorem 111.9.5 we know that a JJ f ( x , y)dVa(x)dVP(y) exists also. 11. Double Integrals with Respect t o Functions of Frkchet Bounded Variation
In connection with the expression for the most general continuous bilinear form on the space C, x C , of products of functions continuous in x on [a, a’]by functions continuous in y on [b, b’], the following theorem is important: 11.1. THEOREM. If a ( x , y ) is of Frtchet bounded variation on J = [a, a’; b, b‘], then the weak norm double integral wkJJf ( x ) g ( y ) d,d,a(x, y ) exists for all function f ( x ) continuous on [a, a‘] and g ( y ) continuous on [b, b’]. The proof depends on showing that the sums
satisfy the Cauchy condition of convergence. Here az is the subdivision a = x,, < x, < ... < x = a’ with xi-l S xi 5 xi, and a, and y / siI“ milarly for [b, b‘], while Iii= [xi-,, x i ; yj+ y i ] .It is sufficient to show
137
11. DOUBLE INTEGRALS: FUNCTIONS OF FBV
I
that the difference 1 Zu - Zu I approaches zero as 1 u5 + 0 and 1 uy 1 -+0, if u: 2 uZ and uy' 2 u y , since the sums for any two subdivisions uzuy and uz'uy' can be compared with a sum involving uzrfuy'', where uz" 1 uZ and uz', and au" 2 uu and a,/'. Suppose that uZ' results from uz by redividing the interval [xi-l, x i ] by the points xik, k = 0 ... m iand uy' by dividing [y,,, y i ] by the points yiz, 1 = 0, 1, ..., n.. Then : (I
X Y
X
Y
Here w ( g ; uZ) and w(f; u p ) are the maximum oscillation of g andf on the intervals of uz and u y , respectively, M , is the maximum of l f ( x ) I on [a, a'], and M , that of I g ( y ) on [b,b'],while Ai,Biz,Cik,D j z are numbers whose absolute value is less than or equal to unity. For the next step in the proof, we use a lemma on the maximum value of bilinear forms which we shall prove below :
I
11.2. Lemma. The maximum value of the bilinear form q=, ZYz1 a i j x i y i for xi 5 1 and y j 5 1 is attained for [ x i = 1 and I y j I = 1. Since this maximum is necessarily positive, there exist E~ = f 1, and = f 1, such that I Z i p i j x i y j15 Zipti E ~ E ~ for ' l x i 1 5 1 and l y j l S l . Applying this lemma to
I I
I I
%nd
there exist
E
with values f 1, such that
~ E~? ~ ,' , tik", E ~ ~ " ' ,
1
138
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
and E ~ ~ ’E ’
5 A ,a (Iik, J.
Now the right-hand sides of these inequalities are each less than or equal to F V a ( J ) . We therefore have if csx’ 2 csx and our2 uv, then
1 cf 024Y
g q p
-
c fgA,+
UE ’av
’
5 [M,4g;
+ M p f f ; .31
FVQ(J1.
Since f and g are uniformly continuous, this inequality leads to the existence of the weak net integral wk JJ f ( x ) g ( y ) d , d , a ( x , y ) . [Proof of lemma. Suppose that the maximum of L’lpsjxty i is attained for x s = x,’ and y f = y f ’ . This maximum is positive unless a,! = 0 for all i and j . Since XTp,!xf’yj‘I Z j Zta,,x,‘ yi’ it follows that the maximum is attained if we set y ; = E ~ ‘= sgn (Csasix,’).Now
I
1
1
11 1
In and so the maximum is attained for x,’= E % = sgn (L’iasj~i’). other words, there exist c s = & 1 and E ~ = ‘ 1, such that CtjasixsyiI 5 Z ~ , + Z ~ ~for E ~all E ~ ~x , 15 1 and y j 15 I]. It is possible to show that this theorem is reversible in the form: If wk J J f ( x ) g ( y ) d , d 2 a ( xy, ) exists for all continuous functions f ( x ) on [a, a ’ ] and g ( y ) on [b, b ’ ] , then a ( x , y ) is of FrCchet bounded variation on J. For a proof see J. A. Clarkson: On double RiemannStieltjes integrals, Bull. Am. Math. SOC.,39 (1933) 929-936. As suggested in the opening sentence of this section, FrCchet has shown that the most general linear continuous functional on the product space of functions continuous in x on [a, a’] by those in y on [b, b’] is wk J J f ( x ) g ( y ) d a ( x ,y ) , where ~ ( x y, ) is of FrCchet bounded variation. For a proof ste M. FrCchet: Sur les fonctionelles bilineaires, Trans. Am. Math. SOC.16 (1915) 215-234. It would be interesting to know whether the Weierstrass polynomial approximation theorem based on Bernstein polynomials could be used for the proof.
I
I
I
EXERCISES
sJ
1. Show that if f ( x , y ) and g ( x , y ) on J = [a,a’; b, b‘] are such that fdg exists, and f ( x , y)= f ( x ) , independent of y, then SJ fdg =J:Lf(x)d,(g(x, b‘) - g(x, b ) ) . Is this reversible, that is, if f ( x , y ) is independent of y
11. DOUBLE
INTEGRALS : FUNCTIONS OF FBV
and Jrf(x)d,(g(x, b‘) - g ( x , b ) ) exists, can one conclude that exists?
139
sJfdg
2. If g ( x , y ) is of bounded variation on J, under what conditions on g does the total variation v(J) = d,d,g(I) I exist as a norm integral?
JJI
3. If g ( x , y ) is continuous and of bounded variation on J, is it true that if a sJ.fdld2g exists, then N fdld2g exists and has the same value?
SJ
4. Let g ( x , y ) be of bounded variation on J , and f ( x , y ) be bounded.
Let p ( x , y ) and n ( x , y ) be, respectively, the positive and negative variations of g ( x , y ) . If M ( I ) = 1.u.b. of f ( x , y ) on I and m ( I ) = g.1.b. of f ( x , y ) on I define F(a) = Z,[M(I)A,d,p(Z) - m(I)A,A,n(I)] and F(a) = z0[m ( I ) A,A,p(I) - M ( I ) A,A,n(I)]. Show that a fdg = g.l.b.,F(a) = lim,F(a) and a J fdg = l.u.b.,F(a) = lim,E(a).
7
-
5. Show that i f g ( x , y ) is the break function p ( x , y ; x,; hl(y), h,(y)) defined: g ( x , y ) = 0 for a 5 x < x,, g(x,, y ) = h,(y), g ( x , y ) = h,(y) for x, < x 5 a’, all for y on [b, b‘], and if JJ f ( x , y)dg(x, y ) exists, then J:’f(x,,, y)dh,(y) exists and gives the value of the integral. 6. Show that if g ( x , y ) is monotonely monotone, if f ( x , y ) is bounded on
J and
sJf ( x , y)dld2g(x, y ) exists, then
where g c ( x ,y ) is the continuous part of g ( x , y ) . The same type of integral (norm or a) appears on both sides of the equality. Can the result be extended to any function of bounded variation such that g ( x , b ) and g(a, y ) are of bounded variation? 7. Show that if the net integral JJ,f(x)g(y)dld20c(x,y ) exists for every continuous function f ( x ) on [a, a‘] and every continuous function g ( y ) on [b, b‘], then a ( x , y ) is of bounded variation on J.
140
111. RIEMANN INTEGRALS IN TWO DIMENSIONS
REFERENCES
The following articles may be consulted in connection with material covered in the preceding paragraphs relating to multiple Riemann-Stieltjes integrals: J. BURKILL,Functions of intervals, Proc. London Math. SOC. (2) 22 (1924) 275-310.
J.A. (LARKSON, Double Riemann-Stieltjes integrals, Bull. Am. Math. SOC. 39 (1933) 929-936. M. FRECRET, Extension du cas des integrales multiples d’une definition d’integrale due A Stieltjes, Nouv. Ann. de Math. (4) 10 (1910) 241-256. M. FRBCHET,Sur les fonctionelles bilineaires, Trans. Am. Math. SOC. 16 (1915) 215-234. H. LUIKENS, Riemann Stieltjes Integratie dij Functies van tree of meer Veranderlichem, Doctoral Dissertation, Groningen, 1937. R.C. YOUNG,On Riemann integrals with respect to a continuous increment, Math. 2. 29 (1929) 217-233. W.H. YOUNG,On multiple integrals, Proc. Roy. SOC. A93 (1917) 27-41. W.H. YOUNG,Multiple integration by parts and the second mean value theorem, Proc. London Math. SOC.(2) 16 (1917) 273-293.
CHAPTER IV
SETS
1. Fundamental Operations
For convenience, we collect in this chapter nomenclature, operations, and manipulations of sets and classes of sets, some of which have been used in the earlier sections. The concepts presented are basic in connection with the theory of measure and Lebesgue integration. We postulate a basic set W, and concern ourselves with subsets E of W. We assume that there is a way of distinguishing between elements of W and consequently between subsets of W. The empty or null set will be denoted by 0. 1.1. Order. The concept of inclusion, where all elements of a set E ,
belong to E,, sets up a natural order in subsets and in classes of subsets of W. El 5 E, shall mean that E , is a subset of, or contained in E, (other notations in use are E l C E,, and E , E E,). 1.2. Addition. For two or more sets Eu, the sum (or union or join) is the set E consisting of all elements in any E,. For two sets we write E = El E,, for a collection of sets Eu,E = ZuEu.Alternate notations are E = El U E, for two sets and E = U,Eu, for a collection. (The symbol U can be associated with the initial letter of union.) Addition of sets is obviously commutative (El E, = E, E,) and associative [ ( E , E,) E, = E , ( E , E,) = El E, E,]. Further, E E = E for all E, i.e., addition is idempotent, and the relation E , 5 E, is equivalent to E l E, = E,.
+
+
+
+
+
+
+
+
+ +
+
1.3. Multiplication. For two or more sets Eu, the product (or intersection or meet) is the set of all elements common to all EU.For two sets we write E = El - E, or E,E,, for a collection E = ZIuEJ.Alternate notations are: E = E l n E, for two sets, and E = nuEufor a collection of sets. (The symbol ncan be thought of as a rounded off n.) Multiplication is commutative and associative. It is also distributive
141
142
IV. SETS
+
+
with respect to addition in both ways: E,(E, E,) = E,E2 E,E,, and El E,E, = (El E,) ( E , E,). The second of these follows if the right-hand side is " multiplied out." We also have EE = E for all E, and El 5 E, is equivalent to E,E, = E,. Further E,E, = 0 is equivalent to the fact that E , and E, are disjoint.
+
+
+
1.4. Complementation. To every set E in W, there corresponds uniquely its complement in W, the set of all elements of W not in E. We shall denote the complement of E by CE. Another acceptable notation is -E (read: not E). Then for any set E, CE is the set for which E CE = W and E * CE = 0. If we apply the operator C to order, addition and multiplication we find:
+
1.4.1. If El 5 E,, then CE, 2 CE,, 1.4.2. C(ZuEn)= IIuCEa, 1.4.3. C(IIEn) = ZuCEn,
that is, under complementation order is reversed, and sums and products interchanged. . Under the definitions of addition, multiplication, and complementation given, the set of all subsets of a set W form a Boolean algebra. 1.5. Definition. A Boolean algebra is a class X of elements: x, y , z ... consisting of at least two elements, on which there are defined two binary operations : x 0y and x 0y on all pairs of elements (x, y ) of to x. These operations are both commutative and mutually distributive ( x 0 (y@z) = x a y @ x a z a n d x @ y Q z = (x@y)Q ( x ~ z ) ) . Further, there exist two elements 0 and 1 such that x 00 = x and x 0 1 = x for all x ; and for each x there exists an x* in X such that x 0x* = 1 and x 0x* = 0. Then it can be shown that @ and 0 are also associative, that x @ x = x and x 0x = x for all x, and x @ x Q y = x , x 0 ( x @ y ) = x f o r a l l x a n d y . SeeE.V. Huntington: Sets of independent postulates for the algebra of logic, Trans. Am. Math. SOC. 5 (1904) 288-309
x
1.6. Difference. If E, is a subset of El, then El-E, is the set of all elements of El which are not in E,. Since this statement also makes sense even when E, does not belong to El, we use it to define E , - E, in any case. Equivalent definitions are: El - E , = El - E,E, = E,CE,.
1.
143
FUNDAMENTAL OPERATIONS
+
We note that we do not in general have E, (El - E,) = El. For two sets El and E,, there exists an E such that El E = E, only if El 5 E,, and in that case there are many sets E which will serve. We do have El - (E,+ E3) = ( E , - E,) - E,= ( E , - E3)-E, = ECE,CE, and E,(E, - E,) = E,E, - E3E2.
+
1.6.1. T H E O R E M . The product of two sets can be expressed in terms of differences. We have E,E2=El-(El-E2). For: El-(El-E2)=E1C(EICE2) = El(CEl+ E,) = EIE,. If En is a sequence of sets then:
For the right-hand side can be written E,C(E,CE,)C(E,CE,) is equal to
.... This
+ E,) (CE, + E3)... E,E,(CE, + E3) (CE, + E,) ...
El(CE, =
= E,E,E,(CE,
+ E,) ...
=
... =
n
E,L.
n
This formula depends on an order of the sets, but the ultimate set is independent of the ordering. If Er2is any collection of sets, then
An alternative expression relating products to sums and differences is
where E = Z',Eo. This is equivalent to complimentation relative to the set E = ZaEU. 1.7. Symmetric Difference. The symmetric difference of two sets
E , and E, and denoted by E,AE, is defined: E,AE,
=
( E , - E,)
+ ( E , - E,)
=
E,CE,
+ E,CE,.
1.7.1. Obviously: If E , and E , are disjoint, then E,AE, = E , f E,;
if E , = E, = E, then EAE = 0; for any E, E d 0 = E and EAW CE; and for any E , and E,, (CE,)/I(CE,) = E,AE,.
=
144
IV. SETS
1.7.2. As its name implies the operation A is symmetric: E,AE, = E,AE,. It is also associative: (E,AE,)iIE, = E,A(E2AE,). For :
+
(E,AE,)AE, = (E,CE, E,CE,)AE, = (E,CE, E,CE,)CE, C(E,CE, E,CE,)E, = E,CE,CE, E,CE,CE, E,CE,CE, E1E2E3,
+
+
+ +
+
+
which is symmetric in E l , E,, and E,. 1.7.3. Further, for any E, and E,, there exists a unique E such that EdE, = E,. For if E is such that EOE, = E,, then (EAE,)dE, = E2AE,. But (EOE,)AE, = EA(E,AE,) = E d 0 = E, so that E must be E,AE,. Conversely if E = E,AE,, then (E,AE,)AE, = E,. Consequently, EAE, = E, is equivalent to E = E,AE,. 1.7.4. Summarizing, we can say that the class of all subsets of W form a commutative group under the operation A between sets, the operation A being nilpotent.
1 J.5. Multiplication of sets is distributive relative to symmetric difference in the sense that for three sets: E l , E, and E,, we have E, (E,AE:J = (E,E,) A (E,E,) ' For : (E,E2)A (E,E,)
+ E,E,C(E,E,) EIE,(CE, + CE,) + E,E,(CE, + CE,)
= E,E,C(E,E,) =
= E, (E,AE,).
1 J.6. Under the combined operations of symmetric difference ( A ) and multiplication the class of all subsets of a set W form then a Boolean ring. * ,)s(
1.7.7. Addition of a finite number of sets can be expressed in terms of multiplication and symmetric differences.
* A Boolean ring of a set of elements X involves two binary operations @ and 0. Under @, X is a commutative group. Multiplication 0is associative and distributive relative to @, and is idempotent (x 0x = x for all x ) . It can then be proved that @ is nilpotent ( x @ x = 0, where 0 is the identity element of @), and multiplication is commutative. See: C. Caratheodory, Mass und Integral und ihre Algebraisierung, pp. 18-20; M.H. Stone, Theory of representations of Boolean algebras, Trans. Am. Math. SOC.40 (1936) 37-111.
1.
145
FUNDAMENTAL OPERATIONS
For :
+ E, = E,AE, + E,E, = (E,AEJA(E,E,) = E,E,A(E,AE,). For three sets E , + E, + E,, we first obtain an expression for El + E , which in turn yields an expression for (El + E,) + E,, and so on. El
1.7.8. Similarly multiplication of a finite number of sets can be expressed in terms of addition and symmetric difference. We have: E,E, = (El E,,)A(E,AEJ, the result of solving E, E, = E,E,A(E,AE,) for E,E,.
+
+
1.8. Limits of Sequences of Sets. In terms of addition and multiplica-
tion, it is possible to define the notions of limits of sequences of sets. If El, is any sequence of subsets of W , then m
m
C
n
D
m=1
m
n=m
These definitions are analogous to the following definitions of greatest and least of the limits of a sequence of real numbers {x,,}:
@ x,,= g.1.b. (1.u.b. x,,) m n2m
-x, = 1.u.b. (g.1.b. x?,). ; lim n
m
n$m
Alternate definitions are: 1.8.1. For any sequence of sets E,,, G , E , , is the set which consists of the elements in an infinite number of E,,, while -n lim E,, is the set which consists of the elements in El, for n greater than some no, depending on the element. For if the element x is in an infinite number of the sets En, then it belongs to Z2=mE, for all m and so to G n E , . Conversely, if x is in limllE,l,then it is in Z;=,E,, for all m, and so in E,, for an infinity of n’s. Similar reasoning applies to lirn EIL. As an example, if E2mis the set of x such that - 8 5 ~ 1 2 l/m and E2m+,,the x such that - 1 + l / m 5 x 5 i, then lim,E, is the set of x such that - 1 < x < 1, and lim - E,, the x such that - 8 5 x 5 i. 11
11
1.8.2. Obviously C ( E l I E I , )= lim,,CE1,. 1.8.3. The sequence ElZ is said to have a limit if
lim E,, = !& E?,.
If the sequence E,, is monotone nondecreasing, that is,
L E,,
146
1V. SETS
for all n, then lim,,E, = Z,E,); if E,Lis monotone nonincreasing, then lim,E, = L',E,. If for any sequence EPL, we write Em'= 2,"E,, then GT,E,,= lim7nEm',while if Em"= Z7ZE,,, then lim E, = limmETn''. 2. Characteristic Functions
To every set E of a set W, there corresponds a point function, the characteristic function of the set E defined x(E; x) = 1 for x in E and x(E, x) = 0 for x in CE or not in E. Characteristic functions set up a one to one map of the subsets of W o n the class of real valued functions on W taking only the values 0 and 1. We note the following: 2.1. x(0;x) = 0 and x ( W ;
x)
=
1 for all x.
2.2. If El and E, are any two sets, then
x(E,; x)
2.3.
+ x(E,; x) = x(E,+ E,; x) + x(E,E,;
x). If the sets Ea are disjoint, then x(ZaEn; x) = Zax(Ea; x).
2.4. If E is any set, then x(E;
x)
=
1 - x(CE; x).
x(E,; x)x(E,; x). If Ea is any collection of sets, then x(IZ,E,; x) = Dax(Ea;x).
2.5. If El and E, are any sets, then x(E,E,;
x)
=
2.6. If E, and E, are any sets, then
x(E,
-
E,; x)
=
xm,;
xl (1
-
2.7. If El and E, are any sets, then
x(E,AE,; x)
=
x(E,+E,; x)
- x(E,E,; x) =
x(E,;x)).
I xw,; x) m,;x) 1. -
2.8. If EtLis any sequence of sets, then
x(KfdE?l; x)
=
K r (x(E,,; x)
and X(limlLE,;x ) = Iim?, - x(E,$;x).
2.9. If E,, is a sequence of sets, then limTIE,,exists if and only if
lim,, x(E,,; x) exists for all x of W, and this limit is x(lim,lE,l; x). The relations of 2.8. above give additional justification for the definitions of E , , E , fand 1imTfE,,. 3. Properties of Classes of Sets
Each of the operations on sets of the preceding sections suggest a closure property of classes (3 of subsets of W. Thus we have: 3.1. Definition. A class (3 of sets E is additive: A , if the sum of any finite number of sets of (3 belongs to &; (3 is sequentially additive: s-A,
3.
PROPERTIES OF CLASSES OF SETS
147
if the sum of any denumerable number of sets of Q belongs to Q; Q is totally additive: t-A, if the sum of any collection of sets of Q belongs to Q. The class of closed sets of points on a straight line or of a topological space is additive but not s-additive. The class of open sets is totally additive. If the function u ( x ) is monotonic on [a, b ] then the class of sets of points of a-measure zero is s-additive. 3.2. In the same way, the class (3 is multiplicative: M , if the product of any finite number of sets of any Q belongs to Q, sequentially multiplicative: s-M, if the product of a denumerable number of sets of (3 belongs to Q; and totally multiplicative: t - M , if the product of any collection of sets of Q belongs to 0. The class of closed sets on a straight line is totally multiplicative. 3.3. A class
Q of sets E has the complementary property (C) if with
the set E, Q also contains its complement CE. 3.4. A class of sets Q has the subtractive property ( S ) if the difference of any two sets of Q belongs to Q. 3.5. A class of sets Q has the symmetric diflerence property ( A ) if the symmetric difference of two and so of any finite number of sets of Q belongs to Q. These properties are not independent. We have, for instance:
3.6. If Q has the complementary property then additive properties are equivalent to the corresponding multiplicative properties. If Q is subtractive, then it is multiplicative, which follows from the identity EIE! = E , - ( E , - E,). If Q is subtractive and sequentially (totally) additive, then it is sequentially (totally) multiplicative, a result of the formula 17cLE,= zl,Ecl - Zo(.ZuE~x - EJ. 3.7. Definition. Certain combinations of these properties have proved to be of interest especially in the theory of measure. A class Q of sets is said to be a ring of sets if it is additive, multiplicative, subtractive and has the symmetric difference property. It is said to be an s-ring if it is also s-additive and s-multiplicative. A class of sets Q is called an algebra if it is a ring which has the complementary property, an s-algebra if it is an w i n g with the complementary property. Since the properties of classes of sets are not independent we can reduce the ring conditions as follows :
148
IV. SETS
3.8. THEOREM. The class Q of sets is a ring if it has any of the fol-
lowing four pairs of properties: (1) (2) (3) (4)
additive and subtractive additive and symmetric difference multiplicative and symmetric difference symmetric difference and subtractive.
Since each of the properties of a ring is involved in one of these four combinations, it is sufficient to show that (1) -+ (2) + (3) -+ (4) --f (1). (1) -+ ( 2), since E,AE2 = ( E , - E,) (E, - E l ) ; (2) (3) since E,E, = ( E , E,) AE,AE,; (3) -+ (4) since E , - E, = E , E,E, = ElA(ElE2);and (4j -+ (1) since E l E, = (E,E,)AElAE2.
+
+
--f
+
3.9. The class Q of sets is an s-ring if it is s-additive and subtractive
since these two properties imply also that has the symmetric difference property.
(3
is s-multiplicative and
3.10. Similarly the class Q of sets is an algebra if it has the complementary property and is either additive or multiplicative, it is an s-algebra if it has the complementary property and is either s-additive or s-multiplicative. 4. Extensions of Classes of Sets Relative t o Properties
In general, a class Q of sets of a fundamental space W will not have a given property P.In such a case, it may however be possible to extend (3 by the adjoining of additional sets so that the extended class has the property P and is the smallest class containing Q which has the property P. The resulting class will be called the class (3 extended to have property P and be denoted by Qp, when it exists. In a way, the class Q and the property P might be said to generate the class Q p which has property P. 4.1. Definition. A property of classes of sets of W is said to be
extensionally attainable, if for every such class Q, there exists the extension QP having property P.
a property P be extensionally attainable are: (1) that the class C$ of all subsets of W have the property P,and (2) the product class or greatest common subclass of any collection of classes having the property P also have property P .
4.2. Necessary and sufficient conditions that
4. EXTENSIONS
149
OF CLASSES OF SETS
For the necessity we note that if P is extensionally attainable then since there is no class Q larger than E, E itself must have property P. Further if for some set of classes, each of which has property P, the product of these classes Eo does not have the property P, then there is no smallest class having property P and containing E0. That the two conditions are sufficient follows from the observation that for any (3 an extension Q p having property P exists if there exists a class E,, containing E which has property P, and the product of any set of classes having property P also has property P. 4.3. We note that if a collection of properties Priare each extensionally
attainable, then any property which is a combination of these properties is also extensionally attainable. 4.4. For the properties A , s-A, t-A, M , s-M, t-M, S , C and A discussed
in IV.3, it is a simple matter to verify the conditions for extensionally attainability of the above theorem. It follows then that any combination of these properties, in particular, ring, s-ring, algebra, s-algebra are also extensionally attainable. The actual construction of extensions relative to given properties may involve difficulties. For the properties : additive, multiplicative, symmetric difference, the extensions EP of any class & of sets is obtained by adjoining to C all finite combinations of subsets El o E., 0 ... 0 E,I, where o stands for +, ., and A , respectively. To secure the complementary extension of a class &, we adjoin to (3 all sets CE which are complements of sets of &. The subtractive extension of a class involves complications. For instance, if Q consists of two sets El and E,, then the class Qs consists of El, E,, E, - E,, E., - E l , E,E,, and 0. If G consists of three sets El, E,, E,, then by aAalogy, GS will need to contain additionally at least E,E,CE,, E,CE,E,, CE,E,E,, E,CE.,CE,, CE,E,CE,, CEICE,E,, and E1E2ES,seven mutually disjoint sets.- These sets can be obtained by multiplying out (El CE,) ( E , CE,) ( E , CE,) = W, which gives the totality of disjoint sets of W determined by El, E,, E,, and discarding CE,CE,CE, = C ( E , E, E,). If we set A , = E,E,CE,, A , = E,CE,E,, A , = CE,E,E,, A,, = EICEZCE,, A , = CE,E,CE:,, A , = CE,CE,E:,, and A , = ElE2E3,then El = E,(E, CE,) ( E , CE,) = A , A , A , A,. Similarly, E, = A , A, A, A, and E, = A , + A , + A , + A,, that is, E l , E,, E , are each expressible in terms of four of the disjoint sets A , ... A , . It follows that the other
+
+ + +
+ + +
+
+
+ + +
+
I50
IV. SETS
sets belonging to Qs can be obtained by dropping out one or two sets in the expressions for El, E,, and E,. In this way we find that Qs consists of: 0 , El, E,, E,, A , , A,, A,, A,, A,, A,, A,, A , A , A , , A , A , A,, A , A , A,, A , A , A,, A , A , A,, A , A , A,, A , A , A,, A , A , A,, A , A , A,, A , A, A,, A7 A, A,, A , A , f A,, - 4 7 A,, A7 f A,, A7 f A,, A , + A , , A , + A , , A , + A , , A 7 + 4 A , + A , , A , + A , , A , A,, A , A,, A , A, A,, A , 4,A , A,, a total of 38 sets. Obviously any one of these sets can be written as a combination of El, E,, and E,. The same procedure can be set up for any finite number of sets: El, E,, ..., E,,. We determine first a basic block of sets by expanding ( E l CE,) ... (E,, C E J and dropping CE, ... CE,, from this expansion. We then express E , ... E,, each in terms of these new sets, each expression containing 2n-1 sets, and then discard successively all collections of one, two, ..., 2"-l - 1 sets from these expressions. All sets so obtained together with the null set: 0, will constitute 6, for Q = ( E , ... E?,). The number of sets resulting increases very rapidly with n. To obtain Q,9 for any collection of sets, it is necessary to collect all extensions of any finite number of subsets of Q. To extend @ to be s-additive, we adjoin to Q, all sums of a denumerable number of subsets of 6. For instance if (l consists of all open subintervals of the open interval a < x < b, then Qs-A4 is the set of all sums of a finite or denumerable number of open intervals that is, the set of all open subsets of a < x < b. The s-multiplicative extension of a class can be obtained in a similar way. In general, if a class Q has the property P I , the extension Qr, relative to a property P, need not have the property PI,in other words the extension Q p I p , may differ from ( ( f p I ) p , . This observation is important in connection with the problem of extending a class of sets to be a ring. As we have seen in IV.3.8 this can be done by extending Q by any of the four double extensions ( A S ) , ( A d ) , ( M A ) , (SA). We note the following instances where (l(,,,,) = (QP1)p2:
+ + + + + + + +
+ + + + + + + + + + + + + + + + + + + + + + 4
9
+
+
4.5. If the class of sets
6 is multiplicative, then the extension 6,4is also multiplicative, and so = For if E , E, + ... + Ek and El' E,' ... E?,' are two sets from Q A , then ( 2 ? E t )(ZiEi') = ZtiEfEi' belongs to QA4, since E,E,' belongs to 6 for all i a n d j . Similarly by interchanging addition and multiplication, we have :
+
+
+ +
4.
Q is additive, then the extension Q,,,is also
4.6. If the class of sets
additive and so
151
EXTENSIONS OF CLASSES OF SETS
=
QA4.,
Q is subtractive, then the extension GA is also = (6,).l.
4.7. If the class of sets
subtractive, so that For (Ca E a = (Ci E i )
CiE J
=
(Xi E , ) C ( X j Ei')
ni xi
'
CE,'
=
( E i n i CE,')
Now the sets EiniCEi' are formed from elements of Q by subtraction and so belong to Q, so that SlEp(Z7jCE,')is in Q A .
Q is multiplicative, then the extension Q is also multiplicative so that = For because of the associativity and distributivity of A , (E,AE, ... AE,) (E,'AE,' ... AE,,,') = E , E , ' A E , E ~ ' . I E , E...~ .AE,E,,,', ,' and so belongs to &,. 4.8. If the class of sets
4.9. If the class of sets
Q is additive, then
is multiplicative. For any two sets E , and E,, the set E,E, = ( E l E,)AE,AE, will belong to Application of the reasoning in the preceding theorem IV.4.8 yields the multiplicativity of &,,. From the preceding theorems and the fact that if Q is subtractive then it is also multiplicative, we have the following four ways of obtaining a ring extension of a class of sets: 4.10-
q i n o = (@**)A
=
(Q.l*)/l
=
(@*).I
=
Q,l
+
(QS).I.
Note that the order of extension is important. In order to extend a class of sets Q to be an algebra, we note that any algebra contains the whole space Was well as the empty set 0. Consequently, to obtain the extension of Q to be an algebra, we adjoin W and apply the methods of ring extension to the resulting class. When it comes to the setting up of a constructive procedure for extending a given class of sets Q to be an s-ring or an s-algebra we run into some difficulty. We start by extending Q to be s-additive, and then the resulting class to be subtractive. Call the resulting class G,. Extending G , to be s-additive, and the resulting class to be subtractive leads to G2.And so on, being ( ( G J - A ) S . If after a sequence of such extensions the resulting total class Gw = ZrlG,, is not an s-ring, we repeat the same operations on Gw. By using the Cantor set of transfinite ordinals, we can in this way obtain an ordered set
152
IV. SETS
of classes Grlfor all transfinite ordinals 01 of the second class. The class of sets y consisting of all sets in any G,L: G = CciGuis then the class & extended to be an s-ring. For every Grl is contained in the existent s-ring extension of 6. On the other hand, G is an s-ring. For if E,, ..., El,, ... is a sequence of sets belonging to G,there will be a minimal u 7 , such that El, belongs to Gfin . Since u,( is denumerable, there exists an ordinal ,b’ greater than or equal to u,, for all n, so that CI1ElL will belong to Gfi,and then G is s-additive. Similarly, for two sets El and E , in G,there exist 0 1 ~and 0 1 ~such that E l is in Gu and E, in Gel, so tha; E, - E, is in the G(’ for 01 the greater of u 1 and 01,). This gives the subtractivity of G. A loose way of describing the class (5 = CrIGa,is to say that it consists of all sets which can be obtained from & by applying the operations of sequential addition and taking differences a denumerable number of times. The same class G also emerges, if we replace the subtractive extension by the ring extension throughout. This means considering the to be s-adclass %,i, where 9i,i+1is obtained from 92‘ by extending ditive and then extending the resulting class to be a ring. For G a s 5 9iu 5 (Ga)s-A4S Ga+,. In a similar way, the s-algebra extension of & involves adjoining the fundamental set W to & and then applying successively the pair of operations of s-additive extension followed by complementation. The procedure outlined above for the s-ring and s-algebra extensions of a class of sets is the basis of the Borel theory of measure (see E. Borel, “ LeCons sur la ThCorie des Fonctions,” Gauthier-Villars, Paris, 1914, pp. 228-230). His fundamental set W is the closed interval 0 5 x 5 1. The initial class of sets Q consists of all open subintervals of (0, 1). %ci
4.11. The measurable sets according to Borel consist of all sets belong-
ing to the s-algebra extension of &. As a consequence in any space in which the topology is defined in terms of open sets, the Borel measurable sets consist of all sets in the s-algebra extension of the class of open sets.
EXERCISES 1. Express all of the sets in the subtractive extension of (3 = (El, E,, E 3 ) in terms of El, E,, and E,.
4.
153
EXTENSIONS OF CLASSES OF SETS
2. Show that if Q is any class of sets, then the algebra extension of be constructed as (((Q,W )C ) M ) A .
Q can
3. What is the ring extension of the class @ if Q consists of the closed subintervals of the closed interval a 5 x 5 b? Same question for two-dimensional closed intervals of [a, a ' ; b, 6'1. 4. What are the ring and algebra extensions of the class of all closed subintervals of - co < x
Q if Q consists
5. What is the algebra extension of @ if Q consists of all half open intervals < CQ? What is the s-algebra extension of the same class @ ?
a 5 x < b of - co < x
This Page Intentionally Left Blank
CHAPTER V
CONTENT AND MEASURE
1. Content of a Linear Bounded Set
For sets of points on a bounded linear interval: a 5 x 5 b, we are accustomed to attach to subintervals [a,, b,], the length of the interval l ( I ) = b, - a, as a measure or content. No distinction is made between closed, open, or half-open intervals. By a natural extension we attach as a measure to a set E consisting of a finite number of disjoint intervals Il ... I , the sum of the lengths of the intervals: Zkl(Ik). The problem of attaching a measure to some subsets E of [a, b ] can then be formulated as follows: To determine a function p ( E ) of some subsets E of [a, b ] having the following properties: (1) for every E for which p ( E ) is defined p ( E ) 2 0; (2) if E, I E, and /.(El) and p ( E , ) exist, then p ( E , ) 5 p ( E , ) ; (3) if El and E, are disjoint and p ( E , ) and p ( E , ) are defined, then p ( E , E,) is defined also and p ( E , E2) = p ( E , ) p ( E , ) ; (4) if E is an interval I (open, closed, or half-open), then !&(I)= l(I). One line of procedure is suggested by the observation that if I is a subinterval of [a, b ] , then the characteristic function % ( I ;x ) , which is equal to unity for x on I and zero for x not on I, is Riemann integrable and J: %(I; x ) dx = l ( I ) . ,Further, if E consists of a finite number of disjoint intervals I , ...I , , then J: x ( E ; x)dx = Zkl(Ik). We consequently consider the class of subsets E of [a, b ] for which the characteristic function x ( E ; x ) are Riemann integrable and the properties of the set function defined by J: x(E;x ) d x . We define:
+
+
+
1.1. Definition. A subset E of [a, b ] has content if and only if the characteristic funtion x(E; x ) is Riemann integrable on [a, b]. The corresponding integral J: x ( E ; x ) dx will be called the content of E and denoted by cont E. The class of subsets E of [a, b ] which have content will be denoted by (5.
155
156
V. CONTENT AND MEASURE
Since for any set E of [a, b ] the function x ( E ; x) is bounded, it is logical to define for any subset E :
J”:x ( E ; x)dx
-
1.2. upper content:
5E
=
1.3. lower content: cont E =
and
s[: x ( E ; x)dx .
-
1.4. Then a set E has content if and only if E
tE =cont E. An application of the definition of upper integral to the special function x ( E ; x) yields the following definitions of upper content in terms of lengths of intervals: 1.2 a. (a) E t E = g.1.b. as to subdivisions o of [a, b ] of Zol(Ir) for all I’ of 0 containing points of E. 1.2 b. (b) c %
E
= g.1.b.
of Lkl(Zk) for all finite sets of intervals
I, ... I?, such that ZkZk2 E. Here we could restrict ourselves to sets of intervals which are disjoint. In the same way, we see that 1.3 a. (a) cont -E = 1.u.b. as to subdivisions o of [a, b ] of Z,,l(I‘r)
for all I ” of o consisting entirely of points of E.
(b) cont E
= b - a - cant
CE. For if CE is covered by a finite set of intervals I , ... I,, then each interval complementary to ZkIk consists entirely of points of E. As a consequence of definition (b) of cont -E, we can say:
1.3 b.
1.4 b. A set E has content if and only if
5E
+E
t CE = b - a.
Obviously any subinterval of [a, b ] has content and cont I = l ( I ) . A set consisting of a finite number of points has content zero. On the other hand, the upper content o f the points of [a, b ] with rational coordinates is b - a, while the lower content is zero. As a consequence, the set of rationals on [a, b ] (or any denumerable dense subset of [a, b ] ) does not have content. Sets consisting of a finite number of nonoverlapping intervals J = I , ... I,, have content with cont J = Zkl(Zk). We shall find use for the following :
J, = I,, ... I,,, and J, = I 2 , ... each consisting of a finite number of nonoverlapping intervals, then
1.5. THEOREM. If
cont J,
+ cont J2
= cont
(J, + J J
+ cont J,J,.
2.
157
PROPERTIES OF THE CONTENT FUNCTIONS
+
The set J , J , can be expressed as the sum of a finite number of intervals and is of the same character as J , and J;,. The set J1J2 will consist of a finite number of intervals and perhaps a finite number of separate points and so will also have content. Now
x(J,; x)
+ xv,; x)
=
x ( J , + J,; x)
+ x(J,J,; x)
If we apply Riemann integration to this identity, we get the relation involving content. 2. Properties of the Content Functions
We have : 2.1. For any set E : 0 5 cont E
5 E t E;
2.2. If E , 5 E,, then cont coiit E, and c x t E , 5 c -E I -I -
+
+
x E,;
+
2.3. For every E , and E,: c 3 ( E , E J 5E,E, 5 E t E , E t E, and cont ( E , E.,) cont E,E, 2 C t E , cont E,. 2.1 and 2.2follow direcily fromthe definitions. For 2.3e find for e > 0, interval sets J , 2 El and J., 2 E., such that cont J , 5 cant E,+e and cont J? 5 COnt E, e. Then” since J , J , 1E l E, and J , J , 2 E,E, :
+
+
+
G F t ( E l + E,) =
cont J ,
+
+
+
+ COIlt E , E , 5 coiit ( J , + J,) + cont J , J ,
+ cont J , 5 5E L+ %t
+
E,,
+ 2e.
+
Consequently, 5( E l E,,) K t E , E , 5 K t El we replace El by CE, and E , by CE, we get :
cant ( C E , + CE?) + 5( C E , CE,,) 5 *
CE,
+ %c
E,. If
+
CE,
and so b
-
a
cant ( C E , + CE.,) + b - a- cant ( C E , * CE,) 2-b - a -cant CE, + b - a - COnt CE.,
-
or
-E,E, cont
+ cont -( E , + En) 2 cant E , + cant E,.
As a n immediate consequence of 2.3 above we have:
+ E, + El) +
2.4. If E l and E, are subsets of [a, b ] with content, then E l
and E , E , also have content, and cont E , cont E,E,.
+ cont E,=
cont ( E l
158
V. CONTENT AND MEASURE
As a result of 2.1 and 2.3 above we have cont E,
+ cont E,- I -cont (E, + E,) + cant E,E, 5 + 5E,E, 5 cont E, + cont E,.
(El
+ EJ
Consequently, the inequalities become equalities and we have cont (E,
-+ E,)
-
+
GTZ (E, E,,) = COnt E,E, - cant E,E,.
Since the right-hand side of this equality is 5 0 and the left 2 0, it follows that they are both zero which proves the theorem. Since the condition that a set E have content is symmetric in E and CE [since C(CE) = El we also have: 2.5. If the set E has content, so does CE and cont E Combining these results we obtain :
+cont CE
= b - a.
2.6. THEOREM. The class (3 of subsets of [a, b ] ,which have content,
is an algebra of sets (additive, multiplicative, and complementary). Moreover, the content function on sets of (3 is additive in the sense that if E, and E, belong to (5 and are disjoint, then
+ cont E, = cont (El + E J .
cont El
The additive-multiplicative part of this theorem could also have been demonstrated as follows: If E, and E, are in (5, then x(E,; x) and x(E,; x ) are Riemann integrable. Consequently, x(E,E,; x) = x(E,; x) x(E,; x) is R-integrable and so x(E, E,; x) = x(E,; x) x(E,, x) -- x(E,E,; x) is also R-integrable. The relation between the contents of the sets involved follows from the last relation between the characteristic functions and the linearity properties of integration. E and cont E as upper and lower inteJf we remember that grals of the characteristic functions, are limits in the a-sense, the basic set of inequalities between greatest and least of limits of sums in 1.4.6 give us in the case when E,E, = 0, that is, when
+
+
X(E,
+ E,; x)
=
x(E,; x)
+ I x(E,; x)dx + I Jl x(E,; x)dx + Jb -a
x(E1; x)dx
Jb
+ x P , ; x):
Ib
x(E,; x)dx I x(E, -k E?; x)dx --a -b
--a
-b
J
x(E,; x)dx I x(E,
--a
-b
x(E,;x)dx.
+ E,; x)dx
2.
159
PROPERTIES OF THE CONTENT FUNCTIONS
Rewritten in terms of upper and lower content, we have: 2.7. If the sets E, and E, are disjoint then:
-E, cont
+ cont E , I -(El + E,) 5 cant El + 5E, cont 5 ToYt (E, + E,,) 5 5El + E2. -
I
Suppose now that M is a set which has content, and E is any set. Then E t E cont M 2 5( E M ) =EM
+
+
>_=(E-EM)
+
+contM+KtEM,
since E - EM and M are disjoint sets whose sum is E cont -M = cont M , we obtain
+ M. Since
~E2~cont(E-EM)+~EM. On the other hand,
contE5 K
t
(E
-
EM)
+ %t
EM.
Hence : 2.8. T H E O R E M . If M is a set having content and E is any set, then
EZl E = GS E M + GZ (E - EM). If E is the basic interval [a, b ] in this identity, we have the condition on a set M that it have content. The validity of this identity for all E is then a necessary and sufficient condition that a set M have content. The identity also asserts that if we divide a set E into two sets by means of a set M with content and its complement CM, then cant is an additive function on this division. It follows that if the sets M , ... M,, each have content and E is any set then K t E(CkMk) = 2, cant EM,. In terms of content, necessary and sufficient conditions that a function . f ( x ) on [a,b ] be Riemann integrable can be expressed as follows: 2.9. T H E O R E M . A function
f(x) is Riemann integrable on [a, b ] if and only if (a) f(x) is bounded on [a, b ] and (b) the points of discontinuity of f(x) can be expressed as the sum of a denumerable number of closed sets E,,, each of content zero. In view of 11.13.16 we need only show that condition (b) is equivalent to the statement that the set of discontinuities of f ( x ) has
1 60
V. CONTENT AND MEASURE
measure zero based on a ( x ) = x on [a, b ] . The set of points of discontinuity of any function f ( x ) is then sum of the sets EtI consisting of the x's for which ~ ( fx), 2 l/n, each of which is closed. If the set of discontinuities of f ( x ) is of zero measure, then the E,/ are also of zero measure. Since by the Bore1 theorem, any covering by intervals of a bounded closed set can be replaced by a covering consisting of a finite number of these intervals, it follows that any closed set of zero measure also has zero content. Then cont E,, = 0 for all n. Conversely, we note that a set of zero content is also of zero measure. If then the set of discontinuities of , f ( x ) is the sum of a denumerable number of sets of zero content, the total set of discontinuities of f(x) will be of zero measure. As a means for assigning a measure to sets, the notion of content has a number of drawbacks. For instance, we have: 2.10. If E is any subset of [a, b ] ,and E' is its derived set, then c X t E = K t ( E El). For if E is covered by a finite number of closed intervals I , ... I,,, then no point of E' can lie in the intervals which form the complement of J = ZJk. Since K t of a set does not depend on whether the covering is by open or closed intervals, we have E t ( E E ' ) 5 c Z E and so equality holds. It follows at once that:
+
+
2.11. If E is any set on [a, b ] which has content, then E
content and cont ( E The inequalities
+ E ' ) = cont E.
contE5+(EfE')5G23(E+Et)
+ E' also has
= S E
lead at once to the conclusion of the theorem. In effect, a set can have content only if its closure E E' as a closed set also has content, which is rather restrictive. This is particularly unfortunate because a large class of closed sets do not have content. For instance, a nondense perfect set has content if and only if it has content zero. For if a set is nondense on [a, b ] , then cont -E = 0, since E contains no intervals. On the other hand, if E is perfect, then c x E = b - a - L',l(btI- a t ) , where (a,,, b J are the open intervals which make up the open set complementary to E. These bits of awkwardness are due among other things to the fact that the class 0 of sets is not sequentially additive. For if E consists of the single point
+
2.
PROPERTIES OF THE CONTENT FUNCTIONS
161
x, then cont E = 0. But a sequence of points dense in [a, b ] does not have content. The extension of Riemann integrals to Stieltjes integrals, particularly with respect to a function of bounded variation a ( x ) suggest consideration of the following set functions :
aN(E) = N
I”
x ( E ; X)da(X)
a
a,(E)
=
x(E; x)da(x)
0 a
a y( E) = Y J b x(E;x ) d u ( x ) a
where a c ( x ) is the continuous part of a ( x ) and it is assumed that the integrals exist. Each integral gives rise to a class of sets, on which a corresponding set function is defined. We have : 2.12. The class of sets for which a ( E ) is defined is an algebra (additive, complementary) of sets of [a, b]. For each class, the corresponding
set function is finitely additive in the sense that if E , and E, are in the class, then a(E,)
+ a(EJ
=
a(E,
+ E,) + a(E,E,).
These integrals lead to a positive valued function of sets, if and only if u ( x ) is monotone nondecreasing.
EXERCISES 1. Show that if E consists of a single point x, then E belongs to the class determined by a y ( E ) and a,(E) = a(x 0) - a(x - 0), but that a,(E) and a J E ) are defined only if a(x) is continuous at x.
+
< x < b,, a half-open interval a, 4 x < b,, a closed interval a, 2 x 5 b, belong to the class of sets defined by aN(E) or a,(E) or a,(E)? Determine the values of these functions when they
2. When does an open interval a, QT
are defined. 3. Suppose a(x) is monotonic nondecreasing on [a, b]. Is it possible to define upper and lower functions Z,(E) and a,(E) by the use of enclosing intervals as in the case of upper and lower content?
162
V. CONTENT AND MEASURE
3. Borel Measurability
It seems sensible to assume that if we define the length of an open interval (a, b ) as b - a, then a proper length or measure for an open set G consisting of the open subintervals (a,(, b,) of a finite interval [a, b ] would be Zn(blL- a J . Further, it seems sensible to attach to a closed set F the complement of G relative to [a, b ] the measure b - a - Z?l(bpl- a,). These considerations involve two principles: (a) if E,, is a sequence of disjoint sets, each of which has a measure ,u(E,), then CllEllalso has measure (or is measurable) and ,u(Z,E,) = C,,p(E,) ; (b) if E has measure, then the complement of E relative to the fundamental set [a, b ] also has measure and ,u(CE) = b - a ,u(E). These principles applied to subsets of the linear interval and the assumption that an open subinterval is measurable, yield that the minimal class of such measurable sets is the s-algebra extension of the class of open subintervals of [a, b]. The same procedure can be applied to open subintervals of - 00 < x < co to yield a class of ‘cmeasurable” subsets. In either case the s-algebra so obtained is called the class of Borel measurable subsets, or Borel sets (see IV.4.11). A finite measure can be attached to sets of this class only if the basic interval [a, b ] is bounded, and then one can presumably proceed according to the second part of each principle stated above. For open and closed sets, such measure is easily computed, but for sets determined after a number of steps, the computation may become rather complex. However, as so often happens in the case of good mathematical notions, H. Lebesgue by a different approach derived a simple procedure which not only assigns a measure to Borel measurable sets, but a measure for an s-algebra which contains the s-algebra of Borel measurable sets. The Borel sets occupy an important position in this theory in that every Lebesgue measurable set is essentially a Borel measurable set of low order. The derivation of Lebesgue measurability is similar to that of content, based on the enclosure of a set by intervals. The main difference in the two theories is that in defining upper content of a set E, E is covered by a finite number of intervals, while for upper measure coverings by a denumerably infinite set of intervals is allowed. We have already encountered such enclosures in the definition of sets of measure zero relative to a monotonic function (see 11.13.12).
+
4.
163
GENERAL LEBESGUE MEASURE OF LINEAR SETS
4. General Lebesgue Measure of Linear Sets
Linear measure is based on the interval function l(I) = b - a, for I the interval [a,b ] with or without end points. This interval function is obtained from differences of the point function y = x. In the following considerations with Stieltjes integrals in mind, we replace y = x by y = a ( x ) , where a(x) is a bounded monotonic nondecreasing function on X = - co < x < co. Then linear measure considerations for a finite interval [a, b ] are based on the monotone function a ( x ) = a, for x 5 a, a ( x ) = x for a 5 x 5 b, and a ( x ) = b for x 2 b. We shall set a (- co) = limz+-m a ( x ) and a(+ co) = limz+ + a ( x ) . The case where a ( x ) is unbounded will be taken up later. The basic class of sets will be taken as the open intervals I : a < x < b or (a, b ) and we shall define a(I) = a(b - 0 ) - a(a 0), with a(b - 0) = a(+ co), if b = co and a(a 0) =,a(- co) if a = - co. Then a ( X ) = a(+ co) - a ( - a ). In order to obtain an upper measure function on sets E, we take as covering of a set E, a finite or denumerable number of open intervals. Now for linear sets, the sum of the points in a denumerable number of open intervals is a general open set and this in turn is equivalent to a sequence of disjoint open intervals. If {In}is the original set of covering intervals and { J k } are the intervals defined by ZnIT,, then Zkcr(Jk) 5 Zna(In).Consequently, if we are interested in greatest lower bounds of sums of the form Zna(I,&),we can replace the sequences of open intervals by general open sets. For any open set G, consisting of the open intervals: Jk = (ak, bk), we define:
+
Q,
+
+
+
+
a(G) = Zka(Jk)= Zk(a(bk- 0 ) - .(ak 0)). In terms of the function a(G) on open sets, we can in turn define an upper (or outer) measure a*(E) for any set E in X by the statement:
4.1.
4.2. Definition. The upper measure cr*(E)
o f a set E is the greatest
lower bound of a(G) for all open sets G covering E in the sense that all points of E are in G; or, a*(E) = g.l.b.[cr(G) for G 2 El.
We have already indicated that one gets the same value for a*(E) if we replace this definition by:
1 64
V. CONTENT AND MEASURE
4.2.1. a*(E) = g.1.b. [L’p(IJ for all {I,,} such that L‘J,l
2 El, the
I , being open intervals. Further, we note that if we order open sets G 2 E by inclusion, then since a ( G ) is monotone on this directed set of G, we also have: a*(E)
4.2.2.
= 1imQzEa(G).
In order to obtain properties of cz*(E), we first note: 4.3. The function a ( G ) on open subsets G of properties :
(1) 0 5 a ( G ) 5 a ( X )
X has the following
< co for all G.
(2) If GI I G2, then a ( G , ) 5 a(G2). (3) For any two open sets G I and G,, we have:
+ a(GJ
a(GJ
=
a(G,
+ G,) + a ( G , .G,)
+
where G, G,, the sum of two open sets, and G , * G,, their product, each is again an open set. (4) If { G , l ] is any sequence of open sets, then “ ( C 1 , G , ,5 ) &4G,,).
Property (1) is obvious. Property (2) follows from the fact that if (a,,, h,J is any sequence of disjoint open intervals on (a, b ) , then
+
z 7 , ( a ( b , 1 0)
-
a(a,, - 0 ) ) 5 a ( b -0)
-
a(a
+ 0).
For the monotone function y = a ( x ) maps the open intervals (a,,, b,) onto the nonoverlapping intervals (a(a,l 0), a(b,, - 0)) on the Y-axis, which are subintervals of (.(a 0), a(b - 0)). For property (3), we assume first that G, and G , each consist of a finite number of open intervals. Then the mapping y = a(x) yields two sets J , and J2, each consisting of a finite number of disjoint intervals on the Y-axis: J , = Z1r(a(a,,I 0), a(b,,, - 0), and J , = L’,,,(a(a,, 0), a(bZm- 0). Then G, G, maps into J , J , and G, - G2 into J , - J.,, while the statement
+
+ +
+
cont J ,
+
+
+ cont J2 = cont (J1J2)+ cont ( J , + J , )
translates into u(G,)
+ a(G,)
=
+ a ( G , + G2).
a(G, * G 2 )
4.
165
GENERAL LEBESGUE MEASURE OF LINEAR SETS
If G , consists of a denumerable set of intervals, while G , of a finite number, then for any e > 0, we can divide G, into two disjoint sets G I , and GI,, where GI, consist of a finite number of intervals while a(G,,) < e. Then a ( G , G,) - a ( G , , G,) 5 a(G,,) < e, since in passing from G,, G , to G , G, the increase is confined to the intervals and parts of intervals of GI,. Similarly, a(GIGB) a(G,,G,) I a(G,,) < e. Then
+
+
+
+
+
But since G I , and G, each consist of a finite numbers of intervals a ( G l , G,) a(G,, . G,) - a(G,,) - a(G,) = 0, so that a ( G , G,) a ( G , -G,) - (.(GI) a(G,)) 5 3e. Since this inequality holds for all e, we obtain a ( G , G,) a(GlG2) = a ( G J a(G,), for the case when G I consists of an infinite set of intervals, while G , consists of a finite set. A repetition of this reasoning leads at once to the validity of the formula when G, and G, are any two open subsets of X . For property (4), let us assume that the open set ZTIGhconsists of the open intervals (am, b,). Then for fixed m and for any e > 0, it is possible to find points am < a,' < b,' < b, such that a(am') a(a, 0 ) < e/2", and a(b,, - 0) - a(bTn')< e/2". Now the closed interval [a,', b,'] is covered by intervals from the sets G , , and by the Bore1 theorem a finite number of these I,, ... 17nk,will suffice. Then
+
+
+
+
+
I
1
+
+
+
+
ff
(bW1
-
0)
-
"(a,
+ 0) 5 a(b,')
-
a(a,')
+
+ e/2"-l
5 C k a ( I r k ) e/2m-1. Consequently,
since the Imkare only part of the intervals which occur in the various G,,. Since the inequality holds for all e, we have a(Z?,G,,)5 Z 7 1 a ( G T , ) . The properties of a ( G ) lead at once to:
166
V. CONrENT AND MEASURE
4.4. THEOREM. The upper measure a*(E) has the following prop-
erties : (1*) For all E :0 5 a*(E) 5 a ( X )
= a(+co)
- a(--co) ;a*(O) = 0.
(2*) If El I E,, then .*(El) I a*(E,). (3*) For any two sets El and E, .*(El)
+ a*(E,) h a*(E, + E J + a*(E, E.J.
(4*) For any sequence of sets { E J I )
a * ( C , l E J I5) C , l a * ( E , , ) . Properties (1*) and (2*) follow from the definition of a* and the corresponding properties of a (G) . For property (3*), to any e > 0, there exist sets G, L E, and G, I E,, such that a*(E,) I @ ( G I )- e, and a*(E,) h a(G,) - e. Then for every e > 0,
+
+
a*(E,) a * ( E 2 )h a(GJ f a ( G J - 2e = a ( G , G2) a(G, G,) - 2e 2 a*(E, E ) ff*(E,E,) - 2e,
+ since G , + G , h El + E,
+ ,+
and G,G, h E,E,. This leads at once to
(3*). For property (4*), to e > 0 and E,Lthere exist G, h E,, such that a*(E,J I a(G,) - e/2?’. Then for all e > 0: Z p * ( E , ) 2 C n a ( G , , ) - e 2 a ( Z n G , J - e 2 a*(Z,E,l)
- e,
from which (4*) follows at once. 5. Lower Measure
As in the case of content, we introduce a lower measure (or inner measure) a,(,??) by means of the equality: a ( X ) - a*(CE) = a(+ CO) - a ( - CO) - a*(CE). For a , ( E ) we have a set of properties analogous to those for a*(E) :
5.1. a,(E)
=
5.2. THEOREM. The lower measure a,(E) defined on all subsets
of X has the following properties: (0) a*(,??)1 a* ( E ) for all E.
5.
167
LOWER MEASURE
0 5 a,(E) 5 a ( X ) for all E.
(1,)
(2,) If El 5 E,, then a*(E,) 5 a*(E2).
+
+ +
(3,) For any E, and E?: " * ( E l ) at(E2) 5 a*(E, E,) "* (E,E,) * (4,) If the sets ET1are disjoint, then a* (ZnE,) 2 Z,a* (E,).
+
Property (0) follows from the inequality a ( X ) 5 a*(E) a*(CE), a consequence of (3*). Property (1,) is obvious. Property (2,) follows from the definition of a,(E), and the fact that El 5 E, implies CE, 2 CE,. Property (3,) follows from (3*). For if El and E , are any sets then for CE, and CE,, we have u*(CE,)
+ a*(CE,) 2 a*(CE, + CE,) + a*(CE,
*
CE,).
Consequently,
a(X)
-
a*(CE,)
+ a ( X ) - a*(CE,) (= a ( X ) - "*(CE, + CE,) + a ( X ) - a*(CE, . CE,).
The use of the definition of a,(E) then yields (3*). Property (4,) follows from (3*). For if E,, ..., E,, then
... are
disjoint,
Since this holds for all m, we have a* (L',E,) 2 Zna* ( E n ) . We call attention to the fact that (4*) is not deducible from (3*), but depends on the property (4) of a(G) for G open. It is possible to deduce some additional inequalities from (3*). If we rewrite (3*) in the form: "*(E,E,)
-
a*(E2) 5 a*(El) -
* (E, + E J
and add a ( X ) to both sides we obtain: (3**) a*(E,E,) a*(CE,) 5 (.*(El) a * (C (E 1 4- E,)) = "*( El) a,(CE, * CE,). Let E,' and E,' be any two sets and set E, = El' E,', CE, = E,'. Then C E, - CE, = 0, ElE2 = E,'CE,' and
+ +
+
+
a*(E,'CE,')
+ a,(E,')
5a*(E,'
+ E,').
168
V. CONTENT AND MEASURE
If we drop the primes, we can say that for any two sets El and E.,:
a*(E, - El)
+ a*(E,) 5 a * (E , + E J ,
If in addition E l and E, are disjoint, then this inequality becomes:
“*(El)
+ a*(EJ Ia*(El + EZ). +
On the other hand, if we set CE, = E,‘ E,‘ and E l = El‘ in (3**), then E,E, = 0, CE, - CE, = E,’CE1’ and
a,(E,’
+ E2‘) 5 a*(E,‘) + a*(E,‘CE,‘) 5 “*(El‘) + a*(E2’).
Consequently, if El and E., are disjoint, then these inequalities together with (3*) and (3,) give 5.3.
(€3,)
+ a* (E,) 5 a* (El + E J 5 &*(El)+ a* ( E J 5 a*(E, + E,) S “*(El) -t a*(EJ.
We note that only the first and third inequalities require that E l - E, = 0, the second and fourth are true for all E l and E,. 6. MeasurabiI ity
We now define: 6.1. Definition. A set E of X i s a-measurable if and only if a*(E) =
(E). We will denote the class of measurable subsets of X by !lR. The common value of a*(E) and a,(E) will be denoted by a(E). The condition a*(E) = a,(E) is equivalent to a*(E) a*(CE) = a ( X ) . This is symmetrical in E and CE since C(CE) = E. Hence:
ff*
+
6.2. If E is a-measurable, so is CE, the complement of E and a(E)
+
a(CE) = a ( X ) . Properties (1*) and (2*) of a*(E) lead to corresponding properties of or(E). Applying (3’) and (3,) to the case when E l and E, are a-measurable gives us
6.
169
MEASURABILITY
+
+
Or a* (EIE,) - a,*(E E ) = a*(E, E,) -a* ( E l E J . Then each ! 2 of these expressions is zero, since a* ( E ) - a* ( E ) I 0 for all E. It follows that El E, are measurable. Then:
+
6.3. If E , and E, are measurable so are El
a(EJ Consequently :
+ a(E2)
=
a(E,
+ E? and EIE, and
+ E,) + a(E,E,).
of u-measurable subsets of X is additive, multiplicative and complementary and consequently an algebra. The function a on Dl is additive. We note that:
6.4. THEOREM. The class
6.5. Definition. A set function
P on a class Q of sets is additive, if for
El and E, disjoint in Q , such that E ,
+
+ E, belongs to Q we
have
+
P(E, E,) = P(EJ P(E2). It follows that if /3 is additive on Q, and El and E, of Q are such that E2 5 El and E, - E, is in Q, then P(El) - /3(E,) = P(E, - E,). If Q is an algebra of sets, then P is additive on Q if and only if for all El P(EIE,) = P(EJ LYE,). and E, of Q we have P(E, E,) Properties (4*) and (4,) of a* and a* applied to a sequence {E,) of disjoint a-measurable sets gives
+
+ +
Cfia(EJ 5 a*(C?LEn)5
(En',)
5 Cna(E?i)*
Since the end terms of this series of inequalities are equal, it follows that the inequalities become equalities, that is ZnE,f is a-measurable and a (Z71E71) = Zr1a(E7,). Now the sum of any sequence of sets can be expressed as the sum of a denumerable number of disjoint sets by the formula Z N E ,= El ( E , -. E l ) ( E , - (El E2))+.... If the sets E , are a-measurable, then E, - El, E, - (El E,), ... are also a-measurable and disjoint. Then :
+
+
+
+
6.6. THEOREM. If (Ell)is any sequence of a measurable sets, then
ZnE,Lis measurable, or the class Dl is s-additive. In view of the complementary character of foz, the class Dl of a-measurable sets forms an s-algebra. Moreover the measure function a ( E ) on !IN, is s-additive. We define: 6.7. Definition. A set function /? on a class of sets Q in s-additive if for any sequence of disjoint sets E,, of Q such that C,E, belongs to Q, we have B(Z,E,) = Z,LB(Er,)*
170
V. CONTENT AND MEASURE
Since we assume here that p is finite valued on &, and any reordering of E, leads to the same set C,E,, it follows that if p is s-additive, then the series Z J ( E , ) is absolutely convergent. The terms completely additive, or totally additive are sometimes used in place of s-additive. Since K , E , and lh?lE,l are each expressible in terms of s-sums and s-products, it follows that : 6.8. If {EPL} is a sequence of a-measurable sets, then E r L E Iand I
lim E, are also a-measurable, and so is lim,,E,( when it exists.
-12
6.9. If E , is any monotonic nondecreasing sequence of a-measurable sets (E, 5 En+J, then Z,,EII= lim,,E, is a-measurable and a(lim,E,) = lim,a(E,). For
lim I1 E
/I
+ ( E , - E l ) + (E, - E,) + ...,
= E,
a sum of disjoint measurable sets. The s-additivity of a gives
( Y ( C , ~ E=~a(E,) )
+ a(E, - E,) + a(E, - E,) + ... = lim,,p(E,,).
Similarly : 6.10. If (En}is a monotonic nonincreasing sequence of a-measurable sets (En2 En+J, then n,E, = lim,E, is measurable and limna(E,) = afnrlElr)= aflim.E,). This follows from the identity nnE,= El - ((El - E,) (E, - E,) ...). The fact that a(E) is finite valued plays a role in this theorem. Assuming that the sets E, are a-measurable, consider
+
+
, 3 2 3 0
n
C E,.
E,, = m=l
Then
n=m
a(cEJ. co
a(& n
E,,) = lim
n=m
Then (e > 0, me,m 2 me): co
a ( G E,) h n
a(C Ell) - e 2 a(E,,,) n =m
-
e.
It follows that a ( f i n E n )1 L n a ( E J . So: 6.11. - If En is any sequence of a-measurable sets, then a(G,E,) 2 1im71a(E7L)'
7.
171
EXAMPLES OF MEASURABLE SETS
This theorem contains the Arztla lemma (11.15.8) as a special case, where Enconsists of a finite number of disjoint intervals and a(1) = / ( I ) . By taking complements with respect to X , or by similar reasoning it follows that: 6.12. If En are a-measurable sets, then a(lim,E,)
5 lim,a(E,J.
Combining these results gives : 6.13. T H E O R E M . If a ( x ) is a monotonic nondecreasing bounded
function on X , and the sets Ell are a-measurable and such that 1im?,El, exists, then a(limT,En) = limp(EJ. 7. Examples of Measurable Sets
The null set 0 and the basic interval X are each measurable since a*(O) =o. A set E consisting of a single point x is measurable and a(E) = ~ ( x 0) - a ( x - 0). For if E consists of x, then cr*(E) = a(x 0) - a ( x - 0). On the other hand, CE consists of the open intervals (- co,x ) and ( x , co), so that a*(CE) = a(x - 0) a ( - co) a(+ co) - a ( x 0). Since !IN is s-additive, it follows that:
+ +
+
+
+
7.1. If E consists of a denumerable set of points { x , } , then E is a-measurable, and a(E) = Z l t ( a ( x T I 0) - a ( x , - 0)). If E is an open interval (a, b ) = a < x < b, then E is a-measurable and a ( ( a , b ) ) = a(b - 0) - a(a 0). For a*(E) = cr(b - 0) a*(CE) = a ( a 0) - a( - co) a( 00) a(a 0) while
+
+
+ +
+ +
a(b - 0). Consequently, if G is any open set consisting of the open intervals (a,, b n ) ,then G is a-measurable and a(G)=Z',(a(b,-0) a(a, 0)); in other words, a ( G ) has the same value with which we started. If F is any closed set, then P is a-measurable, and a ( F ) = a ( X ) - a(G) where G is the set complementary to F relative to X . Moreover, any Borel measurable set E is a-measurable. We call attention to the fact that:
+
7.2. T H E O R E M . The class of a-measurable sets relative to any bounded monotonic function a on XE - co < x < co always includes the
+
s-algebra extension of the open subintervals.of X , that is, the class of Borel measurable subsets of X . In particular, every open subset G and every closed subset F of X is measurable relative to any bounded monotonic a.
172
V. CONTENT AND MEASURE
8. Sets of Z e r o Measure
If a*(E) = 0, then a* ( E ) = 0 also, so that E is a-measurable, and its a-measure is zero. A set of a-measure zero (or a-null set) can then be defined by the condition that for every e > 0, there exists a sequence of open intervals I , such that L'n171 2 E and L',,a(In)< e. Here a(Zp,)= a@,, - 0 ) - a(a,, 0 ) .
+
8.1. T H E O R E M . The class R of sets of a-measure zero is an s-ring.
For if E n are such that a(E,) = 0, then a(ZnEn) I ZrLa(En)= 0, so that ZnEnbelongs to R also. Also if E belongs to R and El I E, then a ( E l ) 5 a(E) = 0, so that El belongs to R also. Consequently, if a ( E ) = 0, and El is any set then a ( E - E l ) = 0, or E - El is in R. Then R is s-additive and subtractive, i.e., R is an s-ring. We note that we have shown that if E is of a-measure zero, then every subset of E is also of a-measure zero. From another angle, we can say, that if E is is in 9,and E , is in X , then E * El is in R. Following algebraic terminology, a class (3 of sets which is a ring and closed under multiplication by subsets of W can be called an ideal relative to W . The class R of sets of a-measure zero is an ideal relative to X . 8.2. T H E O R E M . If E , is a-measurable and E, differs from El by a
set of a-measure zero in the sense that E,dE, is of a-measure zero, then E, is also a-measurable and a(E,) = a(E,). For if a(E,dE,) = 0, then a*(E, - E,) = a*(E, - E l ) = 0 also. Then E, = (El (E, - E , ) ) - (El - E,) expresses E, as sum and differences of a-measurable sets and is consequently also a-measurable. The equality a ( E , ) = a(E,) follows immediately from the additive properties of a. EXERCISES 1. In 11.13.12 a set E on a finite interval [a, b] was defined to be of a-measure zero, if for any e > 0, there exists a sequence of intervals I, = [a,, b,l enclosing or covering E, such that Zna(In)= .Zn(a(bn) - a(a,,)) < e. Show that this definition yields the same class of sets of a-measure zero as the class R above. 2. Show that if for any pair of subsets E of X we define a distance function by the condition 6(E,, E,) = a*(EIAE,) = a*(EICE, E,CE,), then 6 satisfies the metric conditions: 6(El, E,) 2 0 for all El and E,; 6(E, E ) = 0 for all E ; 6(El, E,) = 8(E,, El) for all El and E,; and 6(El, E3) I 6(E,, E,) 6(E2,E,) for all El, E, and E,; but 6(El, E,) = 0 is equivalent to the statement: El and E2 differ by a set of a-measure zero.
+
+
+
10.
ADDITIONAL CONDITIONS FOR MEASURABILITY
173
9. Additional Properties of Upper and Lower Measure
By definition, a*(E) = g.l.b.[a(G) for G 1El, G being open. Consequently, there exists a sequence of open sets G , such that a(G,) 2 a*(E) 2 a(G,) - l/n. Then IInGT,2 E so that a(IITIGIJI a*(E) - a(G,) - l / n 2 a(DmGm) - l/n. It follows that for G , = II,G,, 2 we have a*(E) = a(G,). Then: 9.1. For every set E, there exists a G, set, the product of a denumerable
number of open sets, containing E such that a(G,) = a*(E). Since the complement of an open set is a closed set, and a closed set is a-measurable, we can revise the definition of a* (E) = a ( X ) a*(CE) to read as follows: For any set E the lower measure a,(E) = 1.u.b. [a(F) for F contained in El, where F stands for a closed set. This definition contrasts with that of upper and lower content, where both of these concepts were defined by using the same class of sets: a finite number of intervals. Following through as in the case of upper measure we have : 9.2. For every E, there exists an F, the sum of a denumerable number
of closed sets, such that F, is contained in E and a(Fa) = a,(E). 10. Additional Conditions for Measurability
We have defined E to be a-measurable if a*(E) = a,(E) = a ( X ) a*(CE), or a*(E) a*(CE) = a(X). Since we always have a*(E) a*(CE) I a ( X ) , for measurability, we need only prove that a*(E) a*(CE) 5 a ( X ) . We give in this paragraph additional necessary and sufficient conditions for measurability. From the preceding section we have: -
+
+
+
10.1. The set E is a-measurable if and only if there exists a G , and an Fa such that G , 2 E 2 F, and a(G, - Fa) = 0. For if E is a-measurable, there exist G, 1E and F, 5 E such that a(G,) = a(E) = a(F,), and so a(G, - F,) = 0. On the other hand, from a(G, - Fa) = 0 and G, 2 E 2 Fa it follows that a*(G, - E ) = 0 and a*(E - F,) = 0, so that Ediffers from the (Borel) measurable sets G, and F, by a set of a-measure zero, and is consequently a-measurable. We can strengthen this theorem as follows: 10.2. A necessary and sufficient condition that the set E be a-measurable is that for every e > 0, there exists an open set G , 2 E such that
1 74
V. CONTENT AND MEASURE
a*(G, - E) < e. Also E is a-measurable if and only if for all e > 0 there exists a closed set F, 5 E such that a*(E - F,) < e. We prove the first part only, the second part being proved in a similar way. If E is measurable, then for e>O, there exists G, such that a(G,) 2 a ( E ) > a(G,) - e, so that a*(G, - E ) = a(G, - E ) < e. For the sufficiency take e = 1In, and let G , be such that a*(G,, - 4) < l / n . If G, = nllG,, then a*(G, - E ) 5 a*(G,, - E ) < l / n for all n, so that a*(G, - E ) = 0. Then E agrees with the a-measurable set G, except for a set of a-measure zero, and is by V.8.2 a-measurable. 10.3. The set E is a-measurable if and only if for every e > 0 there exist open sets G , containing E and G2 containing CE such that a(G,G,) < e. This is essentially a reformulation of the preceding theorem. If E is measurable, then for e > 0 there exists open G, L E and closed F , 5 E such that a*(G, - E ) < e/2 and u*(E - F,) < e/2. Then a ( G , - F,) = a*(G, - F,) 5 a*(G, - E) a*(E - F,) < e. The open set G, = CF, will enclose CE and a(G, - %,) < e. Conversely, if for G , 2 E and G, 2 CE we have a(GIG,) < e, and if I; = CG, then F S E and a(G, - F ) < e. Then a*(G, - E ) < e and E is a-measurable by the preceding theorem.
+
10.4. A necessary and sufficient condition that a set M be a-measurable is that for every E, we have a*(E) = a*(ME) + a * ( E - ME). This condition is sufficient since for E = X we would have a * ( X ) = a ( X ) = a*(M) a*(CM), the definition of measurability. To prove the necessity we note that since for all E : a*(E) 5 a*(ME) a*(E - M E ) , it is only necessary to show that if M is a-measurable then a*(E) 2 a*(ME) + a * ( E - M E ) . Now
+
a*(E)
+
+ a ( M ) 2 a*(E + M ) + a*(ME) = a*(M + ( E - M E ) ) + a*(ME) 2 a ( M ) + u*(E - M E ) + a*(ME), +
+
where we use the inequality a*(El E,) 2 .*(El) a*(E,) from V.5.3 for E , and E, disjoint and take El = M and E, = E - ME. Since a ( M ) is finite, we have a*(E) 2 (**(ME) a*(E - M E ) . The equality a*(E) = a*(ME) + a*(E - M E ) virtually says that if we cross any set E by a measurable set M , then a* is additive on the portions of E : EM and E - EM. Then if M , and M , are disjoint measurable sets, we have a*((M, M J E ) = a*(M,E)-+ a*(M2E).
+
+
11.
THE FUNCTION a(X)
175
IS UNBOUNDED
This extends at once to a finite number of disjoint a-measurable sets. We even have: 10.5. If ( M A }is a sequence of disjoint a-measurable sets, then for any set E : a*(EZ,M,) = CI l a * ( E M , ) . This means that a* is s-additive if you cut across any set E by a sequence of disjoint a-measurable sets. By way of proof set M = Z,M,,. Then since ZyMn and M - ZTMn are a-measurable, m
m
a*(EM) = a*(E C M?,)4- a*E(M
-
1
1
Now limma*(E(M- T M , , ) ) 5 lIm,a(M
a*(EM)
=
a*(E C Mil)
=
n
m
MJ
= 1
- ZYM,) = 0.
Hence
2 .*(EM,). n
EXERCISES
+
1. Suppose El and E, are disjoint and such that a*(E,) a*(E,) = .*(El E J . Show that there exist disjoint a-measurable sets Ml and M , such that (El E,)Ml = El and (El E,)M, = E2. More generally, if En is a sequence of disjoint sets such that a*(Z,,E,,) = Zna*(En),then there exist disjoint measurable sets M,, such that (Z,,En)Mm= Em, for each m.
+
+
+
2. Prove that if M is a-measurable, then a* ( E ) = a* ( E M )
+ a* ( E - E M )
for all E. 3. Prove that if M n are disjoint measurable sets, then for any E : a*(EZnMn) = Zna,(EMn). 4. Is a theorem corresponding to exercise (1) true for lower measures
a* ( E ) ?
11. The Function a ( x ) I s Unbounded
Up to the present time, we have assumed that a ( X ) is finite, that is the monotone function a ( x ) giving rise to our interval function is bounded on - 00 < x < + 00. We now consider the changes in our theory of measure if the boundedness condition is dropped and we allow a ( X ) = 00. We assume that a(x) is finite valued on - 00 < x < co, but it may happen that limz+m a ( x ) = - co and/or
+
176
V. CONTENT AND MEASURE
+
lim,,,, a(x) = co. For any finite open interval I = (a, b ) we still have a ( I ) = a ( b - 0) - a ( a 0), but if I is (- co, a ) or (b, co) or (- co, a),then a ( I ) may be 00. Obviously if G is any open set on X , then co is a permissible value for a ( G ) .
+
+
+
+
+
11.1. The fundamental properties of a on open sets G are still valid,
+
to wit: (1) 0 5 a ( G ) 5 co; (2) if G, 5 G,, then a(G,) 5 a(G,); (3) a(Gl G,) a(GI *G,) = 4 G , ) a(G,); and (4) a ( q I G 7 , 5 ) C,,ff(G,). For (3) we note that if either a(G,) or a(G,) is co, then a(G1 G,) = co also. If both a(G,) and a(G,) are finite, then the former reasoning is valid. Similarly, if Zl,a(Gll)= co, there is nothing to prove; if L',ta(G,) < co, the reasoning for the case when a ( X ) < co applies. There is no change in the definition of upper measure, that is: a*(E) = g.l.b.[a(G) for G 2 El. Moreover, the properties (l*), (2*), (3*), and (4*) of V.4.4. continue to hold. The reasoning involved is the same as for a ( G ) above. The definition of a,(E) in terms of a ( X ) and a*(CE) is no longer usable. Consequently, the definition of measurability must be discarded also. We then have a choice of definitions of measurability which do not involve lower measure a* ( E ) or a ( X ) . We shall use the condition of V.10.4 as our basis of measurability:
+
+
+
+
+
+
+
+
11.2. Definition. A set M is a-measurable if for every set E of X ,
we have
CU*(E) = a*(EM)
+ a*(E - EM).
The objectionable feature in this definition of measurability is that it requires the verification of an equality for a large group of sets: all subsets of X , while measurability for a ( X ) < a3 requires validity of the equality only for the single set X . It is, however, possible to reduce the number of verifications involved. In the first place because of the inequality a*(E) 5 a*(EM) a*(E - M E ) , we need consider only those E for which a*(E) < co. Further, we have:
+
11.3. THEOREM. A set M is measurable if and only if for all open intervals I, = (- n, n), we have a ( I J = u*(I,M) a*(I, - I n M ) . If we let U,~(X)= a(x) for - n < x < n, a,,(x) = a ( - n 0) for x 5 - n and an(x) = a ( n - 0) for x 2 n, then a*(I,E) = aTL*(E) for all E, and the theorem asserts that M is measurable if and
+
+
11.
THE FUNCTION a(X)
177
IS UNBOUNDED
only if M is measurable relative to a,, for all n, that is relative to a restricted to the open intervals (- n, n). The condition of the theorem is obviously necessary. To prove the sufficiency, we demonstrate first the following : 11.4. Lemma. For any set E :
a*(E)
=
=
xrn Z,-,)E xrn(a*(Z,E) a*(Z?8-l)E)) a*(Zrn-
= limp*(Z7,E).
-
It is assumed that Z, = 0. If we use the functions a,,(x), then since are a,,-measurable we have by V.10.5: the sets I , n
a*(I,,E) = aIl*(E)= C a,l*(Z,,l m=l
?I
-
ZTn-,)E = C a*(ZTn - I,-,)E. m=l
Consequently, for all n
2 .*((I,
m=l
x Go
-
Z,-,)E) = a*(I,,E) 5 a*(E) 5
a*((Z,
-
Zrn-JE).
m=l
These inequalities lead at once to the lemma. If M satisfies the condition a(ZJ = a*(I,,M) a*(Z, - Z,M) for all n, then as already indicated, M is measurable relative to each C C , ~ . Consequently if E is any set then
+
+
a*(Z,,E) = a n * ( E ) = CY,~*(EM)all*(E - EM) =
a*(Z?,EM)
+ a*(Z,,(E- EM)).
The fact that lim,,a*(Z,E) = a*(E) for all E now gives a*(E) = a*(EM) a*(E - EM), or the a-measurability of M. Obviously the open intervals Zl1 can be replaced by any monotone sequence of open intervals (a,,, b,,) with a,l+,< a,, b,,, > b,, lim,,al, = - co, and limIlb,,= co.
+
+
11.5. As a result of this lemma and theorem, most of the properties of a-measurable sets can be extended to the case when a ( X ) = 4-00. In particular: (a) if E is measurable so is CE, follows from the definition of measurability, (b) if El and E, are measurable then E, E, and E,E2 are measurable and a(E, E,) a(E,E,) = a ( E J a(E,), follow at once from the theorem and the corresponding finite case; if Ell are a-measurable, then L',AE,l is measurable, follows
+
+
+
178
V. CONTENT AND MEASURE
Other theorems on a-measurability and a-measure based on these theorems are immediately available. In particular, since every open interval is a-measurable, every open, closed, or Bore1 measurable set is measurable relative to every monotonic a ( x ) . We can then define lower measure in terms of closed sets: 11.6. or,(E) = 1.u.b. [a(F) for all closed F 5 El. However, the condition a * (E) = a* (E) guarantees measurability of E only if a * (E)
E,
= E, -
( ( E l - EJ
n
+ ( E , - E3) + -.-)
m
As a consequence the additional assumption: there exists an m such that a(E,) < co is indicated. That this assumption is essential follows from the instance: a ( x ) = x, E , = [x > n ] , for which L',E,l = 0, but a(E?,)= co for all n. Similarly the theorem: if E,, are measurable, then a ( K n E ? J2 & a ( E J requires the additional 4 limf(En) condition: for some m : (Z,"=,E,,)
+
12.
LEBESGUE MEASURE
179
11.7. THEOREM. If En is a sequence of measurable sets such that lim,Ell exists, and if for some m : a(Z,"=,En) < co, then lim,cr(E,) = a (limnE,,). By slight changes, the theory presented in this section can be adapted to the case when X is an open interval (a, b) and a is finite valued monotone but unbounded on (a, b).
EXERClSES 1. What properties of lower measure a,(E), defined as the 1.u.b. of a ( F ) for all closed F 5 E, are valid if a ( X ) = co? 2. Show that the condition a*(E)
3. Show that even when a ( X ) =co, the following three conditions are each necessary and sufficient for a-measurability of a set E :
(A) There exist G6 and Fa such that Gd2 E 2 Fa and a(Ga - F,) = 0. (B) For every e > 0, there exists an open set G , 2 E such that a*(G, E) < e.
(C)For every e > 0, there exist open sets GI 2 E and G, 2 CE such that "(GIG,) < e. 4. Show that if M , is a sequence of a-measurable disjoint sets and E is any set then a*(Z,M,E) = Zna*(M,E), even if a ( X ) = 03. 12. Lebesgue Measure
For the case when a ( x ) = x, the value of a ( I ) reduces to the length of the interval I and the resulting measure function and measurable sets are then generalizations of the length or measure of intervals and their measurability, The process by which we have deduced a measure function from an interval function defined by differences of a monotonic point function is an adaptation of that used by Lebesgue for the special case when a(1) = l ( Z ) . We shall consequently call a set of points E on an interval Lebesgue measurable, or L-measurable or simply measurable if it is measurable relative to a ( x ) = x. We obtain Lebesgue measurable subsets of a finite interval [a, b ] if we set a(x) = a for x S a ; a(x) = x for a 5 x 5 b, and a ( x ) = b for x 2 b.
180
V. CONTENT AND MEASURE
If p * ( E ) and p u , ( E ) are the upper and lower L-measure of E on [a, b ] then cont E 5 p.,( E ) 5 p*(E) 5 5 E. Consequently, if a set E has content it is also L-measurable and cont E = p ( E ) . We recall that any L-measurable set differs from a Borel measurable set by a set of zero measure, in particular, that for any measurable set E, there exist G , and F, such that G , 2 E L F, and p(G, - E ) = 0 and p ( E - F,) = 0. Since the class 9 of sets of measure zero has the property that if E belongs to 9, then for every E, 5 E, E , also belongs to 9,the class of sets of measure zero consists of the totality of subsets of G, sets in 9. The totality of L-measurable sets can be obtained by adjoining to the class of Borel measurable sets any set which is the sum of a G, and a subset of 9. Since in Lebesgue integration many situations involve the statement “ except for a set of measure zero,” the Borel measurable sets are important. The definition of measurability on a finite interval depends on the equality of upper and lower measure. This suggests the question : do there exist sets which are not measurable, where the upper and lower measure are unequal. For the case of L-measurability, this question is answered in the affirmative, provided we allow ourselves the validity of the well ordering theorem for points on a linear interval (or the axiom of choice). We note that the measure function in L-measure has an invariant property: if the interval I is translated, the length function I(I) remains unchanged, or if E x represents the set obtained from E by adding x to each point of E, then 1(1 x) = l(1). Since upper and lower measure depend on interval lengths, it follows that p * ( E x) = p * ( E ) and ,uu,(E x) = pu,(E) for all E and x. If we think of the set of real numbers as a group under addition, then Lebesgue measure is an instance of measure invariant under this group. We give the construction due to G. Vitali of sets which are not Lebesgue measurable (see F. Hausdorff: “Grundzuege der Mengenlehre,” Veit, Leipzig, 1914, pp. 401-2). The invariance of measure under translation plays an important role. The points of the interval 0 5 x < 1 are divided into a denumerable number of disjoint sets. To obtain the basic set E,, assume that the points 0 S x < 1 are well ordered. Retain 0 in E, and delete all rational points from the well ordered set. Let x, be the first remaining point and add it to E,. Delete all points of the form x, r, where r is any rational. Add x2,the first remaining point to E,, and delete all points of the form x2 Y. Continuing in this way, we obtain a set E,, such that if x’and x” belong to
+
+
+
+
+
+
12.
181
LEBESGUE MEASURE
+ +
E,, then x’- x” is not rational. We define E , = E, r n (mod l), where r,L is the nth rational in some denumeration of the rationals with 0 < r,f < 1, and x r,, - 1 belongs to E,Lif x r n > 1. If we think of the interval 0 I x < 1 wound on the circle of radius 1/2 n, then E,, will be obtained from E, by translation by a length Y,, on this circle. Consequently, p*E,, = p*E,and p * E , = ,uu,E,. Moreover, ZrLE?, = [0 S x < 11. Consequently, 1 = p*(C,EJ 5 ZJL*E,,, so that p*E, > 0. On the other hand, since the EILare disjoint 1 = ,u*(ZnEIL) 2 Zltp*E,,.Since ,u*E,, are all equal it follows that p * E I L= 0. Then the sets EILare not measurable. The problem of existence of nonmeasurable sets for a general monotonic nondecreasing function u(x) is somewhat more complicated. As a matter of fact, there exist monotone functions relative to which every set is measurable. We take u(x) as a monotone bounded pure break function on [0, 11, namely:
+
+
where a(y 0) - a ( y - 0) 2 0 and zero except for a denumerable set of points xIl, at which a(x) is discontinuous. It is assumed that
2
OS:r$l
(a(x + 0 ) - u(x
-
0 ) ) < co.
Then every subset of [0, 11 is measurable and if {x,,} are the points of discontinuity of a , then a*(E)
=
a,(E) =
C rn inE
+
(~(x,, 0 )
- u(X,, - 0 ) ) .
For if E contains no points of discontinuiy of a , then a ( E ) = 0. To see this, for e > 0 we select m so that Zz==, (a(x,, 0 ) a(x,,- 0)) < e. Then G, the open set which is complementary to the set of points consisting of (x,... x,,~-,) will enclose E and a ( G ) = YJ; (a(xIL 0 ) - a(x,,- 0 ) ) < e, so that a ( E ) = 0. Since any sequence of points is a-measurable, it follows that all subsets of [0, 11 are a-measurable. Essentially, the Vitali example shows that there does not exist a s-additive measure function on X = [0, I], such that p(X) = 1, which is invariant under translation. S. Banach (“Theorie des Operations Lineaires,” Warsaw, 1932, p. 32; reprint, Chelsea, New York, 1955, p. 32) has shown that there exists a function ,rc defined for all
+
+
182
V. CONTENT A N D MEASURE
subsets E of the circumference X of the circle of radius 1/2 n,which satisfies the conditions: (1) p ( E ) 2 0 for all E ; (2) if E , and E, are disjoint then p ( E , E,) = p(E,) p(E,) ; (3) p ( E ) is invariant under translation of E on the circle; (4) p ( X ) = 1; that is, a finitely additive measure function invariant under translation. The same result is not valid on a spherical surface. (See F. Hausdorff: " Grundzuege der Mengenlehre," Veit, Leipzig, 1914, p. 469.)
+
+
EXERCISES I. Show that there exists a monotone decreasing sequence of sets En of [0, 13 such that n,E, = 0, but lim,p*(E,) # 0, where p* is upper Le-
besgue measure. 2. Suppose Eq is a collection of sets defined for q on the directed set Q .
Define
i&
Eq = 4
c Eq and lim Eq c -
9,
PR4,
4
=
9,
PR9,
Eq ,
Show that lGqEqand @,E, may be nonmeasurable, even though Eq are measurable for all q. If, however, Q has sequential character, then measurability of E4 for all q implies measurability of G q E qand h , E , . 13. Relation between a-Measure and Lebesgue Measure
+
If a ( x ) is a monotonic nondecreasing function on - co < x < co, then a maps the X-axis on the portion of the ,Y-axis between a ( - a) and a(+ co), if we agree that to any point of discontinuity xo of a, there corresponds the closed interval [ a ( x o 0), a ( x o - 0)] on the Y-axis, while an interval of constancy of a will map into a single point on the Y-axis. Open intervals on the X-axis will map into a point, or an interval (open, half-open, or closed) on the Y-axis, so that an open set on X will map into a sequence of nonoverlapping intervals together with a sequence of separate points on the Y-axis. We denote by a-'(y) the inverse map to a ( x ) . It is of the same character as a ( x ) . We assert:
+
13.1. THEOREM. If E, is any set on the X-axis, and E , the map of Ex on the Y-axis induced by the monotonic function a ( x ) , then a*(E,) = ,u*(E,), where p*(E,) is the upper Lebesgue measure of E,. Suppose that a*(E,) < co. Then for any e > 0, let G,, be an open set such that a*(Ex) 2 a(G,) - e. Under a ( x ) , G, maps into G,, consisting of a sequence of nonoverlapping intervals and points on
14.
183
RELATIONS BETWEEN a-MEASURABLE SETS
the Y-axis, containing E,. If I , denotes the nth interval or point of this sequence, then for the same e > 0, we can extend I , at the ends by an amount at most e/2"+' to an interval I,' so that Z,)(In) 2 Zl1,l(I1,') - e, and the open set Z,LI,l'2 L'J, = G,. Consequently, a*(E,) 2 a(G,) - e = ,ZJ(Irr) - e 2 L',,l(I,') - 2e L p*(E,) - 2e, for all e. Then a*(E,) 2 ,u*(E,). On the other hand, let G, be an open set containing E , such that ,u(G,) 5 p*(E,) e. Under the mapping by the inverse function a-l(y), G, will be mapped into a sequence of nonoverlapping intervals and separate points : G, containing E. We discard from any separate point or end point of an interval which is not in E. The end points of intervals of G, belonging to G, and E will be a denumerable set. Let b,, be such a right-hand end point of the nth interval I , of G,. Then extend I,, to a point of continuity b,+ d of a so that a(b,,+ d - 0 ) -u(b,, 0) = a ( b , d) a(b,, 0 ) < e/2"+'. For a left-hand end point we proceed similarly. If x,,is the nth separate point belonging to Gz and E, we find points of continuity of a : x, d', x,,- d", to the right and left of x,, such that (a(xn d') - a ( x , - d " ) ) - ( a ( x , 0) - a ( x , - 0 ) ) < e/2". The sum of the open intervals so obtained will be equivalent to an open set G, which contains G, and such that a(G,) - 2e 5 a (G,). Consequently,
+
c,
+
+
+
+
,uu*(Ey) 2 p(G,) - e
= a@,)
+
+
- e 2 a(G,)
-
3e 2 a*(E) - 3e,
for all e > 0. Then ,u*(Ey)2 a*(E,) and so ,u*(E,) = a*(E,). If a*(E,) = co, and if I,,, = (- n, n ) , then
-
a*(E,) = lim,,a*(E, I,lx)= lirn',,u*(E,(a(n- 0), a( - n
+0))
= p'E,.
Then for all E, we have a*(E,) = ,u*(E,). This leads to: 13.2. THEOREM. A set E, is a-measurable if and only if its map E, via a ( x ) on the Y-axis is Lebesgue measurable and u(E,) = ,u(E,).
Note that an arbitrary set E, on the Y-axis may be not-l-measurable and still have its map on the X-axis via a-' measurable relative to a. This is shown by the case where u ( x ) is a pure break function relative to which every set Ex is a-measurable. 14. Relations between Classes of a-Measurable Sets
The class N ! of a-measurable sets depends on the particular function a(x) on which measurability is based. As we have already noted, every
184
V. CONTENT AND MEASURE
such class !INu contains the class of Bore1 measurable sets, the s-algebra extension of the class of open subintervals of X . We consider relations between the classes !INnn depending on relations between the corresponding u. Let a,(x), a,(x) be two monotone nondecreasing functions on X such that for every finite interval [a, b ] : a,(b) - .,(a) 2 a,@) .,(a). Then we also have for all (a, b ) : a,(b - 0 ) - u,(a 0) 2 u,(b - 0) - a,(a 0). Consequently, for any set E and open interval I , = (- n, n ) we have al*(I,,E) 2 a,*(I,E), and so a,*(E) = limpI*(I,,E) 2 lim~,a,*(I~,E) = cc,*(E). From this we derive:
+
+
14.1. T H E O R E M . If a,(x) and a,(x) are monotonic nondecreasing on X , and if for every finite interval (a, b ) : a,(b) - a,(a) 2 a,(b) a z ( a ) , than any a,-measurable set is also a,-measurable. For if E is a,-measurable, there exists for every I,, = (- n, n) and e > 0, open sets GI2 I,/Eand G , 2 I,, - I;E such that a,(G,G,) < e. Consequently, a,(G,G,) 5 e also, so that by V.10.3 E is a,,-measurable on I,, and so on X . If we order monotone functions by the condition that a l ( 2 ) a z if and only if for every finite interval I = (a, b ) : a,(b) - .,(a) 2 a,(b) - a,(a), this will induce the same ordering on the corresponding upper measure functions. If we consider the corresponding classes of a-measurable sets !INa, then a l ( 2 ) a 2implies !INu1 5 !INa2. 14.2. T H E O R E M . If a,(x) and a,(x) are monotonic nondecreasing
on X , then E is measurable relative to a,+ 01, if and only if E is both a,-measurable and a,-measurable, or !INal+u2 = !INul *!INa2. Since a I a , (2) a 1 and 0 1 , a , ( 2 )a,, it follows from the preceding theorem that if E is measurable relative to 01, a , it is also a,-measurable and a,-measurable. For the converse we note that for any E : (a1 a,)* ( E ) = a,*(E) + a,*(E). For if G is any open set, then ( a I a,) (G) = a,(G) + a,(G). Further from the definition of upper measure it follows that a*(E) = limGZEa (G), where the class of open sets G 2 E is a directed set ordered by inclusion. Then by the additive property of directed limits:
+
+
+
+ +
(a,
+ a,)*(E) = lim ( a , + a,)(G) GZE
=
a,*(E)
= lim GZE
+ a,*(E).
a ,(G)
+ lim a,(G) G2E
Applying the condition for measurability to this equality gives
US
14.
185
RELATIONS BETWEEN M MEASURABLE SETS
the result desired: if E is measurable relative to a 1 and a2, then it is measurable relative to a , a2. In this connection it is desirable to consider briefly measurability relative to functions of bounded variation. We shall assume that the functions considered are of bounded variation on every finite interval (B.V.F.I.) of X = - co < x < CQ. Then V ( x ) = da determines a monotone point function on X . It is customary to define:
+
+
Ji I
I
14.3. Definition. A set E is measurable relative to an a of B.V.F.I.
if and only if it is measurable relative to the monotone function V ( a ) . If we define P(x) = $
(1: I da I + a ( x )
and
N(x)
=
4
(1' I da I
-
~(x)),
then P ( x ) and N ( x ) will both be monotonic nondecreasing on X with V(x) = P(x) N ( x ) and a ( x ) = P ( x ) - N ( x ) . Moreover, V ( I ) = P(I) N ( I ) where I = [a, b ] and V ( I ) = J: da It follows that a necessary and sufficient condition that E be measurable relative to the function a ( x ) of B.V.F.I. is that E be measurable relative to both P ( x ) and N ( x ) . The class of functions of bounded variation on every finite interval is linear. We have
+
+
I I.
14.4. If a l ( x ) and a 2 ( x ) are in B.V.F.I. and E is measurable relative
+
to both a1 and a2 then it is also measurable relative to c,a, c2a2, where c, and c2 are any two constants. For if E is measurable relative to a, then it is also measurable relative to ca for any c. From the inequality.
it follows that if E is measurable relative to a , and a , it is also measurable relative to al a2. The converse does not need to hold. For instance, let a by a monotone pure break function on [a, b ] , and set a , ( x ) = a ( x ) x and a 2 ( x ) = x , then any subset E of [a, b ] is measurable relative to a , ( x ) a 2 ( x ) , but need not be measurable relative to a2(x).
+
+
14.5. If C X , ~ ( X )is any sequence of monotone nondecreasing functions on X , and if a ( x ) is such that for every E of X : lim.a*,(E) = a*(E), then any set E measurable relative to all aT,is also measurable relative to a , or ma >= 17,1"a,.
186
V. CONTENT AND MEASURE
For if M is measurable relative to a,,, then for all E a: (E) = a: (EM) f a,* (E - EM).
+
Taking limits as to n yields a*(E) = a*(EM) a*(E - EM) for all E, the measurability of M relative to a. A stronger condition under which the conclusion of the preceding theorem is valid is: 14.6. If a n ( x ) and a ( x ) are monotone nondecreasing functions on X such that lim, JLz d ( a , - a ) = 0, then for any set E we have
I
1
lim,an*(E) = a*(E) and so any set measurable relative to all is also measurable relative to a. For for any interval I = [a, b ] we have
I a n ( U - f f f l ) 15 ID 1 for any open set I an(G) - a(G)
- a)
cyl,
I.
Then 15 J, I d(u, - a ) 1, and by taking limitsas to GZE,(a:(E)-a*(E) l S J x l d ( a n - a ) I. On the other hand, in case a 7 , ( x ) and u ( x ) are bounded on X we can state: 14.7. If a n ( x ) and a ( x ) are bounded on X and such that limrla,t(G8) = a(G,) for all G, subsets of X , products of denumerable open sets, then lim,a:(E) = ar*(E) for all E and so any set measurable relative to all a,1 is also measurable relative to a. For if E,( is any subset of X,then a:(E) and a*(E) will be finite and so there exists for each n an open set G,, 2 E such that
a,,(G,) -l/n 5 a,*(E) 5 a,,(G,) and a(G,)-l/n
S u*(E) 5 a(G,).
The sets G, can be taken to be the same for a7,and a , since if GI, serves for a,, and G,, for a, then G,, * G2,1will serve for both. Let G, = I7,G,. Then for all n : an(G,) - l/n 5 a:(E) 5 av,(G,), and and cr(G,) - l / n 5 a*(E) 5 a(G,). Then a(G,) = a*(E) and lim,,a: (E)
= lirnp,/(G,) = a(G,) = u*(E).
The question arises what conditions does the hypothesis limrla,l(G,) a(G,) for every G, impose on a n and a. For any G, there exists a sequence of open sets G, monotonic nonincreasing in rn, such that lim G, = G,. Then lim,an(G,) = an(G,) for each n. Since any open T? set is also a G,, it follows that the condition lim,a,,(G,) = a(G,) =
14.
RELATIONS BETWEEN a-MEASURABLE SETS
187
for all G is equivalent to limnlim,an(Gm) = limmlimnan(Gm)for all monotone decreasing sequences G,, all limits indicated existing. Since the double sequence a,(G,,) is monotone in m for each n, it follows from the converse 1.7.8 of the iterated limits theorem 1.7.4 that lim7,a,(Gm) = an(IlmGm)uniformly in n and limna,(Gm) = a(G,) uniformly in m. By the iterated limits theorem either of these conditions is also sufficient to guarantee that lim,,a7L(G,)=a(G,) for all G,. Then: 14.8. If a,(x) and a ( x ) are bounded monotonic functions on
X,
then a necessary and sufficient condition that lim,a:(E) = a*(E) for all E of X is that for any monotone decreasing sequence of open sets G, either lim7pn(Gm)= a(G,) uniformly in m,or limman(Gm)= a7,(IlrnGm)uniformly in n. We note that the condition lim,an(Gm) = all(I17,.Gm)uniformly in n for any decreasing sequence of open sets carries with it the condition lirn7,an(Z~IJ= 0 uniformly in n for any sequence Ik of disjoint open intervals. This condition in addition to the condition that lim,apl(I) = a(1) for any open I is sufficient to ensure that lim,a,(G) = a(G) for all G. In these theorems, there is nothing assumed explicitly concerning the convergence of a,(x) to a ( x ) , since upper measures depend on difference. We could then assume, for instance, that the a,(x) are normalized in such a way that for some point of continuity xo of all arrand a, lim,apl(xo) = a(x,). Then from the condition lim,p,(I) = a ( I ) applied to open intervals xo< x < x, 0 and x, - 0 < x < x,, it follows that lim7pn(x, 0) = a(x, 0) and lim,a,(x, - 0) = a(x, - 0) for all x,.
+
+
+
EXERCISES 1. Which of the theorems on the measurability of E relative to a(x) on the basis of its measurability relative to the each of the sequence of functions a,(x) are valid if the sequence a , is replaced by a set of functions a,(x) with q on a directed set Q? 2. Show that if for a sequence of monotone functions a,(x) on a finite closed interval [a, b], and a(x) on [a, b], we have lim,a,(x) = a ( x ) , lim,a,(x 0) = a(x + 0), and limna,(x - 0) = a ( x - 0) for all x, then a , converges to a uniformly on [a, b]. Is the same theorem true if the sequence a , is replaced by a set ap,q on directed Q?
+
188
V. CONTENT AND MEASURE
15. Measurability of Sets in Euclidean Space of Higher Dimension
A theory of measurable sets and measure can be formulated for higher dimensions along the lines carried through in the one-dimensional case in V.4-V.10. An examination of the procedure shows that the theorems for the one-dimensional case are based on the definition of an a-measure for open sets G satisfying the following conditions:
(1) For all G we have a(G) 2 0; (2) G, 5 G, implies a(G,) 2 a(G,) ; (3) a(G, G,) a(G1 G,) = a(G,) a(GJ for all G, and G,; (4) a(L'?,GJ 5 Z,p(G,) for every sequence of open sets G,.
+
+
-
+
We shall indicate how functions on open sets having these four properties can be defined for the two-dimensional space: - co < x, y < co. The theory for n-dimensions: - co < x,, x,, ..., x, < 00 follows through at once by proper change of terminology. The main difficulty in this program for higher dimensions is that of finding a suitable definition of a-measure for an open set G, which in two or higher dimensions does not have the simple structure it does in one dimension. We make the following observation. If for the monotone function a(x) we define a ( I ) for the interval [a, b ] as a(b) - a(a), then for the open set G, a(G) is the 1.u.b. Z,,a(In) for all finite collections of (closed) nonoverlapping intervals I,, ..., Ik such that Z J , L is contained in G. In particular, if I,,is the open interval a < x < b, then a ( I J = a(b - 0) - a(a 0). In two dimensions we start with a function of two variables P(x, y ) which satisfies the monotoneity condition of Vitali,
+
+
+
d*dyP= P(x+ Ax, y
+ 4)-P(x + tgx, y ) -Pfx, Y + AY) +P(x, Y ) 2 0
for all (x,y ) and Ax 2 0, dy 2 0, and in addition the condition that P(0, y ) = P(x, 0) = 0 for all x and y. Such a function gives rise to a positive function of rectangles I = [a, b ; c, d ] = [a 5 x 5 b ; cI y 5 dl, namely, P(I) = P(b, d) - P(b, c) - P(a, d) P(a, b). In terms of this function of intervals we can define a function of interval sets.
+
15.1. Definition. An interval set J will be assumed to consists of a finite number of intervals Il ... I,t having at most edges or parts of edges in. common. Then:
15.
MEASURABILITY OF SETS IN EUCLIDEAN SPACE
15.2. The class
189
5 of
J is finitely additive and finitely multiplicative. To see this, we observe that every interval I is equivalent to the interval set obtained by any subdivision of I into subintervals. If then for the interval sets J , and J,, we cut across any interval of J , by lines which are extensions of sides of intervals of J,, and any interval of J , by lines which are extensions of sides of rectangles of J , , then J , j , will consist of the resulting finite set of nonoverlapping intervals, each of which is a subinterval of J , or J,, while J , J , consists of those subintervals which belong to both J , and J,. We now extend the interval function P(1) to interval sets J by the condition that if J = I , ... Z,,, then P(J) = ZkP(Ik).If I , ... I,! is a subdivision of the interval I, then because of the additive character of P(I) it follows that P(I) = .T#(Ik). Further, if J , and J , are equivalent in the sense that there exists a set of subdivisions of the intervals constituting J , and one of the intervals of J , which yield the same collection of intervals, then the additivity of P(I) assures us that P ( J J = P(J,).
+
-
15.3. As a function of interval sets J ,
P has the following properties:
(1) P(J) 2 0 for all J.
(2) J , 5 J, implies P(J,) 5 P(J,). (3)
P(J,
+ J,) + P(J,J,)
=
P(JJ
+ P(J,).
Property (1) is obvious. For (2), we note that J , 5 J , means that every interval of J , is a subinterval of some interval of J,, and use the monotone character of ,B. For (3) we follow out the above construction leading to interval sets equivalent to J,, J,, J,+J,, J,J,. If we express p(J,), p(J,), P(J, J,), and P(J, * J,) in terms of this particular group of interval sets, the identity P(J,) P ( J J = P(J, J,) P(J,J,) emerges since p(J,J,) covers the terms which appear in both N J , ) and PtJJ. In order to define a function of open sets, we notice that any open set G can be expressed as the sum of a monotone sequence of interval sets J,, (with J , 2 J,-,). This leads to the definition:
+
+
+ +
15.4. a(G) = 1.u.b. [a(J) for all J S GI.
If we think of the interval sets J 5 G ordered by inclusion, then these interval sets J f G are a directed set, and we can also define: 15.5.
w(G) = limJsG P ( J ) since P(J) is monotone in J .
190
V. CONTENT AND MEASURE
We show that a(G) has the four properties (l), (2), (3), (4) listed at the beginning of this section. Properties (1) and (2) are immediate consequences of the corresponding properties of p ( J ) . For property (3) we note that if either a(G,) or a(G,) = 00, then a ( G , G,) = co and then (3) is valid. Assume then that both a(G,) and a(G,) are finite. Then for e > 0 there exist interval sets J1 5 G, and J , 5 G,, such that a(G,) 5 p(J,) e, and a(G,) = ,J(J,) e, so that
+
+
+
a(G,)+a(G,) 5 B(J1)+B(J,)+2e 5 a(G,+G,)+a(G,
+
-
+
B(J,+J,)+B(J, G2)+2e.
=
-
*
J2)+2e
+
since J , J , 5 G , G, and J , J , S G , * G,. Then cr(Gl) a ( G J 5 a ( G , G,) a(G, * G,). On the other hand, for any e > 0 there exist interval sets J' and J" such that J' 5 G , G, and J" 5 G , G,, e. Here we and a ( G , G,) 5 B(J') e and a(Gl G,) 5 /3(J") can assume that J' 2 J" since otherwise we need only replace J' by J' J". Unfortunately, the intervals in J' will usually not belong entirely either to G , or to G,. Consider an arbitrary interval I of J ' . Then since I is contained in G , G,, for each point P of I, including the boundary there will exist an interval Ip lying entirely in G , or G,. Since I is a closed bounded set, a finite number of these intervals Ip will cover I and each of them will lie either in G , or G,. The sides of this finite number of intervals extended across I will determine a subdivision of I, such that each interval I' of this subdivision cr will be interior either to G I or G,, and such that p ( I ) = zIdp(I'). This procedure can be extended to all intervals in J'. In this way we replace the interval set J' by an equivalent interval set J,' such that each interval of J,' is interior either to G , or G,. It follows that J1' can be expressed as the sum of two interval sets J ( l )and 5('),where J ( l )is the set of intervals of J,' lying in G , and J(') the set of intervals of J,' lying in G,, so that J' = J ( l ) J(') and J ( l ) J(') L J". Consequently,
+
+
+
+
+
-
+
+
+
-
+
+ G,) + a ( G ,
o(G, d p(J(')
5 u(G,)
+
+
+
G,) 5 B(J') B(J") 2e J ( ' ) ) + /?(J(') J ( * ) ) 2e = /3(J(')) P(J(')) *
+ a(G,) + 2e.
-
+
+
+
+
-
+ 2e
+
We can conclude that a(G, G,) a(Gl G,) = a(G,) a(G,). It follows that if G , ... G , is any finite collection of open sets, then o (L':G,) 5 L': o (GJ .
16.
SOME ABSTRACT MEASURE THEORY
191
The procedure for property (4) is much as in the one-dimensional case. Suppose a(Zy G,) < co. Consider for e > 0 any J set such that J 5 Z,G, and a(Z,G,) 5 P ( J ) e . Now J is a bounded closed set of points covered by the denumerable set of open sets {G,). Consequently, by the Borel theorem a finite number of the G,: Gn1,G, .. . G f l kwill suffice to cover J. Consequently,
+
Then a(L',G,) 5 Z,p(G,). If a(ZV1Gfl) = co, then for every N there exists a J5 ZflGrlsuch that ,B(J) 2 N . Applying the Borel theorem again, we have N 5 P(J) 5 Z,a(G,) for all N , which implies that ZJ~(G,) = co. Consequently the inequality of (4) always holds. If X is the total set - 00 < x, y < co, then X is a G and consequently a ( X ) is defined. It is equal to limrl,B(Ifl)where I,,= [- n, n ; - n, n]. On the basis of the function a (G) defined on open sets G we can now proceed as in the one-dimensional case, define an upper measure a*(E) =g.l.b. [a(G) for all G 2 El, and develop a theory of a-measurability, so far as this theory uses the properties (I), (2), (3), and (4) of cr(G). We note that in general for a closed interval I : a ( I ) will not agree with P(I) but be larger. This is also true in the one-dimensional case since the same procedure applied to the function a ( I ) = a(b) - a ( a ) gives a(b 0) - a(a - 0) as the value for a ( T ) where 7 = [a 5 x 5 b].
+
+
EXERCISE Show that if Z is any open interval = ( a < x < b ; c < y < d) where may be infinite, then Z is measurable relative to the a function on open sets defined above from the monotone function P(X, y ) .
a, b, c, d
16. Some Abstract Measure Theory
The preceding constructions of measure functions and measurable sets suggest some lines along which an abstract theory of measure can be developed, the advantage of which would be to establish certain theorems on the basis of few postulates, so as to avoid proofs for individual cases. We outline such an abstract theory, with the usual
192
V. CONTENT AND MEASURE
reservation that it is valid in a particular case only if the case conforms to the general situation in both postulates and definitions. The measure theories developed for the one and two dimensional Euclidean spaces, center in an " upper measure " function, which is defined and positive valued for all subsets of a fundamental set. We consequently assume a basic set X together with the class (3 of all of its subsets E. We then assume: 16.1. Postulate (A). The function p * ( E ) is defined for all E of (3 with 0 5 p * ( E ) I co. We select from the class 6 a subclass Im of sets M which satisfy the " measurability " condition :
+
16.2. 132 = [class of sets M such that for all E :
p*(E)
=
p*(EM)
+ p*(E
-
M)
=
p*(EM)
+ p*(ECM) 1.
We note that the class W may be vacuous or consist of all of (3 as in the case when p * ( E ) = 0 or co for all E. We proceed to determine properties of the class Im. In case M belongs to Im, we shall write P * ( M ) = AM). If we assume that roZ contains at least one set M , and put E = 0 in the measurability condition, then
+
p*(O) = p*(O * M )
-+ p*(O
*
C M ) = p*(O)
+ !yo).
Then p*(O) =O or +a.If we put E = M in the same condition, then p*(M)
=
p*(M * M )
+ p*(M
*
C M ) = p*(M)
+
If p * ( O ) = 00, then for every Moftm: p * ( M ) sensible to add the additional postulate:
= +a.It
16.3. Postulate (B). If E is the null set 0, then p * ( O ) As an immediate consequence we have :
then seems
= 0.
thenIm contains at least the null set 0. For then p * ( E ) = p*(E 0 ) + p*(E X) = 0 p*(E) for all E. Further because of the relation C ( C M ) = M , the measurability condition is symmetric in M and CM, so that: 16.4. If p * ( O )
= 0,
+~'(0).
-
-
+
W has the complementary property, i.e., with M it also contains CM. Consequently, if p*(Oj = 0, then 0 and X both belong to Im. 16.5. The class
16.
193
SOME ABSTRACT MEASURE THEORY
+
Suppose next that M , and M , belong to !lX, and consider M I M,. By applying the measurability condition to M I and M,, we find:
+ M,) + p*EC(M, + M,) p*(E(M, + M 2 ) M I ) + P*(E(M, + M2)CMl) + P*(ECM,CM,) = p * ( E M , ) + p * ( E M 2 C M , )+ p*(ECM,CM,) p*E(M,
= p*(EM,)
=
+ p*(ECM,)
=
p*(E).
Consequently :
+
16.6. If M , and M , are in !Dl, then M , M , is also in %TI, and because of the complementary property M I - M., is also in Im. It follows that the class !Dl i s an algebra of sets. Moreover, if M , and M , are in !lX, then for any E
p*(E(M,
+ M,)) i-p * ( E M , M 2 )
= P*(E(M,
+ M,)M,)
f p*(E(M, -k M,)CM, -I-p*(EM,M,) =
p*(EM,) -t p*(EM,CM,)
=
p*(EM,)
+ .u*(EM,).
If in particular we take E
+
+
=
+ p*(EM,M,)
,'A then:
+
16.7. p ( M , M,) p(M,M,) = p ( M J p ( M , ) . As a consequence, the function p on the class Im is additive. If M , ... M , are disjoint, then p ( 2 T M k ) = Z ~ , u ( M , ) .Further, we note that if M I and M , are in Im, and if M , 2 M,, then M , - M , is in !Dl also and p+(EM,) 2 p*(EM,) for all E. If further p*(EM,) < fa,then p*(EM, - EM,) = p*(EM,) - p*(EM,). These results may seem somewhat surprising until one realizes that the measurability condition is rather strong so that the class 9J is limited, and may if p * ( O ) = 0, consist only of 0 and X . In order to obtain the s-additive property for the class !Dl, the function p* must be restricted further. The following upper semiadditive condition is suggested by the properties of upper measure: 16.8. Postulate (C). For all sequences of sets E,, we have ,u*(Z7,E,) 5 .5',P* (E,).
Since this also holds for a finite number of sets, we obtain at once the monotoneity property of p* on &, namely: 16.8.1. If E I S E,, then p * ( E , ) 5 p * ( E , ) , and if p * ( E , ) @*(El- E,) 2 p*(E,) - /d*(E,).
<
+ a,then
194
V. CONTENT AND MEASURE
We note in passing that postulate (C) is equivalent to the condition : If E 5 Z,E,, then p*(E) 5 Z,p*(E,,). To show that the class W has the s-additive property, we derive first two lemmas:
16.8.2. (C‘).
16.9. Lemma 1. If { M 1 l }is a sequence of disjoint subsets of W ,and E is any set, then p*(EZ,M,) = 2?,p*(EM,,). Note that we do not assume that ZnM,, is in W . Since E ZYM,, 5 EL’YM,, we have
2 p*(EM,)
oc
m
=
1
cc
P * ( E C1 M7,) 5 C p*(EM,).
p * ( E C M,) 5
I
1
Since this holds for all m,it follows that p*(E 2: M , ) = 2: p*(EM,). 16.10. Lemma II. If M , and M are in W ,M,, are disjoint, and Z1,M,,5
+
M , and if p*(EM) < 00, then p*(E(M - Z T M J ) = p*(EM) 2: P*(EM,,).. We have Z:p*(EM,) = p*(.ZYEM,,) 5 p*(EM). Since p * ( E M ) < 00, it follows that Z,p*(EM,) converges. Then
+
m
IU*(E(M -
2 M , ) ) 5 P*(E(M
-
1
2 M,,)) 1
= /u*(EM) -
2 P*(E,,) 1
for all m. Consequently,
On the other hand, by Lemma I
Then p * ( E ( M - 2: M , ) ) = p * ( E M ) - L’Tp*(EM,,). As an immediate corollary we have:
W and such that M , 2 M , 2 ... L M,, L ..., then for all E such that p*(EM,) < 00, we have
16.10.1. If {M,) are in
p * ( E n Mil) n
= p*(Elim n
MI,)= lim p*(EM,,). n
16.
SOME ABSTRACT MEASURE THEORY
195
For n , M , = M , - (2,"( M , - M n + J ) with M , in tm and M,l M , + , disjoint in !Dl since Im is an algebra. Obviously the condition p*(EM,) < co can be replaced by the condition there exists an m such that p*(EM,)
+
m
since the M , are disjoint and so CM , 2 M , for n 2 2. But since ,uU*(ECM)5 ,&(E) < co, Lemma I1 applies and m
p*(E(CM, - C M , ) )
m
=
P*(ECM,)
?
-
C P*(EMJ. ?
Then P*(E(2 M,))
+ P*(EC&
M,J)= P*(EM,) +P*(ECM,) = P*(E),
1
1
so that Z,"MP1is in Im. To show that ZnM,, is still in tm even when M , are not disjoint, we use the fact that tm is subtractive to express ZnM,, as the sum of a sequence of disjoint sets of tm, namely Z n M , = M , (M, - MI) ( M , - (MI M,)) The identity p(,ZnMn)= ZII,u(M,I) for M,, disjoint follows from Lemma I by setting E = X . We summarize our results in the theorem:
+
+
+
+
a*.*
16.12. THEOREM. If p*(E) is a function on the subsets of a set X , which satisfies the conditions: (A) 0 5 p*(E) 5 co for all E ; (B) ,u*(U) = 0; and (C) p*(Z,E,,) 5 C,lp*(E,l), then the class of subsets M of X for which ,u*(E) = ,u*(EM) p*(ECM) for all E is an s-algebra of sets, and the function p is s-additive on tm.
+
196
V. CONTENT AND MEASURE
EXERCISE Show that if the function p*(E) on the subsets of X satisfies the conditions (A), (B) and (C), then any set N such that p * ( N ) = 0 belongs to !Ill, the class 9 of sets N for which p*(N) = 0 is an s-ring which has the property that if for some N , the set E is contained in N , then E is in I also; if the set M is in tm and E is such that MAE = MCE ECM is in, 9,then E is in tm also and p ( E ) = p ( M ) .
+
17. Upper Semiadditive Functions
The assumption of the upper semiadditivity property for the function ,u*(E)on all subsets E of X , suggests the problem of determining such a function. In the study of measurability in the one-dimensional case, this property was induced by the values of and properties of the function a ( G ) on open sets, and these in turn .depended on the value of a(G) on open intervals. In terms of a(G) we defined a*(E) for any set as the greatest lower bound of a(G) for G 2 E. However, we would have obtained the same value for a*(E) if we had defined a*(E) as the greatest lower bound of Z,a(I,) for all sequences ( I , } of open intervals such that ZnIncontains or covers E.Guided by these remarks, we assume a class B of subsets G of X , which contains the null set 0, and is such that for any E of X , there exists a sequence of sets {GPL} such that ZnG,, 2 E. Further, we assume given the function a ( G ) on B to 0 5 y S 00, such that a ( 0 ) = 0. We define then
+
,u*(E)= g.1.b. [Zna(GPL) for all (G,} such that ZnG,, 2 El. We could drop the condition that for every E there exists {G,} such that ZnG, 2 E,if we define p*(E) = co,for all sets E for which no such sequence exists. Then: 17.1.
+
17.2. THEOREM. The function p * ( E ) satisfies the conditions (A): 0 5 p*(E) 5 + a ; (B): p * ( O ) = 0; and (C): if E 5 Z n E , , then
P*(E)5 q L P * ( E n ) . That (A) and (B) hold is obvious. As for (C) we need only consider the case when Z,,u*(E,) < co, so that ,u*(E,) are all finite. Then for e > 0, and n, there exists a sequence {Grim} in B, such that ZmG,, 2 E,, and p*(E,) 2 Zma(Gn7,J- e/2". Then
e 2 p*(E) -
17.
UPPER SEMIADDITIVE FUNCTIONS
197
since CnZ, G,, 2 E. Since this holds for all e, property (C) holds for p * ( E ) . it follows that the function p*(E) defines a class W which is an s-algebra of sets. There are at least two difficulties with this construction: (a) it is not certain that for any set G of @, we have a(G) = p*(G); and (b) the sets G may not belong to the class W of measurable sets. With respect to (a) we note that for any G p*(G) 5 a(G). That the inequality is possible is illustrated by points on the linear interval [0, 11, where (3 is the class of all subsets of [0, 11 and a(G) = K t E. In this case p*(E) is the upper Lebesgue measure of E. For a denumerable dense set on [0, 11 we would have a(G) = 1, but p*(E) =O. By using the definition of p*(G), we see that a necessary and sufficient condition that p*(G) = w(G) for all G of @, is that 01 satisfy the condition : 17.3. For all sequences {G9(}and G of (3, such that Z,,G?{h G, we
have Z7,a(G,) 2 a(G). Such a condition might be called cover-additive. i n case @ is s-additive, it is equivalent to upper semiadditivity. To ensure that any set G of (3 belongs to the class W determined by ,u*(E) it would be necessary that for all GI of @, we have p*(G,) = p*(G,G) p*(G,CG). if (Y is multiplicative and subtractive and p*(G) = a(G) for all G, this condition becomes a(G,) = a(G,G) a(G,CG), that is any G is a-measurable relative to @. Sufficient conditions to cover this situation are (1) @ is a ring of sets and (2) 01 is (finitely) additive on @. We can then prove:
+
+
is finitely additive on @, then any set G of @ is in the class 2X determined by p*(E). We need to show that for any E of Q and G of @, we have p*(E) = p*(EG) p*(ECG). In view of the upper semiadditivity property of p* we can limit ourselves to the case when p*(E) < co. Then for e > 0 there exists {G,,}such that L',G,, 2 E, and p*(E) 2 Z J X ( G ~-~e.) Now a(G,) = (w(G,,G) + a(G,CG) for all n. Then for all e > 0 17.4. If @ is a ring, and
01
+
+
P*(E) 2 C p ( G , , ) - e 2 C , [ 4 G 7 , G ) a(G,CG)I - e 2 p*(EG) p*(ECG) - e.
+
Since for all E : ,u*(E) 5 pU*(EG)+ p*(ECG), it follows that equality holds.
198
V. CONTENT AND MEASURE
We note that if @ is a ring of sets and a is additive on @, then the cover-additive condition: for any sequence {G,} and G of 8 such that G 5 ,Z,G,, we have a ( G ) d ZClla(G,) is equivalent to s-additivity: is in @, then Zna(Cn) = a(2,GJ. if G, are disjoint and C,G,, For from the additive and cover additivity condition on a, we have for all rn:
and so a(Z:Gn) = Z F a ( c n ) . On the other hand, if {Gll}is a sequence of sets of (3 with G 5 ZnG,,, and if we set GTl= G ( G , - 2,”-’G J , then G, are disjoint in @ and G = 2$n, so that
We can summarize: 17.5. THEOREM. If @ is a ring of subsets of X , and u(G) on (3 to 0 5 y 5 00 is s-additive on @, then there exists an upper measure p * ( E ) on Cf to 0 5 y 5 a,which is upper semiadditive, and such that the class tllz of measurable sets M is an s-algebra, which contains @ with p ( G ) = a(G) for all G. The class ! l l is then an s-algebra extension (?f preserving measure. More detailed expositions of abstract measure theory can be found, for instance, in:
+
+
C. CARATHEODORY, ‘‘ Reelle Funktionen,” Teubner, Leipzig, 1918, Chapter V. “ Mass und Integral und ihre Algebraisierung,” Birkhaeuser, Basel, C. CARATHEODORY, 1956 Chapter V. H. HAHNand A. ROSENTHAL, “Set Functions,” Univ. of New Mexico Press, Albuquerque, 1948, Chapter 11. P. HALMOS, “Measure Theory,” Van Nostrand, New York, 1950, Chapters I1 and 111. J. v. NEUMANN, “ Functional Operators,” Princeton Univ. Press, Princeton, New Jersey, 1960, Chapters 11-IV.
CHAPTER VI
MEASURABLE FUNCTlONS
1. Semicontinuous Functions
Before discussing measurable functions, we make a slight detour and treat briefly semicontinuous functions and some of their properties. We shall limit ourselves to functions f(x) on X = [a, b ] to finite reals, although many results carry over to metric spaces. The continuity of f(x) at a point x, requires or
( e > 0, dexo,I x
-
x, I < dexJ : I f(x)
-
f(x,,)I < e,
+
f(x,,)- e <.f(x) <.f(x,,) e*
I
+
If only the inequality: f(x) < f(x,) e holds for x - x, I < dero, then .f(x) is said to be upper semicontinuous at x,,, if f(x,,)- e <.f(x), then f(x) is lower semicontinuous at x,. Now the statement ( e > 0, dexo,I x - x, I < d e x o ): ,f(x) < f(x,) e, is equivalent to limx+xof(x) 5 f(x,), Then alternatively we can say :
+
1.1. Definition. The function f(x) is upper semicontinuous at xo if and only if lim,+,o.f(x) 2 f(x,,) and ,f(x) is lower semicontinuous at x,,,if and only if limFrof(x) Zf(x,,). If f(x) is both upper and lower semicontinuousat a point, then it is continuous. A function is upper (lower) semicontinuous on an interval [a, b ] if it is upper (lower) semicontinuous at every point of the interval. A function all of whose discontinuities are of the first kind (breaks) 0) and ,f(x - 0) for all x. is upper semicontinuous if ,f(x) 2 f(x In particular, f(x) on [0, 11 with f(x) = 0 for x irrational and f(x) = l / q for x = p / q , p and q relatively prime, is upper semicontinuous on [0, 13. A monotone nondecreasing function on [a, b ] is upper semicontinuous if f(x + 0) =,f(x) for all x, that is .f(x) is continuous on the right. The negative of an upper semicontinuous function is lower semi-
+
199
200
VI. MEASURABLE FUNCTIONS
continuous since = f ( x ) = - lim (- f ( x ) ) . The sum of two upper semicontinuous functions is again upper semicontinuous since 6 ( f , ( x ) f , ( x ) ) 5 lim f , ( x ) f , ( x ) . The space of upper semicontinuous functions is positively linear. A function upper semicontinuous on a closed interval [a,b ] can also be characterized as follows :
+
+
1.2. THEOREM. A function f ( x ) is upper semicontinuous on [a, b ] if and only if for every constant c, the set E of x for which f ( x ) 2 c, is closed. We denote by E [ f ( x ) 2 c ] the set x for which f ( x ) 2 c. Let f ( x ) be upper semicontinuous on [a, b ] and let x , be a limiting point of E [ f ( x ) 2 c]. Let { x , , } in E [ f ( x ) 2 c] be such that lim,Lx,= x,). Then c 5 & , , f ( x , ) 5 6 x - + r , , f ( S,f(x,,), ~) so that x , is in E [f ( x ) 2 c]. Conversely suppose that for all c: E [ f ( x ) 2 c] is closed. Let x(, by any point of [a, b ] and the sequence {x,} such that lim,x,, = x,, and f ( x ) . Now for each m, the set E [f ( x ) 2 - lim,, f ( x , , ) = Gwx0 limz-ro f ( x ) - l / m ] is closed, and contains the points x,, for n large enough. Hence it contains x,, so that f ( x , , ) 1E T + r o , f ( x )- 1 /m for all m, and so independently of m. The interesting thing about this characterisation is that it makes the property of semicontinuity on an interval depend on the fact that the set of x’s for which f ( x ) 2 c belongs to a certain class of sets for all c. By taking complements, we can also characterize upper semicontinuous functions on an interval by the condition that E [ f ( x ) < c] is open for all c. For a lower semicontinuous function E [ f ( x ) 5 c] is closed and E [ f ( x ) > c] is open for all c. Since a continuous function is both upper and lower semicontinuous, it follows that f ( x ) is continuous on [a,61 if and only if the sets E [ f ( x ) 2 c] and E [f ( x ) 5 c] are closed for all c or the sets E [f ( x ) > c] and E [ f ( x ) < c] are open for all c. Since the product of two open sets is open and a continuous function is bounded on a closed interval, the latter condition is equivalent to the statement that for all c1 and c,, the set E [c,
1.
20 I
SEMICONTINUOUS FUNCTIONS
dense set c,, on the real line.” For for any c not in {c,,}, the set E [ f ( x ) 2 c ] will be the product of the closed sets F,, = E [ f ( x ) 2 c,,] for c,, < c, and will consequently be closed also. A similar type of reasoning leads to:
f(x) is finite valued and upper (lower) semicontinuous on [a, b ] then f ( x ) is bounded above (below) and assumes it least upper bound (greatest lower bound). If P,, = E [ f ( x ) 2 n ] , then the product Zl,,F,, is vacuous. Consequently there exists an m such that for n > m, F, is vacuous, and so f(x) is bounded above. If B = 1.u.b. [f(x) for x on [a, b ] ] ,then E [ f ( x ) > B ] is vacuous but F,, = E [ , f ( x )2 B - l / n ] contains elements x for each n so that E [ f ( x ) I B ] = IZ,,F,, is not vacuous, and B is the maximum value of J f x ) on [a,b ] . It follows from this theorem that a sufficient condition that a function f ( x ) have a maximum on [a, b ] is that f ( x ) be upper semicontinuous on [a, b ] . 1.3. If
1.4. If f , , ( x ) is a monotone nonincreasing sequence of upper semicontinuous functions, then .f(x) = lim,,f,,(x) is also upper semicontinuous. For all c, the set E [ f ( x ) 2 c ] = ZI,,E[f,(x) 2 c ] . For if f ( x ) 2 c, then f , ( x ) 2 f ( x ) 2 c. And if f , , ( x ) 2 c for all n, then f ( x ) = lim, f , , ( x ) 2 c. Then E [ , f ( x ) 2 c] as a product of closed sets is also closed. Similarly a monotone nondecreasing sequence of lower semicontinuous functions converges to a lower semicontinuous function. As a consequence : 1.5. A nonincreasing sequence of continuous functions converges to an upper semicontinuous function and a nondecreasing sequence of continuous functions to a lower semicontinuous function. The last statement is reversible in the form :
f(x) is a finite valued upper semicontinuous function on [a, b ] , then there exist nonincreasing sequences of continuous functions f , , ( x ) 2 f v l + l ( x2) f ( x ) such that lim,f,(x) = f ( x ) for all x. For any function , f ( x ) bounded above we can define the functions 1.6. If
f , , ( x ) = 1.u.b. [f(x,) - n I x - x, I for x , on [a, b l l . Obviously f , ( x ) Z f , + , ( x ) 2 f ( x ) for all x and n. Further if x and x’ are two points of [a, b ] , then for all x1
f,,W 2 f f X J
Ix
+ I x’- x, I)
x, I 2 f ( x , ) - n(l x - I =f(x,) - I? Ix‘- x, I - n Ix - x‘ 1 . -
-
202
VI. MEASURABLE FUNCTIONS
1
tinuity of f ( x ) at x gives (e > 0, d,,, x‘ - x < d,,): < d,,, then f(x) e. If for n > n, we have x , , ~ - x
+
1
f(x‘) 5
EXERCISES 1. For any bounded function f(x) there are defined the functions
M(f, x)
=
g.1.b. [l.u.b. (f(y) for y in I,) for all I, containing x]
= the larger of
m(f, x)
lim f(x‘),
x’+x
and f(x);
k.1.b. (f(y) for y in I,) for all I, containing x] smaller of lim - f(x’) and f(x);
= 1.u.b. = the
X ‘ *
Wff;X )
=,
g.1.b.
[w(A I,) for I, containing XI, where w(fi I) is the oscil-
2.
203
MEASURABLE FUNCTIONS
lation off on the interval I. Show that M ( f , x ) and w ( f , x ) are upper semicontinuous functions while m(f, x ) is lower semicontinuous. 2. Let f ( x ) be bounded on [0, 11. For x = m/2", m = 0, 1, ..., 2" - 1 set f n ( x )= the 1.u.b. of f ( x ) on (m - 1 ) / 2 " 6 x 5 (m 1)/2", and linear between m/2" and (m 1)/2". Show that f n ( x ) hfn+l(x) 2 f ( x ) for all n and x and if f ( x ) is upper semicontinuouson [0, 11 then limnf n ( x ) = f ( x ) for all x .
+
+
2. Measurable Functions
The closing lines of the preceding paragraph suggest a procedure for defining " measurable " functions in terms of " measurable " sets. Since the basic properties of such functions depend only on those of the class of measurable sets and not the measure function defining such sets, we proceed abstractly, with the observation that the developments apply in the one and two-dimensional cases considered in the preceding chapter. We assume a basic set X and Q the class of all subsets E of X. Further we postulate a class W of measurable sets M , which is an s-algebra of sets, that is it has the s-additive, s-multiplicative, subtractive and complementary properties. Real valued functions f ( x ) will be assumed to be either finite valued, i.e. f is on X to Y = (- co < x < co) or in many cases co and - 03 will also be allowed as values (called the extended real line). We define:
+
+
+
X to Y = (- 00 4 y 5 00) is measurable relative to the classW if for every y of Y, the set E [ . f ( x ) h y ] is in W or measurable. This definition applies for instance to the case where X = (- co < x < a),a ( x ) is monotone nondecreasing on X , and Im is the class of a-measurable sets. If W is the class of Borel measurable subsets of (- 00, co), we shall call f ( x ) Borel measurable. For convenience, we shall in the sequel omit the x in E [f(x) L y1 and assume that E [ f h y ] means the set of x such that f(x) 2 y , with corresponding interpretations of similar symbols. We note that the following sets could also have been used for the definition of measurable functions : 2.1. Definition. The function f ( x ) on
+
or the three variants of (4) if 5 is replaced by <. We have:
204
VI. MEASURABLE FUNCTIONS
2.2. THEOREM. The conditions: for ally: (0) E [ f l y ] ,( 1 ) E [ f < y ] ,
(2) E [ f 5 y ] , and (3) E [ f < y ] belong to Im are all equivalent so that each could be used for a definition of a measurable function. If f(x) is finite valued the condition (4) : E[y‘ 5 f5y”] for all y’ 5 y” is also equivalent to (0) for all y . We show that (0) implies (1) implies (2) implies (3) implies (0), where (0) means: for all y : E [ f 2 y ] belongs to W ,and similarly for (11, (21, and (3).
(0) implies (1) since E [ f < y ] = C ( E [ f z y ] ) ,and W is closed under complementation. (1) implies (2) since E [ f 5 y ] = n , , E [ f < y s-multiplicative property.
+ l / n ] and Im has the
(2) implies (3) since E[f> y ] = C ( E [ f S y ] ) . (3) implies (0) since E [ f Z y ] = 17,,E[f>y
-
l/n].
To relate condition (4) to the others, we note that E[y’ 5f5y”] = E [ f 2 y ’ ] E [ f 5 y ” ] , so that iff is measurable then the sets E[y’ 5 f5y”] or their variants are in ?1J1 for all y’, y”. Further iff is finite valued, then E [ f Z y ] = Z,E[y 5f2n ] , so that f is measurable if (4) is in Im for all y’ and y”. If f i s not restricted to be finite valued, then f ( x ) will be measurable if and only if the set E[y’ 5 , f 5y”] is in W for all - co < y’ 5 y“ < co, and either E[f= co] or E [ f = - co] is in W.For E [ y 5 f < co] will belong to W and E [f 2 y ] = E[f= 001 E [ y 5 f < a ] . Since E[f= y ] = E [ f Z y ] . E [ f 5 y ] it follows that:
+ +
+
+
2.3. If f(x) is measurable relative to W ,then the sets E [ . f ( x ) = y ] are in 93 or measurable for every y . For the measurability of a function f ( x ) it is not necessary to verify that the sets E [ f z y ] are in W for all y . We have: 2.4. If [ y , ]is a denumerable dense set on Y (for instance, the rational
numbers), then f ( x ) is measurable if and only if the sets E [ f 2 y,,] ate in Im for all n. If y is not in the sequence { y , } , then E [ f 2 y ] = n n E [ f > = V,] for the y,, > y. Consequently, E [ f z y ] is in W for all y . We note in passing that any point function on X to Y determines the sets EY = E [ f Z y ] , a function on Y = (- co 2 y 5 00) to
+
3.
205
PROPERTIES OF MEASURABLE FUNCTIONS
the subsets of X. This collection of sets has the following properties: (1) if y' 2 y" then E v , S Ev.; (2) E, = U y , ( E yfor , y'< y ) ; (3) E-, = X. If f ( x ) is finite valued then Zy(Ey for y > - co) = X and E,, = II,(E, for y < co) = 0. The set E [ f > y ] is given by EY=Zy,(Ey, for y' > y ) , and consequently E [ f = y ] = E, - E,.
+
2.5. The procedure is reversible. If we have a function EY on (- co 5 y 5 + co) to subsets of X satisfying the conditions (I), (2) and (3), then E, gives rise to the point function defined: f ( x ) = y if x belongs to the set E, - EY= E , - Zy,(Ey,, for y' > y ) , and E, = E [f ( x ) 2 y ] . If we had started with a point function and the sets E, defined as E [ f > y ] or E [ f S y ] or E[f< y ] , we get variants of the properties (I), (2) and (3) which in turn define point functions. If the sets E, satisfying (l), (2), and (3) belong to D then the corresponding point function is measurable relative to D. 3. Properties of Measurable Functions
The following theorems hold when the functions involved are on a set X to reals, and measurability is with respect to an s-algebra of subsets of X . 3.1. I f f is measurable so is
-
For E [ - f 2 y ] = E [ f S
-
f. y].
3.2. Iffis measurable and c is any constant, then cfis also measurable.
If c > 0, then E[cf 2 y ] = E[f2 y / c ] and if c < 0, then E[cf 2 y ] [ f S y/c].
=E
3.3. Iff is measurable, so is
If
If
I.
For if y < 0, then E [ I 2 y ] = X , and if y 2 0, then E [If E [ f 2 yl E [ f S - yl.
+
Ip
I2y]
1-
(f for p > 0. For if y < 0, then E [ 2 y ] = X , if y 4 0, then E [I f 2y]= E [ f 2 y l l p ] E [ f S - y l / p ] .If p is an odd integer, then a slightly altered proof yields that f p is measurable iff is.
3.4. Iffis measurable, so is
+
If I p
Ip
3.5. Iff is measurable and f f 0 on X , then I/fis measurable. If y > 0, then E[l/f2 y ] = E[O
w/f2y ] = E[l/f> 01 + E[l/f= 01 + E[O > l / f 2 y l E[f> 01 + E[f + co] + E [ f = co] + E [ - co c , f S l/y]; and if y = 0, =
=
-
206
VI. MEASURABLE FUNCTIONS
+
+ +
then E[llfh 01 = E [ + co >f> 01 E[f= co] E[f= - co], all of which are measurable sets i f f is measurable. The condition f# 0 can be dropped if we define l!O = co.
+
3.6. Iff, and fzare measurable, then .fi V fz,the greater of f, and A fz,the lesser off, and fzare each measurable. For E [ f , V fz2 y ] = E[f,L y ] E [ f z2 y l and E [ f , A f i 5 y1 = E [ f , 5 y ] * E [ f z5 y ] . Consequently, the class of measurable functions forms a lattice under the ordering f , ( S ) f z = f , ( x ) 5 . f Z ( x ) for all x o f X . Ifwe takef, = f and fz= 0, thenfV 0 =f'andfr\ 0 = -fare both measurable iff is. Here ,f+ is the positive part off and may be defined: f + ( x ) =f(x) if f(x) 2 0 and f+(x) = 0 if f(x) S O ; while f- is the negative part o f f , namely f-(x) = 0 if f(x) h 0, and f-(x) = - f(x) if f(x) 5 0. Then f(x) = f +(x) - f -(x) and I.f(x) = f + ( x ) +f-(x) for all x.
fz,andf,
3.7.
+
1 If fi and fzare measurable and finite valued, so is fl + fz.
+
We note that if for a given x: fl(x) fz(x) > y , orf,(x) > y f z ( x ) , then for any rational r between f,(x) and y - f z ( x ) we have f,(x) > r and r > y - fz(x) or f z ( x ) > y - r. As a consequence E [ f , fz> y ] 5 Zr,E[f, > r ] E [ f 2> y - r ] , where r runs over the rationals, that is a denumerable set. On the other hand, if for some r f,(x) > r and fz(x) > y - r, then fl(x) fz(x) > y . Consequently, E [ f , f f 2 > vl = +W, > rl - E[fz> r - ~ 1and , so E [ f , .fz > yl is measurable if .f,and fzare.
+
+
3.8. Iff, and fzare finite valued and measurable, then f,
+
- fzis meas-
urable, that is the class of finite valued measurable functions is multiplicative, and forms a ring. For f, fz= & [(f, f z ) 2- (f,- . f J 2 ] , where in view of the preceding theorems, the right hand side is a measurable function if f,and fzare. An alternative proof can be made by applying the method of the preceding theorem to the case whenf,(x) 2 0 andfz(x)20 for all x, leading to the general case by expressing f , and fiin terms of their positive and negative parts.
+
3.9. Iffr(x) is a sequence of measurable functions, then 1.u.b.rL.fn(x)= g ( x ) and g.1.b. J,(x) = h ( x ) are each measurable. For E [ g ( x ) > y ] = E[l.u.b..f,(x) > yl = 2 $ [ f n ( X ) > Y l and E[h(x) < y ] = E[g.l.b.,f,(x) < y ] = n n E [ f i l ( x )c y ] , and the s-additivity and s-multiplicativity of !Ul gives the desired result.
4.
EXAMPLES OF MEASURABLE FUNCTIONS
207
3.10. If each of the functions f,(x) is measurable, then =,f,(x) and @,f,(x) are each measurable. If lim,f,(x) =f(x) exists for each x, thenf(x) is measurable. = l.u.b., For lim,f,(x) = g.1.b. m(l.u.b. n 8 m f,(x)) and W,f,(x) (g.l.b.n2mf,(x)), and the preceding theorem applies. If we limit ourselves to finite valued functions, then the sequence f,(x) must be bounded for each x. The proofs of the last theorems involve the fact that we ate dealing with sequences of functions. It is then to be expected that corresponding results are not always valid for sets of functionsfJx) where q is on a directed set Q. For instance if X is the interval [0, 11, the class !N is the class of Lebesgue measurable subsets of X , Q is the class of elements q consisting of finite subsets of a nonmeasurable set E of X , ordered by inclusion, and fJx) = x(q, x), the characteristic function of q (0 for x not in q and 1 for x in q), then lim,f,(x) = 1 on E and 0 on CE, and is not measurable although all f,(x) are. We summarize the principal results of the preceding theorems in the statement: 3.11. THEOREM. The class of finite valued measurable functions is
a linear lattice, multiplicative and closed under pointwise sequential convergence. 4. Examples of Measurable Functions
Suppose that X is the linear closed interval [a, b ] , and f is on X to Y : - co < y < co. Let !N be the class of measurable subsets of X determined by the monotone nondecreasing function a on X to Y. Then
+
4.1. Every continuous function on X to Y is a-measurable.
Iff(x) is continuous then for all y , the set E [ f l y ] is closed and consequently belongs to the class N ! determined by any monotone nondecreasing a. 4.2. Any function on X to Y having at most a denumerable number
of discontinuities is a-measurable. For suppose E [ f l y ] is not closed for a particular y . Then there will exist a limiting point x, of E such thatf(x,) < y. Then lim,,Jf(x) #f(xo), since for a sequence x , such that lim,,x, = x, andf(x,) 2 y , any value approached byf(xn) would be greater than or equal to y .
208
VI. MEASURABLE FUNCTIONS
So that x, is a point of discontinuity off. It follows that the set E [f 2 y ] differs from its closure E E' by a denumerable set of points. Now a denumerable set is a-measurable for every a. Then E[f I y ] as the difference of a closed set and a denumerable set of points, that is of two a-measurable sets, is a-measurable. As a corollary we have:
+
4.3. Any function of bounded variation on X is a-measurable relative
to any monotonic nondecreasing function ~(x). 4.4. Any upper semicontinuous function on X to Y is a-measurable,
and the same thing is true for lower semicontinuous functions. For i f f is upper semicontinuous, then the set E[f 2 y ] is' closed for all y , and so measurable. We could also use the fact that any-upper semi-continuous function is the limit of a monotone nondecreasing sequence of continuous functions, that is of a sequence of a-measurable functions. As a matter of fact, any function which is the limit of a sequence of continuous functions on X will be a-measurable. Such functions studied by Baire, are called Baire functions of the first class. Since the Baire functions of higher classes are limits of sequences of Baire funcions of the lower classes, it follows that all Baire functions are measurable, actually Borel measurable, since the continuous functions which initiate the series are Borel measurable. 4.5. Definition. A function f(x) on a set X will be called a step function on X if {E,} is a sequence of disjoint subsets of Xsuch that Z,E,= X , and f(x) is constant on E,,, that is f(x) = y , for x on E,. If the sequence E , consists of a finite number: of sets, f(x) will said to be a jnite-step function.
tm is an s-algebra of subsets of X , and M , is a sequence of disjoint subsets belonging to tm such that Z n M , = X , then any step function constant on each M , is measurable relative to tm. For the set E [f 2 y ] will consist of the sets M , for which the corresponding values o f f , that is the y,, are 2 y , and so will be in W . The strength of this theorem is in the fact that: 4.6. If
4.7. Any finite valued measurable function is the uniform limit of a
sequence of measurable step functions. For suppose that the function f(x) is measurable. Then the sets E??!?, = E[(m - I)/n
5.
209
PROPERTIES OF MEASURABLE FUNCTIONS
and negative integers are all measurable for any n, and C,Emn = X for all n. If we set f(x) = m / n for x on Em,, then limnf , , ( x ) = f(x) uniformly on X . This observation is of importance for the Lebesgue approach to integration. It can also serve as an approach to measurable functions, by defining a measurable function as the point wise limit (or uniform limit) of elementary measurable functions such as step functions. 5. Properties of Measurable Functions Depending on the Measure Functions
In the preceding paragraphs, we have derived properties of measurable functions which depended only on the fact that the class !IN of measurable sets is an s-algebra : s-additive, s-multiplicative, subtrsctive, and complementary. In this section we assume that 93 is defined in terms of an upper measure a * a function on all subsets E of the set X which has the properties: 0 5 a*(E) i co for' all E, and if E 5 C,E,, then a*(E) 5 Z,a*(E,). These imply also monotoneity: if El 5 E , then a*(El) 5 a*(E,). The class 93 consists of the sets M which satisfy the measurability condition a*(E) = a*(EM) a*(ECM) for all E. This induces a subclass 32 of !IN of sets such that for all E of 32: a*(E) = 0. As indicated in the exercise of V.16, the class 9 is s-additive and subtractive, that is an s-ring, and has the socalled hereditary property: if E is in 32 and El 5 E , then El is also in Y. Moreover if M is in 9.X and E differs from M by a set in 32 that is a*(MilE) = a*(MCE E C M ) = 0, then E is also in !lJ and l, a ( M ) = a ( E ) . We shall call the sets of 32, sets of zero a-measure or a-null sets. We say that two functions fl(x) and f,(x) difer at most by an a-null set or are equal almost everywhere (a.e.) if a*(E[ - f, > 01) = 0, or E [ l f , - f, > 01 belongs to 32. We can then assert:
+
+
+
Ifl
I
I
is a-measurable and f,(x) differs from fi(x) at most by an a-null set, thenf,(x) is a-measurable also. This is almost obvious since E [ f , 2 y ] and E [f,2 y ] differ by a subset of E [ l f l - f, > 01, that is at most an a-null set and so are simultaneously measurable. A consequence is:
5.1. If f,(x)
I
5.2. If f,(x) are a-measurable, and limnfn(x) = f(x) for all x except
an @-null set, then f(x) is a-measurable.
210
VI. MEASURABLE FUNCTIONS
For f(x) will differ from the a-measurable function & , f , ( x ) at most at the points where lim,f,(x) does not exist, that is on an a-null set. Note that in a sense the values of f(x) are undetermined at the points of the a-null set where limnf,(x) does not exist. In case a(X) < coy it is possible to intensify the nature of the convergence of a sequence of a-measurable functions convergent except for an a-null set.
a(X) < coy and f n ( x ) on X are a-measurable, and such that lim,f,(x) = f(x) except for an a-null set, then for every e > 0, there exists a set M e of !IN, such that a(M,) < e, and lim,f,(x) =f(x) uniformly on X - M e . 5.3. If
Let M , be the set on which f,(x) does not converge to f(x), so that a(M,) = 0. By the preceding theorem f(x) is measurable. Then for fixed n and e the set E [ ( f ,- f I > el is measurable, and so M,, = C,"O_,E[ - f I > el is measurable for each e and n. Now M , , is the set of all x such that for some m greater than n: If,(x) - f(x) I > e. For any x not in M,, lim,f,(x) = f(x), or (e, n,, n 2 n,) :If,(x) f(x) I 5 e, that is x is not in M , , for n 2 n,. Consequently, n n M , , = M , for all e. Then by V.6.10 lim,p(M,,) = a(M,) = 0, for all e. This is equivalent to (d, ride, n 2 ride) :a(M,,) 5 d. Let em = d, = 1/2", and set n, = nde for d = e = d,. Also set Mk = .Zz-kMn e,. Then 00 00
If,
m I..
I,.
so that limka(Mk) = 0. We show that lim,,f,(x) = f(x) uniformly on X - M k for all k. For suppose that x belongs to X - Mk for fixed k. Then x is not in M,,,, for m 2 k. Then for m 2 k , and n 2 n,, and all x in X - Mk we have If,(x) - f(x) 5 em = 1/2". This means uniform convergence of f,(x) to f(x) on X - Mk, since the n, does not depend on 0, we have completed the proof of the theorem. x. Since limka(Mk) =~ Observe that this theorem does not assert that if the sequence of a-measurable functions f , ( x ) converges to f(x) except for an a-null set, then the convergence is also uniform except for an a-null set, that is on X - M,,' for some M,' such that a(M,') = 0. In the preceding theorem, we showed that if M , , = Zz E [ l f , f > el, then lim,a(M,,) = 0 for all e. As a consequence lim,a(E[lf, - f > el) = 0 also, if the sequence f, converges to f except at an m u l l set. This condition suggests a weaker type of convergence in
I
I
I
5.
21 1
PROPERTIES OF MEASURABLE FUNCTIONS
which the functions need not even be assumed to be measurable relative to a. 5.4. Definition. The sequence of functions f n ( x ) on X converge to f ( x ) in the a* sense, or relative to a*, if limna*(E[(f,,-.f I > el = 0 for all e > 0. We denote such convergence by f, f ( a * ) . In a way this type of convergence shifts the burden of convergence from the functions to the measure of the sets En,, where E,, = E[ If' - f I > el, that is where f, differs from f by too much. The following theorem gives an indication of the amount of convergence induced on the sequence f,, by the convergence relative to a*. --f
5.5. If f, +f ( a * ) , then there exists a subsequence f,, such that lim,fn m (x) =f(x) for all x of x except an a-null set.m Let em and d, be sequences of positive numbers such that Zmem and Zmdmare convergent, for instance em = d, = 1/2m. Choose n, so that a*(Enmc,)< d,, and set Ek = Z;En m m, with E ,m ,m = E[I f,, -f I > em].Then since
it follows that limka*(Ek) = 0. Consequently, if E, = IIkEk:we have a*(E,) 5 a*(Ek)for all k, so that a*(E,) = 0. Suppose x is in X E, = CEO. Then since Ek+l5 Ek, there exists a k, such that for k 2 k,, x is not in Ek. Consequently, for m 2 k,, we have Ifnm(x)f ( x ) 1 5 en,, which means that fnnl (x) converges to f ( x ) . Then lim, fnm(x) = f ( x ) for all x in X - E,, that is excepting for an a-null set. a*-convergence of a sequence of functions f n ( x ) gives rise to a Cauchy condition of convergence, stated : 5.6. A sequence of functions f , , ( x ) converges relative to a* or satis$es - f, + O(a*), a Cauchy condition relative to a*, if and only if
If,
I
or if En,, = E[I f n - f, I > el, then limmna*(E,~L,) = 0 for all e > 0. The following Cauchy theorem of convergence is then valid:
5.7. If the sequence of functions f n ( x ) satisfies the Cauchy condition relative to a*, then there exists a function f ( x ) such that f,, +f(a*). Select again d, and em sequences of positive numbers such that Z,d, and Zme, converge, In virtue of the Cauchy condition on f , ,
212
V I . MEASURABLE FUNCTIONS
it is possible to select an increasing sequence of integers n, that if EWL+lem = E[lfTIm(X) -f,,,+, ( x ) > em]' then a * ( Et'm"m+lem1 < d,. Set W E , = ~ "Em " m + l e m * Then W W 5 ff*(E,/,//m+,em) dwj,
such
I
,
2k
k
so that limka*(Ek) = 0. Consequently, if E, = 17!Ek, then a*(E,,) 5 a*(Ek) for all k, and a*(E,) = 0. Consider a point x in X - E,, = CE,. Since Ek+l5 Ek, there exists a k , such that if k 2 k,, then x x is not in Ek and so not in Ell , m + l e m for m 2 k,, or I f,,,(x) fTlm+l(X) I5 em. Then for m" > m' 2 k,: TIL
Then the sequence f ( x ) is a Cauchy sequence of numbers and "m lim,f,, m ( x ) exists equal to say f ( x ) for all x not in E,, which is an a-null set. We show next that f , m +f ( a * ) , that is that lim,a*(E[( f "lm - f > e l ) = 0. Suppose x is in X - E,, where Ek is the set defined above. Then as we have shown, for m' 2 m 2 k :
1
Since X - Ek is contained in X - E,, f , , , ( x ) converges to f ( x ) so m that by taking limits as to m' we find for m 2 k that If, m ( x ) f(x) 5 L ' : ep. If now for m 2 me : L': e p< e, then for x in X - Ek, m 2 me, m I k, we have f , m ( x ) - f ( x ) 15 e. Consequently, no points of the set Em, = E [ l f T i m - f > el belongs to X - Ek, or Erne is contained in Ek for m 2 me and m 2 k . But lim,a*(E,) 1 0 . Consequently lim,a*(E,,) = 0 for all e or f , ,m +f ( a * ) . Finally to show that f,, +f ( a * ) , we observe that
I
E[
1
If n
since if
-f
If,,
I
I > el 5 E [ I f , - f 1 > e/21 + E [ I f, I,,
-f,,,IS e / 2 and If,,, - f 15 e / 2 then
m
If, ,
-
f I > e/21,
-
f 15 e. But
5.
213
PROPERTIES OF MEASURABLE FUNCTIONS
limn,,.*(E[lf,, -fJ > e/21) = 0 and lim,a*(E[lf,l - f ( > e / 2 l ) = 0, so that lim,a*(E[lf, el) = 0, or f,, +f ( a * ) . These results apply to any sequence of functions on a set , 'A if an upper measure is defined on subsets of X . Incidentally it should be noted that a sequence can converge relative to an a* without converging at a single point of X . For instance, suppose X is the open unit interval 0 < x < 1, with a*(E) Lebesgue upper measure. Let the functions f,,(x) with 0 5 m < n, be the characteristic functions of the open intervals (m/n, ( m l)/n) : 1 for m/n < x < (m l)/n and 0 elsewhere. These functions can be arranged as a single sequence if (m', n ' ) < (m",n") when n' < n" and for n' = n", when m' < m". Then the resulting sequence converges to the zero function relative to Lebesgue upper measure, but does not converge for any point x with 0 < x < 1. However, the subsequence f , , J x ) converges to zero for all x of 0 < x < 1. If we consider functions measurable with respect to the class !ll of measurable sets M determined by the measurability condition applied to a*, then we have:
-fl>
+
+
5.8. If f,(x) are a-measurable, and f,, + f ( a ) , then f is measurable.
For iff,, + f ( a ) , then there exists a subsequence f , m of a-measurable functions converging to f excepting for an a-null set, so that f is a-measurable. On the other hand:
f, except at an a-null set, then f,, +f(a). We have already seen that in this case, if E , = C,,nE[lf, - f > el, then limna(E,) = 0, so that limna(EII f , f > el) = 0. The measurability condition here is necessary since there exist sequences of nonmeasurable functions converging to zero everywhere, but for which it is not true that lim,La*(E[lf,, I > el) = 0. For let X = [0, 11, a ( x ) = x,a* be upper Lebesgue measure, E,, the disjoint nonmeasurable sets such that a*(E,) = a*(E,) for all m and n, and ZnEn= X , defined in V. 12. Let fm(x) be the characteristic functions of the sets LffPE,. Then limmfm(x) = 0 for all x, but for e < 1 and all m : a * ( E [ f , > el) > a*(Em)= c, a fixed positive constant. In the case of sequences of numbers a,,, we have the theorem that if for every subsequence a,,m,there exists a subsequence a,, con-
5.9. If f , ( x ) are a-measurable, and f , converges to
~
1
I
*k
214
VI. MEASURABLE FUNCTIONS
verging to a, then l i m p , = a. This theorem does not carry over to convergence of measurable functions, if the sequence converges except for an a-null set. We can, however, say: 5.10. Iff, is a sequence of a-measurable functions on X , and the function f is such that for every subsequence f n , off,, there exists a subsequence f , such that lim, f n (x) =f ( x ) except for an a-null set mk
then
mk
f,+f ( a ) .
For it would then follow that f n +f(a) so that lim, a(E[I f n mk mk f I >el) = 0 for all e. Then for any e, the real number sequence an= a ( [ EI f , -f I > el) has the property that for any subsequence a,, there exists a subsequence a, such that limka, = 0. Then limnan= 0 or
fn
+f(a).
mk
mk
EXERCISES 1. Show that the a*-convergence of a sequence of functions f n ( x ) on X has the linearity properties: (a) if f,+ f(a*),and a c is a constant then cfn cffa*); (b) if f i n +fi(a*) and f i n +fi(a*), then f i n + f i n + +fda*). +
fi
2. Shown that if f,,-+ f(a*),then f,, satisfies the corresponding Cauchy condition of convergence f, -A,, -+ O(a*), or lim,,a*(E[I f,,- f,( > el) = 0 for all e > 0.
3. If f,(x) is a set of functions on X with q on the directed set Q, a* an upper measure function on all subsets of X , and if f(x) is such that f,+ f(a*),or lim,a(E[I f, - f I > el) = 0 for all e > 0, then there exists a monotone sequence of elements qlLand an a-null set E, such that limnfg*(x) = f(x) for all x on X- E,,. If f,(x) are a-measurable, then f ( x ) is a-measurable. Note that the sequence q,, need not be cofinal with Q. 4. If f,(x) is a set of functions'on X with q on a directed set (9, satisfying a Cauchy condition of convergence: lim, a*(E[ 1 f q l - fg2I > el) = 0 1 2 for all e > 0, then there exists a monotone sequence of elements qn, a function f ( x ) and an a-null set E, such that limnf q n ( x ) = f(x) on X - E,. Further f , +f (a*).
,
5. Show that there exists a directed set Q of elements q, a set f,(x) of Lebesgue measurable functions on [0, 11 and a measurable function f ( x ) such that lim, f,(x) = f(x) for all x , but lim,a(E[I f , - f 1 > el) = 1 , for all e < 1, a being Lebesgue measure.
6.
215
APPROXIMATIONS TO MEASURABLE FUNCTIONS
6. If Q is a directed set with sequential character (there exists a sequence qn of Q cofinal with Q), f J x ) a set of a-measurable functions on X , f ( x )
such that lim, f p ( x )= f ( x ) except on an a-null set, then f ( x ) is measurable and f,+f( a). 6. Approximations to Measurable Functions. Lusin’s Theorem
We have already seen in VI.4.6 that if f ( x ) is any finite valued function on a set X,then there exists a sequence of step functions f,,(x) such that lirn?,f , ( x ) = f ( x ) uniformly on X.These step functions can be obtained by dividing the Y-axis by the points x = m/n, m ranging over the positive and negative integers, and setting f , , ( x ) = mjn on E[m/n5f < (m 1)j.l. If f ( x ) is any function on X and we define f , , ( x ) = rnjn for x on E[m/n5 f < ( m l)/n] for - n2 2 m < n‘, and f,,(x) = n on E [f 2 n] and f , , ( x ) = - n on E [f 5 - n], then f , , ( x ) will be a finite-step function (assuming a finite number of distinct values), such that 1im1,,f,,(x)= f ( x ) for all x of X, with uniform convergence on the sets E[a 5 f 5 b] for all a and b. In case Im is a class of measurable subsets of X and f ( x ) is measurable, then the functions f , , ( x ) in each case are measurable. If we limit ourselves to the case when X = (- co < x < co), and a ( x ) is a bounded monotonic nondecreasing function on X which gives rise to an a-measure and a class YJI of a-measurable sets M , as expounded in Chapter V, we can go further. Let f(x) be an a-measurable function on X. Set M,,, = E[m/n5 .f < (m+ l)/n], for - n‘5 m < n2, and M,, = Z7,,M,,,, so that X M,, = CM,, = E [f 2 n] E [ f < - n]. Then the sets M,,, and CM,, are mutually disjoint for fixed M , and a-measurable. Since . f ( x ) is finite valued and a ( X ) < a,we have lim,la(CM,,) = 0. By V.10.2 we can then find closed sets F,,,, contained in the corresponding M,,, such that U ( M ~ ~-, ,FTlI,,) , < 1j2k’. Then because of ~ ~X7nF7rl,l) ,, < l/n, the additive property of a, it follows that ( Y ( Z ~ , , Mn‘ 5 m < n’. Set F,, = Z,nFm,l, where ni ranges over the integers since for fixed n, there are a finite number of closed sets F7,i,,,F,, will be closed and a ( M , - F,,) < 11. We define the function f , , ( x ) = m/n if x belongs to F,,,,, and linearly on the open intervals complementary to F,,. If F,, has a maximum point xo on the X-axis, which belongs to F,,, then f , , ( x ) = ni/n for x 2 xo, and similarly if FI1 has a minimal point. Then the functions f,,(x) are continuous on
+
+
+
+
~
216
VI. MEASURABLE FUNCTIONS
X = (- 00 < x < + a). If x, is in the complement of F,, then f,,(xo) is on a straight line part off,,(x) so that f,,(x) is continuous a t xo. If x, is an isolated point of F,, then two straight line portions of f,(x) meet at xo so that f,,(x) is continuous at x,. If x, is a limiting point of F,,, then since the F,,, are disjoint for fixed n, it is a limiting point of only one Frn?[.Consequently there exists a neighborhood V(xo) of x, such that V(x,) - F,, = V(xo) . F,,,. Then f,,(x) is constant on V(x,) if x, is a two sided limiting point of F,,,, or constant on one side of x, and linear on the other side of x,, if xois a one sided limiting point. In either case f , , ( x ) is continuous at x,. *
We show that for the continuous functions f,,(x) we have f,, +f(a). For the set E I I f - f f , , l / n will belong to X - F,,. Now a ( X F,) 5 a(X - M,,) a(M,, - F,,). Since lim,,a(X - M,,) = 0 and lim,La(A4r,- F?,) = 0, it follows that lim,,a(X - FJ = 0, and so lim,cr(E[If,, -f > l/n]) = 0. If e > l/n, then E[lf,, - f > e 12 E [ l f , - f l > l/n]. Then lim,,a(E[lf,,-fl>e]) = O for all e, and
+
I>
1
I
f, +ff.).
Iff,+f(a), then there exists a subsequencef,, such that lim,f,lm(x) m =f(x) except for an a-null set. We have then demonstrated:
- co < x < + 00, a ( x ) monotonic nondecreasing and bounded on X , f(x) finite valued on X and measurable, then there exists a sequence of continuous functions f,(x) such that lim,f,(x) = f(x) except for an a-null set. Now from the convergence properties of measurable functions it follows that for any e > 0, there exists a measurable set M e such that a(A4,) < e, and lim,,f,,(x) = f(x) uniformly on X - M e . Since the f,,(x) are continuous it follows that f(x) is continuous in x if x is restricted to X - M e . Now X - M e contains a closed set F , such that a ( X - M , - F,) < e, so that a(X - F,) < 2e. Then .f(x) will be continuous on F,. Consequently:
6.1. THEOREM. If X is
< + 00, a ( x ) is monotonic nondecreasing and bounded on X, f(x) is finite valued and measur-
6.2 (Lusin's Theorem). If X is - co < x
Essentially f,(x) is continuous on the closed set F,,, if only the values of f,,(xl F,, are considered. By adding to f n ( x ) , values on the complement of F,, we extend the functionf , ( x ) to all X so as to be continuous in x on X . An alternativeway of obtaining such a function is to define gJx) = 1.u.b. [ fn(x') - n I x - x' I, for x' on F,,], which is similar to the functions used in VI.1.6. The functions f J x ) will be continuous for all x , and agree with f,(x) on F,, [see E. J. McShane: Extension of range of functions, Bull. Am. Math. SOC.40 (1934) 8371.
6.
APPROXIMATIONS TO MEASURABLE FUNCTIONS
217
able relative to a, then for e > 0, there exists a closed set F , such that - F,) < e and f ( x ) is continuous if restricted to F,. This theorem asserts in a way that a measurable function is almost continuous. It has been used as a basis for the definition of measurable functions. Like many important results in mathematics, there are several different ways of proving Lusin's theorem. Consequently we think it worth while to give another proof which links up with the definition of measurability of a function. [See L. W. Cohen: A new proof of Lusin's theorem, Fundamenta Math. 9 (1927) 122-3.1 A finite valued function f ( x ) is measurable if for a sequence y,, dense on the Y-axis, the sets E [ f I y,,] or the sets E [f 5 y,] are measurable for every n. Similarly if X is the closed interval [0, 11, then the function f(x) on X is continuous if for a denumerable dense set y , on the Y-axis, both E [ f Z y,] and E [ f S y,,] are closed for all n. If F is a closed subset of X,and F * E [ f 2 y,] and F E [f 5 y,] are closed for all n, then f is continuous on F, using only the values of f ( x ) on F. Let M , = E [f 2 y,] and Mr1'= E [ f 5 y,]. Then there exist closed sets FVlin M , and F,' in M,' such that if M , - F , = R , and M,' F,' = R,', then a(R,) < e/2" and a(R,,') < e/2". Then u(Z,R,J 5 S, a(R,) < e and a(S,R,') 5 .Er1a ( R , ' ) < e. Set R = .E,(R,, 4R,')' so that a(R) < 2e. If M = X - R, then or(X - M ) = a ( R ) < 2e. Now there exists in M a closed set F such that a ( M - F ) 5 e, so that a(X - F ) = a(X - M ) a ( M - F) < 3e. Now f ( x ) limited to the set F is continous on F. For
a(X
+
F . E [f 2 y , ]
=F
*
F * E [f 5 y,]
=F
*
+ R,) = F . F,, F(F,,' + RVl')= F F,l',
M,t = F(F,,
and
M,,' =
*
since Rrl and R,' are in R. Then these sets are closed for all n and so f ( x ) is continuous if restricted to F. Since by the Weierstrass polynomial approximation theorem a continuous function on a closed interval can be uniformly approximated by polynomials, it follows that for the continuous functions f , ( x ) defined relative to the a-measurable function f ( x ) above, there exists a polynomial P , ( x ) such that f , ( x ) - P,,(x) < l / n for all x of - n 5 x S n. Consequently, we can state:
I
+
1
6.3. If X is - co < x < co, a ( x ) is monotonic nondecreasing and bounded on X, f(x) is finite valued and or-measurable, then there exists
218
VI. MEASURABLE FUNCTIONS
a sequence of polynomials P , ( x ) such that lim,LP,L(x) = f ( x ) except for an a-null set. Another type of approximation to a continuous function on a closed interval is by means of staircase or interval step functions, that is functions which are constant on each of a finite number of intervals. For the continuous functions f , , ( x ) above, there exists then a staircase function S , ~ ( X ) vanishing for x > n, and such that f , ( x ) s,(x) < l / n for all x with x 15 n. Then we have:
I
I
I
f ( x ) is finite valued and a-measurable on X = - 00 < x < fa, with a ( X ) < co, then there exists a sequence s,(x) of staircase functions, which are constant on a finite number of intervals and vanish outside of a bounded interval, such that limns,(x) = f ( x ) except for an a-null set. It is possible to prove a theorem of the Lusin type for the case when the finite valued function f ( x ) is on a topological space X, if the s-additive measure function a ( M ) on the s-algebra !Vi of a-measurable subsets of X has the property that closed sets F belong to !Vi and for any M of !IN, and e > 0, there exists a closed set F , belonging to M such that a ( M - F,) < e.
6.4. If
CHAPTER VII
LEBESGUE-STIELTJES I N T E G R A T I O N
1. The Lebesgue Postulates on Integration
Before taking up in detail the theory of Lebesgue or LebesgueStieltjes integration, we feel that it is interesting and instructive to list the properties of an integral which Lebesgue considered fundamental and which in a natural way lead to his definition of integral. In his " Leqons sur l'lntegration " (second edition, Gauthier-Villars, Paris, 1928, p. lOSff), he considers the problem of determining for any bounded function on - co < x < + co and for any finite interval a 5 x 5 b, a finite real number or a real valued functional, which we shall denote by J t f ( x ) d x or J:f, which satisfies the following postulates: (1) For any a, b, and h : Jf f ( x ) d x
(2) For all a, b, and c : J9.f (3) For
=
JlIi f ( x
+ J:,f+ J:f=
-
h)dx;
0.
f, and f i any two bounded functions and all
a, b :
(4) If f ( x ) 2 0 for all x and a 5 b, then J l f 2 0. (5) J,'1 * d x = 1. (6) If f,, is a monotonic nondecreasing sequence of functions converging to the (bounded) function f for all x, then lim,( J $ f , ,= J:J
It is tacitly assumed that the value of J,",f(x)dx depends only on the values of f(x) for a 5 x 5 b. We consider briefly the import and consequences of these postulates.
Postulate (1): for all a, b, h : Jf,f(x) = Jlz:f(x - h ) d x , says that integration is invariant under translation in x.This is connected with the invariance of length of an interval and ultimately measure under translation.
219
220
VII. LEBESGUE-STIELTJES INTEGRATION
+
Postulate (2): for all a, b, c: J f f + Ji f J:f= 0. If we set a = c, then 3 J:f= 0 so that for all a : J:f= 0. If now a = c, then Jff+ 0, or J f f =- J,”f, and so J f f + J: f = J:J This last asserts that the integral J’Bfis an additive function of intervals for fixed f, and consequently is completely determined by the point function F( x) = Jlf, with F(b) - F(a) = J’fJ b
=
J-y=
+
Postulate (3): J: (f,+fJ = Jff, J:f,, says that the integral is an additive functional on the class of bounded functions for fixed a and b. If we takef, = 0, then Jff, = Jff1 Jf 0, so that Jf 0 = 0. I f f = f l = -f,, t h e n O = J ~ O = J ’ ~ f + J f ( - f ) , s o t h a t J:(-f) = - J’fJ If we t a k e f , = f , = . . . = f , , = f , then J f n f = n J ’ f f f o r all integers n, from which we deduce that Jf r f = r J’ff for all rational numbers r. We can then conclude that under postulate (3) the integral J f f i s rationally linear in f for fixed a and b.
+
Postulate (4): If f ( x ) h 0 for all x and b 2 a, then J f f h0 is the statement that J f f i s a positive functional for fixed a 5 b. If we use the linearity condition (3), then this postulate is equivalent to the following order condition : (4 a ) : If f , ( x ) 5 f,(x) for all x, and b 2 a, then
Jlf, 5 J:fi.
We can also demonstrate the theorem:
Jff satisfies the postulates (3) and (4) and if limnfn = f uniformly on [a, b ] then lim,$ = JfJ For then (e > 0, n,, n 2 n,, a 5 x 5 b): f ( x ) - e 5 f , ( x ) 5 1.1. If
Jsf,
+
f ( x ) e. If we assume e to be rational, then by the rational linearity of Jff and the order relation (4 a), we have: J b f -
rf+
e / l 1 J ~bfr,5
e f 1.
Since J: 1 is a finite number, it follows that limn Jffn = JfJ If now c is any real number and rlrany sequence of rational numbers such that lim,Lr,t= c, then for any bounded function lim.r,f= cf uniformly. Then Jf cf= c Jff for all real c. We can then state: 1.2. If J f f i s an additive positive functional on the class of bounded functions on [a, b ] [satisfying (3) and (4)], then it is linear and con-
tinuous in the sense of uniform convergence of functions.
Postulate ( 5 ) : Ji 1 = 1 . We show that in the presence of (I), (2), and (4) this condition implies that J: 1 = b - a for all a and b.
1.
22 1
THE LEBESGUE POSTULATES ON INTEGRATION
From the additive property (2) we have
From the invariance condition (1) it follows that
S,m-l,,nl = J mln
lln 0
1,
for all m,so that J:'" 1 = 1 In. From a repeated use of (2) it now follows that Jr1 l = r 2 - r l 7,
for all rational numbers r, and r2. If we define the function f , ( x ) = 0 for x 5 a and x 2 b, and f , ( x ) = 1 for a < a' 5 x S b' < b, and linear between a and a', and b and b', and if fJ'x) = 1 for all x , then f J x ) 2 f , ( x ) , so that
Then J: 1 2 b < r2, then
Js: 1. If now a and b are any, and rl < a < r,' < rz'<
Then J2 1 = b - a. This postulate then ties the integral to the lengths of intervals. From the point of view of Stieltjes integrals, this postulate would be omitted,
Postulate ( 6 ) : If f , is a monotone nondecreasing sequence on [a, bl converging to the bounded functionf, so that f , ( x ) 5 f , + , ( x ) 5 f f x ) for all x, then limn Jf f , = Jlf. Since by postulate (2): (- f) = - Jff, an equivalent statement of this postulate replaces " monotone nondecreasing " by '' monotone nonincreasing." Further since under postulate ( 3 ) ( f , - f) - Jf f , = Jff, each of these statements is equivalent to :
Js
Js
(6 a ) : If f , are positive or zero for all x , and monotonic nonincreasing in n for each x, with limllfn(x) = 0 for all x , then limn J: f , = 0 for all a and b. The conditions (6) are of the nature of continuity conditions on the functional JfJ In the case when f, is a sequence of Riemann integrable functions converging monotonely to a Riemann integrable
222
VII. LEBESGUE-STIELTJES INTEGRATION
function, the sequence f,is uniformly bounded and Osgood’s theorem 11.5.6 can be used to give (6) if our class of functions is limited to functions Riemann integrable on [a, b]. As a matter of fact postulate (6) is equivalent to a property of the Osgood type, that is, we can state: 1.3. If for every monotone nondecreasing sequence of bounded func-
tions f, on [a, b ] , converging to the bounded function f, we have lim, J,”f,, = Jlf, then for any sequence g,(x) of functions uniformly bounded on [a, b ] , such that lim,g,(x) = g(x) for all x of [a, b ] , we have lim, J:g,, = Jsg; and conversely. The converse is obvious. For the direct theorem we note that if g , is a uniformly bounded sequence of functions, then in view of postulate (3):
s” (g.I.b.,,g,,) 5 g.l.b.,Jb g, 5 l.u.b..lb g,, 5 a
a
a
J ba
(l.u.b.71gn).
If we set (l),l(,x)= 1.u.b. g,(x) m8n
and Y,,(x)
=
g.1.b. g,(x), men
then for all n :
J: Y,,5 g.1.b. j bg, 5 1.u.b. J: g , 5 m8n m8n
Jb
a
a
Now Y , are monotone nondecreasing in n, and clill are monotone nonincreasing in n, so that lim,Yn = l.u.b., Y n= l&,g, and limn@?, = g.l.b.,Oll = lim,,g7,. In view of postulate (6), we then have:
From these inequalities we deduce at once that for any uniformly bounded sequence of functions g,, on [a, b ] we have:
Sb g, 5 lim, J g, 5 J limflgr,. -
Jb a
-n lim g
fl
- lim I -n
b -
b
a
a
a
If we assume that lim,gll exists for all x then Jb
a
lim,g,
= lim,
g,. a
As has previously been indicated (see VI.4.7) any function can be uniformly approximated by a sequence of step functions. In particular any bounded function is the uniform limit of a sequence of finite-step functions, which assume only a finite number of values.
1.
THE LEBESGUE POSTULATES ON INTEGRATION
223
In particular, if c < f ( x ) < d for all x , and we divide the interval [c, d ] into n equal parts by the points c = yon < y I n< y Z n< ... < y,,, = d, and set f , ( x ) = Y,,, for x on Em,L= Eb,, 5.f < ~(,+,),l, then lim,f, = funiformly on X . If x(E; x ) is the characteristic function of the set E (unity for x on E and vanishing for x not on E ) , thenf,(x) = Cmymr,x(Em,z; x ) . If we assume that Jtf satisfies postulates (1)-(5), so that it is a linear functional continuous under uniform convergence, then J l f = lim,, Z7,,y,, x(E,,) for all a and b. In other words if Jfif satisfies postulates (1)-(5), then its value is completely determined if the values of J’l x ( E ; x)dx = a ( E ; a, b) are known for all intervals [a, b ] and subsets E of X . The question is what conditions are imposed on the set function a ( E ; a, b) by conditions (1)-(5). We find:
Jt
1.4. The postulates (1)-(5) on J:f induce the following properties on a ( E ; a, b ) = x ( E ; x ) d x for subsets E of [a, b ] :
Js
+
(1 a ) a ( E h ; a, b) der translation. (2a) a ( E ; a, b )
+
= a(E;
a
-
h, b
+ “ ( E ; b, + a ( E ; C)
+
- h)
or oc is invariant un-
a ) = 0.
C,
+
(3 a ) a ( E , E,; a, b) a(E,E,; a, b) = “(El; a, b) a(E,; a, b ) This is based on the identity x ( E , E,; x) x(E,E,; x) = x(E,; x ) x(E,; x ) . It follows that a ( E ; a, b) is a finitely additive set function for fixed [a, b ] .
+
+
+
(4a ) a ( E ; a, b) 2 0 for all E and a 5 b. (5 a ) If E = 0 5 x 5 1, then a ( E ; 0, 1)
=
1.
lf we consider only subsets of [0, 11, then these conditions require an additive measure on all subsets of [0, I ] invariant under translation, with the measure of the unit interval being 1 . As we have noted in V.12 such a measure for all subsets of [0, 13 has been constructed by S. Banach. Linear content satisfies the conditions (1 a)-(5 a ) only if we limit ourselves to subsets which have content for every finite interval [a, b]. Postulate (6) adds a further requirement on a ( E ; a, b). We note that if E n is a sequence of disjoint sets, then the sequence of functions f n ( x ) = L‘:=lx(Em;x ) is monotonic nondecreasing and converges to x ( E ; x ) = Z n x(E,,; x ) , where E = C7,E7,.Using linearity as well as postulate (6), we find that for any interval [a, b ] we have:
224
VII. LEBESGUE-STIELTJES INTEGRATION
(6 a ) a(S,E,; a, b) = S T L a ( Ea, , ; b). This means that a ( E ; a, b) is s-additive for all [a, b]. We have already seen in V.12 that for the interval [0, 11 no measure function invariant under translation exists for all subsets of [0, 11. It is possible that Lebesgue suspected this, because he abandoned the project of finding an integral for all bounded functions satisfying (1)-(6), and developed the notion of measurable sets as well as a measure function as discussed in Chapter V. This measure function [based on a ( x ) = x ] satisfies postulates (1 a)-(6 a), on measurable sets. By limiting himself to measurable functions defined in terms of (Lebesgue) measurability of the sets E ( c
We take up first the definition of integration due to Lebesgue, which depends on subdivisions of the range of the function. We assume X to be a one-dimensional interval, either u 5 x 5 b or - co < x < 00. The function f ( x ) is assumed to be finite valued, that is on X to Y : - co < y < co. We assume a monotonic nondecreasing function a ( x ) on X,which for the present will be assumed bounded, which gives rise to a class !lR of a-measurable subsets of X, which is an s-algebra, and an s-additive measure function on a-measurable sets, as derived in Chapter V.
+
+
2. THE
LEBESGUE METHOD OF DEFINING AN INTEGRAL
225
We take up first the case when the function f ( x ) is bounded and a-measurable. Let m
is bounded and or-measurable, then limla,+oS,yk'a(Ek) exists. This is defined as the Lebesgue integral: L Sxfdff. The existence of limlal+oZ,y,'a(E,), is a consequence of the following observation. If we define p ( y ) = a ( E [f 5 y ] ) , then ,u(y) is a monotonic nondecreasing function of y , such that p(y,) - ~ ( y , - ~= ) a(E[y,-,
tegral L JX-f d a exists and is expressible as the Stieltjes integral YdP(Y), where P(Y) = f f ( E [ f5 Y l ) . Because of the fact that p ( y ) is zero for y < g.1.b. [f ( x ) , x on XI and constant for y h 1.u.b.[f ( x ) , x on XI, it follows that ;S yd,u(y) has the same value for all M 2 1.u.b. [ f(x), x on X ) and all m 5 g.1.b. [f ( x ) , x on XI. Further, we can apply integration by parts and obtain L JX-f d a = M a ( X ) - J," p(y)dy, so that the Lebesgue integral for a bounded function is expressible in terms of the Riemann integral of the monotone function p ( y ) , on the range of f ( x ) . In case the function f ( x ) is a-measurable, but not bounded, two procedures are available. Following Lebesgue, we divide the infinite interval - co < y < co by a subdivision cry or u, determined by the points ... y-, < y-k+l < ... < yk < ... such that y-, + - a,yk+ a,and I u y I = l.u.b., y , - ykPl < 00, and set up y,'ff(E,), with E, = E[y,_,
s:
+
+
I
when this limit exists. The important result is :
1
Z:zoo
226
VII. LBBESGUE-STIELTJES INTEGRATION
2.3. If f ( x ) is a-measurable on X,and a(X) < co, then a necessary and sufficient condition that Jxfda exist according to either procedure, is that there exist a subdivision u of the Y-axis of finite norm, and a sequence yk‘ such that yk-l 5 yk‘ S y k for which Zkyk’a(Ek) is absolutely convergent. Then limlo,+oL’,yk‘a(Ek), with yk-, 5 yk‘ 5 yk, as well as
both exist and are equal. For the necessity, if limlo,+oYroyk’a(Ek) exists, there exists a 0 with 1 ~ 7 < co, such that Yruyk‘a(Ek)converges for all choices of ykm15 yk‘ 5 y k . This means that
I
+
co independently, the series conexists. Since k‘ + - co and k“ --f verges absolutely. If @ ( X ) < co, and Ckyk‘a(Ek)converges absolutely for a particular choice of the yk’ such that yk--l 5 yk’ 5 yk, then the series converges absolutely for all such yk‘. This follows at once from the fact that for any finite set of k, and yk--l 5 yk’, yk” 5 y,, we have
If
lim (y’, y’)+(-rn,
Jy”
+a)
yd,u(y)
Y‘
exists, then since y‘ and y” are independent, both exist. For any subdivision 0 :
J,“yd,u(y) and .I?, y d d y )
I
so that by the preceding paragraph Zk yk’ la(Ek)
< co and the a-measurable function f ( x ) , there exists a subdivision uo of the Y-axis into intervals such that go < 03, and Zu,yk’ a(EOk)< co, for some y k ’ :yk--l= < Yk’ 5 Y k , then for any
2.4. If for a ( X )
I
I
1 1
2.
THE LEBESGUE METHOD OF DEFINING AN INTEGRAL
227
I
subdivision of the Y-axis such that (T I < co, and any choice of y,’ on [Y,-~, y,], we have ZaI y,‘ a(E,) < GO. If, moreover, o1 and ( T are any two subdivisions of the Y-axis of finite norm then
I
Ic -c 1 s f
10,
I+ I
02
~
I)a(W,
a2
Dl
where Za stands for an expression of the form S,yk‘a(Ek). Suppose that [y‘, y ” ] is any interval of the Y-axis and y’ y1 < ... < y , = y” any subdivision of [y’, y”]. Then
= yo <
where E = E[y‘
where the sums as to cr and cr0 are initially taken for a finite subset of each, and ultimately valid for the entire subdivisions. It follows that Za[ y,’ a(E,) < co for all (T 2 0”. Consider now any subdivision (T of finite norm. Then for the go in the hypothesis of the lemma, the product 0 will be finer than go. Using the inequality demonstrated above for any interval [y’, y ” ] , we can show that
I
where initially the sums are taken for finite subsets of the 0. It follows that X u yk‘ a ( E k ) < co, for all (T such that 0 < co, and all choices
I
I
Of Yk‘ On
Finally, if
Ic Dl
1 I
[yk-l, yk].
and u2 are two subdivisions of finite norm, then
( T ~
Is I
02
c c I + I c c Is -
01
-
a102
a102
(((TI
I + 1 % I)4x),
a2
where o p 2 is the product of the subdivisions, containing all points of both o1 and u2, and Sostands for the sum of the type X, y,’a(Ek). The existence of lim ,a,+oZoyk‘a(Ek) is an immediate consequence of the last inequality of the lemma, leading as it does to the validity of the Cauchy condition of convergence for the approximating sums.
228
VII. LEBESGUE-STIELTJES INTEGRATION
We note in passing that if L Jx- f d a exists, then for any u of finite norm a(X)’ - U yk’a(Ek)
I i,fda c
1 1‘ I We show next that if for some u with I u I < + co, and some yk‘ with ykPl5 yk’ 5 yk (and so for all such yk’) Z6I yk’ I a(Ek) < co, then
JTz yd,u(y) exists. For if 0 < Y,-~ < y’ < y” < y,,, then 1: ydP(y) < 2 Yk(p(Yk)
-
/‘(Yk-l))
=
m
2 Yka(Ek)’ m
1
I
From the convergence of ,T yk’ a ( E k ) it follows that lim
(v‘, v X ) + ( m , m )
s’: ydp(y)
so that l,”yd,u(y) exists. Similarly To show that
15: Ydp(Y)
==
lim
lob0
= 0,
Y
y d p ( y ) exists.
m !J
2 yk’a(Ek) = y ,
Jxfda,
we note that if [y’, y ” ] is any interval of the Y-axis, and u is any subdivision of [y‘, y ” ] , then
I J::
2 Ykla(Ek) 15 1 I c a(Ek) 5 I a 1 a ( X ) * 1 u 1 5 d,) : I J,: yd,u(y) - Soyk‘(Ek)I < e, where
Ydp(Y)
-
C7
k
a
Then (e, d,, the d, is independent of [y’, y ” ] and u is a subdivision of this interval. Since JT,“ y d p ( y ) exists, we have (e > 0, M e > 0, y” > M e , - y’ > M,): ydp(y) 66,y d p ( y ) < e. Chose any subdivision u of (- a, a) such that I u IS d,, and select n so that y,, < M e and y-,,< - M,, while oo
$7
+
+I
1
2 Yk’a(Ek) + l z l y k ’ f f
(Ek)
nfl
I < e‘
-UJ
This is possible since L‘kyk’a(Ek) is absolutely convergent. Then for 10. 15 d,:
2.
229
THE LEBESGUE METHOD OF DEFINING A N INTEGRAL
We note that the definition of an L-integral for an unbounded function involves essentially an iterated limit, one of the nature of convergence of an infinite series, the other as the norm of the subdivision approaches zero. The statement just proved is in an extended way, the equality of iterated limits in two orders. Since the conditions: (1) f(x) is a-measurable, and (2) there exists a ou such that oy < 00, for which Z,y,’a(E[yk-l < f S y,]) is absolutely convergent is sufficient to guarantee the existence of L fda, Lebesgue calls such functions “ sommable.” However, the transliterated word summable has other connotations, so we shall call a function satisfying these two conditions L-integrable relative to a , or Lebesgue-Stieltjes (L-S) integrable relative to a.
I I
Sx
2.4.1. In addition to the bounded a-measurable functions, the class
of L-integrable functions includes also the a-measurable functions which are almost bounded in the sense that there exists a constant M such that the set E [ > MI is an a-null set, since for such functions the sets E[y,-, M and y, < - M . If a(X) < co, a n d f i s a-measurable then C,a(Ek) = Z,k~l(E[y,-~ < f5y,]) converges for any subdivision oy with ou < co. Because of the s-additivity of the measure function a, it follows that if EAM = E [ I f 12 MI, then limMjm cr(E,,) = 0.
If I
I I
2.5. If the function f is L-integrable with respect to a, then we have
the stronger condition limM+mMa(E,) = 0. For by condition (2) on L-integrability, for any u y with 1 oy 1 < co, we have 2, y,‘ a(E,) < co, with yk--l5 yk‘ 5 y,. Take y,‘ = y k if yk-1 > 0, and y,‘ = yk--l if yk-l < 0. If 0 < y n P 1 < M 5 y,,, then
I
z
I
yka(Ek) > M
2 a(E,) > M a(cE,) > M a ( E [ f m
n
2 MI).
n
If Y-?,-~< - M 5 y - , < 0, then
It follows that
We have shown incidentally that iff is L-integrable with respect to
230
VII. LEBESGUE-STIELTJES INTEGRATION
f &M])=O, and lim,,,Ma(E[ f S-M])=O. In each case, the equality sign can be omitted in the definition of the sets E. There is an additional approach to the definition of the L-integral of an unbounded function due to Ch. de La Vallee Poussin [see, for instance, T o u r s d’Analyse Infinitesimale,” I, third edition, GauthierVillars, Paris, 1914, p. 2601. If f ( x ) is a-measurable and we define f ( m , M ; x ) = m if f ( x ) im, f ( m , M ; x) = f ( x ) if m i f ( x ) iM , and f ( m , M ; x ) = M if f ( x ) 2 M , that is, level off f ( x ) at the top and bottom, then f ( m , M ; x) is L-integrable for all m, M . We can then state: a , then lim,,,Ma(E[
2.6. A necessary and sufficient condition that f ( x ) be L-integrable with respect to a , is that
lim
( m , M)+(-m,
+ m)
l I f ( m , M ; x)da(x)
exist, L JJ f d a being the value of this limit. We have E [f ( m , M ; x ) 5 y ] = 0 if y < m ; E [f ( m , M ; x ) iy ] = E [ f S y ] if m 5 y < M , and E [f ( m , M ; x ) 5 y ] = X if y 2 M . Consequently, if m‘ < m iM < M‘ and p ( y ) = a(E[ f 5 y ] ) ,
=
=
+ Sy ydp(y) + M ( a ( X ) a ( E [ f5 M I ) ) m a ( E [ f 5 ml) + 1”ydic(y) + M a ( E [ f > M I ) . m a ( E [ f5 ml)
-
rn
m
We use here the fact that the function a ( E [f ( m , M ; x) 5 y ] ) has a discontinuity of magnitude a(E[ f 5 m ] ) at y = m. Since L Jxfda exists, the first term on the right converges to zero as m + - a, the last term converges to zero as M + co, and the middle term to L J-I-fda as (m,M ) + (- co, a). Consequently
+
+
Suppose, on the other hand, that lim
(m. M)+(-m, +m)
JdY
f ( m , M ; x)da(x)
exists. Consider any subdivision au with < ... < y,, 5 M .
1 0 ~ 1
3.
A RIEMANN-YOUNG TYPE OF INTEGRAL DEFINITION
Then
2
I;= 1
Y,-p(Ek)
23 1
5 J x f (0, M ; x ) d a ( x ) <
for all M , so that Zyk,o Y ~ - ~ C Y <(co. E ~In) a similar way we prove that - Zyk..oy,a(E,) < 00. Then Zok\y,' a(E,) < co, and f is L-integrable with respect to a. Since is a-measurable when f is, an examination of the proof above shows that:
I
If 1
2.7. A necessary and sufficient condition that an a-measurable function f be L-integrable is that Js I f ( - M , M ; x ) d a ( x ) be bounded in M .
I
3. A Riemann-Young Type of Integral Definition. The Y-Integral
In defining a Riemann type of integral of a function f ( x ) on a finite interval [a, b ] , the basic interval is divided into a finite number of nonoverlapping intervals and approximating sums of the form Zf d x determine the vdue of the integral when it exists. One reason for using subintervals in this procedure is that the length of an interval is available as a basis for linear measure. Because of the extension of the notion of measure to measurable sets determined by a (bounded) monotonic function a, we can extend the subdivision idea in two ways: (a) by allowing subdivisions of the fundamental interval into disjoint measurable sets, and (b) by allowing subdivisions into a denumerable as well as a finite number of disjoint measurable sets. Consequently, for this section a% or a shall stand for a subdivision of the set X into a jinite or denumerable number of disjoint a-measurable sets, a collection of sets E, such that E,E, = 0 for n # m, and ZnETL = X. Since a norm cannot be associated with such subdivisions in any obvious way, we fall back on the idea of successive subdivisions or refinements. We introduce order into the class of all subdivisions 0 of X into a-measurable sets, so as to make this class a directed set. We define crl 2 a2if every set in crl is a subset of some set in a2, or equivalently if o1is obtainable from a2by subdividing the sets of a2.This class is directed since, the subdivision a, . a2 consisting of the products of any set in al with any set in cr2, is finer than o1 and a2, or al . a2 L o1 and ol . a22 02. Following the procedure for the definition of Riemann-Stieltjes integrals, we can then set up the following definition for J X f d a of the finite valued function f ( x ) with respect to the bounded monotonic function a :
2JL
VII. LEBESGUE-STIELTJES INTEGRATION
3.1. Jx fda is the a-limit of the approximating sums Zuf ( x , ) a ( E , ) with x , in E n , provided this limit exists as a finite number. There is a difficulty in this definition, which centers in the expression Zuf ( x , ) a ( E , ) . If a consists of a finite number of sets, there is no trouble. Also, if f is bounded on X , then Zuf ( x , ) a ( E , ) is an absolutely convergent series if a is denumerable, since if f ( x ) 15 M , for all x , then L'?,f ( x , ) a ( E , ) 5 M Z n a ( E , ) = Ma(X). If f ( x ) is unbounded on X , then the question of convergence is pertinent, especially in view of the fact that although we denote a subdivision a by E,, ..., E n , ..., the order indicated by n means denumerability rather than an ordering of the sets. However, for P any set of elements p and a p a set of real numbers, we have defined (see 1.6)
I
I
I
where p 1 . . . p , is a finite subset of P and limit is as to the set Q of finite subsets of P directed by inclusion. Such a sum exists if and only if a pvanishes except for a denumerable set of p and if Zn apn < co, p , ranging over the p for which a p # 0. In view of this, we shall attach a number to zluf ( x , , ) a ( E , , ) if and only if Z7,\ f ( x , ) a(E,) < co. We then modify our definition of integral:
I I
I
3.1 .l. Definition. If there exists a subdivision a,, of X into a-measurable sets such that for a 2 a. we have Zuf ( x , ) a ( E , ) absolutely convergent for all choices of x , in En, and if limuZuf ( x f l ) a ( E , ) exists as a finite number (where we need only consider a 2 a,,), then Jxf d a exists and we definite Jx fda = limuzluf ( x , , ) a ( E , ) . The existence of Jx fda implies that
( e > 0,ae, a 2 ae):
jX,.fda
-
e5
xu
f(x,,)a(E,) 5 j x f d a
+ e.
For any particular 0 2a e the expression ZUf ( x , ) a ( E , ) is then bounded for all choices of x,, in E,. Suppose then that for a given subdivision a, Z u f ( x , ) a ( E , ) is bounded as the x,, range over E,l. Then the product f ( x J a ( E , ) is bounded, that is f ( x ) is bounded on all E , for which a ( E , ) f 0 . Let M ( f ; E ) = 1.u.b. [ f ( x ) , x on El and m(f; E ) = g.1.b. [f ( x ) , x on E l . We then show: 3.2. If for a given a, Zuf ( x , ) a ( E , ) is bounded as x , ranges over the
I 1;
E,,,then Z,,M( f E , , ) a ( E , ) < co,provided we define M (
1 f I; E ) a ( E )
3. =0
A RIEMANN-YOUNG TYPE OF INTEGRAL DEFINITION
if a ( E )
= 0.
233
Moreover, for all a' 2 0 with a' = {E,*'},the sums
Z, f(x,')a(E,') are absolutely convergent for all x,' in En' and bound-
ed as x,' ranges over E,,'. The convention that M(I f E)a(E) = 0 if a(E) = 0 might be justified by the fact that M(I f E ) a ( E ) = 1.u.b. f ( x ) a ( E ) for x on El, when a ( E ) # 0, and 1.u.b. f ( x ) a ( E ) , x on El = 0 if a ( E ) = 0. For e > 0, let x , on En with a(E,) # 0 be such that M ( f 1; E n ) 5 f(x,) e. Then for all m :
1;
I
1;
[I
I
[I
I
I
I+
5 C Iffx,,) I 4 E , ) a
+ eff(X).
Since Enf ( x 7 J a ( E n )is absolutely convergent it follows that ZnM( E,)a(E,) is convergent. Let 0' 2 a. If En, are the sets in a' subsets of E,, in a, then
5 C M(If n
1;
1;
1.f 1;
E,l)~(En)>
1;
since M(I f En,) 5 M(I f E,,) for all k and n, and C,a(Enk) = a(E,,). Then Za,f(x,,)a(E,,) converges absolutely for all xPlkin E,,, and is bounded. We remark that if for a fixed 0, Laf(x,)a(E,) is convergent for all x,, in E,, then the condition that Zo f(x,)a(E,) be bounded in the x , is equivalent to the condition that f ( x ) be bounded on each E , for which a(E,) # 0. Moreover, we have 1.u.b. [Zo f(x,)a(E,) for x , on E n ] = ZoM(f; En)a(En) and g.1.b. [Zaf(x,)a(E,) for x , on E n ] = Z p f A E , M E , ) , where Z , W f ; E,)a(E,) and q p f f ;E,) a(E,) are both absolutely convergent since they are dominated by ZnM(I f En)a(E,). We assume as usual that M ( f ; E)a(E) = m ( f , E)a(E) = 0 if a(E) = 0. As in the case of Riemann integration we can now define integrals similar to the upper and lower Darboux integrals.
1;
3.3. Definition. If f ( x ) is finite valued on X , and a ( x ) is monotone bounded on X, and there exists a subdivision a, = {E,,] of X of
a-measurable subsets, such that f ( x ) is bounded on each En, while Zao f ( x n ) a ( E n ) converges absolutely for all choices of x , on En,
234
VII. LEBESGUE-STIELTJES INTEGRATION
rr
then we define the upper integral fda = g.1.b. [Z,M(f, E n ) a ( E n ) for all u L a,] and the lower integral Jx fda = 1.u.b. [Z,m(f; E n )ar(E,) for all o 2 a , ] . Since the sums Z,M(f; E,)a(E,,) and Z,m(f; E n ) a ( E n ) are monotone in u, these integrals exist also as u-limits. Moreover, since for each 0 2u,
we conclude: 3.4. If f ( x ) is finite valued on X , and if a ( x ) is montone bounded on X , then necessary and sufficient conditions that Jx fda exist are (l), there exists a subdivision 0, of X into a-measurable sets, such that f ( x ) is bounded on each set E of go, and Snf ( x , ' ) a ( E , , ) converges absolutely for all x , in E n ; and (2) fda = Jxfdar. 3.4.1. Condition (2) can be replaced by the equivalent condition :
(2 a) g.1.b. [Z,w(f; E,,)a(E,) in 01 = lim,Z,w(f; E , ) a ( E , ) = 0, where w ( f ; E ) = 1.u.b. f ( x , ) - f ( x , ) for x l , x2 on El, that is, the oscillation of f on E. A third replacement of condition (2) comes from the fact that a Cauchy condition of convergence induces the existence of a u-limit, so that we have:
[I
1
3.5. Jgfda exists if and only if condition (1) of 3.4 is satisfied and if the Cauchy condition of convergence: (e > 0, ue, u 2 ue, u' 2 oe): I Z,f(X,,)"(EJ - Z u J ( X , , ' ) f f ( E n '1 )< e holds. We note that if u, = {E?,}is such that f ( x ) is bounded on each E n for which a ( E , ) # 0, then there exists a (T h o0, such that f ( x ) is bounded on each E of (T. For if E is any subset in u, such that a ( E ) = 0, and f(x) is not bounded on E, we can redivide E into the sequence of sets E * En' of x for which n 2 f ( x ) < n + 1, and x on E, which will
also be a-null sets. The idea of using subdivisions of Lebesgue measurable subsets of a finite interval as a basis for defining upper and lower integrals of bounded functions is due to W. H. Young [A general theory of integration, Phil. Trans. Roy. SOC. London 204A (1905) 21 1-252 (p. 243)]. In order to distinguish the integral just developed from the Lebesgue definition: L Jx fda (which assumes that f is ar-measur-
3.
235
A RIEMANN-YOUNG TYPE OF INTEGRAL DEFINITION
able), we shall label the integral of this section as the Y-integral: y Jxfda. We give some instances of existence of Y Jxfda. 3.6. If f ( x ) is an a-measurable step function on X , that is, if X = ZnE7,, where the E , are a-measurable and disjoint, and f ( x ) = c, on En, then Y Jxfda exists if and only if Z, Ic, la(E,)
I
I;
Note that the sums Zu f(x,,)a(E,,) which occur in the definitions of Js f d a can be expressed in the form Jxfoda, where.fu is the step function Z n f ( X 7 J x ( E , x;). 3.7. If f ( x ) is bounded and a ( x ) is monotonic nondecreasing on
b ] and if the Riemann-Stieltjes integral (T J,"fda exists, then the Y J:fda exists also and the integrals are equal. We recall that if a(x) is monotone on [a, b] and if 0 3 (a = x, < x, < ... x, = b ) , then [a,
1 fda
-b
[xi-l, xil)(a(xJ
= g.l.b.,[C,M(f;
- afxi-l)),
where [xi-,, xi]= x. 2-1 5 - x 5 x i . Similarly for J,"fda. Let crl be the subdivision of [a, b ] into a-measurable subs& consisting of the points a, x,, ..., xn-l, b and the open intervals (xi-l, x i ) , i = 1 ... n. Let M i = 1.u.b. [ f ( x ) for x on [xi-], xi]and Mi' = 1.u.b. [ . f ( x ) on fXi-1, xJ1. TJ.len Mi(.(Xi)
-
Lf(X,)(ff(Xi)
-
a(xi - 0 ) )
+ Mi'(&
+ 0)) +f(X,-Jf.fxi-l + 0 ) afxi-l)), C i M i M x J - 4q-J) L + 0 ) afxi + Mi'(& 0 ) + O))I, - "(xi-l
- 0)
-
and so
CJf(Xi)(.(Xi
-
-
- 0))
- .I(Xi-,
where a(a - 0) = .(a) and a(b + 0) = a(b). But the right-hand side of this inequality is the upper sum corresponding to the subdi-
236
VII. LEBESGUE-STIELTJES INTEGRATION
7
vision ul. Taking greatest lower bounds, we find that (T f f d a 2 Y fda. Similarly, (T J f d a 5 Y Jfda. Consequently, if u J f d a exists so that (T f f d a = u SfiEOr, then-Y J fda exists also and agrees with (T Jfda. Since theexistence of the N Jf fda implies the existence of the u Jffda, we also have: 3.8. If a is monotonic nondecreasing and the Riemann-Stieltjes
N J: fda exists, then the Y Jffda exists and the integrals are equal. Since when the N J: fda exists, the function f is a-measurable, it follows that the L Jf fda exists also. For then f will be bounded on a finite.set of nonoverlapping intervals and on the complement of this set the function a is constant, that is, f is bounded if one excludes an a-null set of [a, b ] . In view of the next theorem Y Jffda exists also, giving us another proof of the present theorem. 3.9. THEOREM. If f ( x ) is a-measurable, then the existence of the L J x f d a implies the existence of the Y Jxfda and conversely, and the integrals are equal. If the L Jxfda exists, and uy is a subdivision of the Y-axis with IoYl
12
yiafEk)
-
LjfdaI 5I
'y1
% I
where Ek = E[yk-,
=
I
Yk'xrrafEnk)
I I51 I
-x.ffx~~k)afEnk)
- x.ffx,,)afEnk)
(Ty
Consequently, if ulz 2 ( T ~ , then
I
jxfda I I
xffxnk)a(Enk)612
xffx&)afEnk)-
+ I x yk'a(Ek)
-
J x f d a 15
x
y,'afEk)
I
UY
UlX
I
'y
I a(X)*
UY
If we let I by I 0,it follows that Y Jx fda = L Jxfda. Conversely suppose that f is a-measurable and that Y Jx fda exists. --f
3.
A RIEMANN-YOUNG TYPE OF INTEGRAL DEFINITION
237
Then to prove that L S,fda exists, it is sufficient to show that for some subdivision au of Y with I av I < co we have
c Iv i I
< f5 u k l ) <
"fEIYk-I
UY
We take ay so that y k = k, for all positive and negative integers k. Since Y J f d a exists, there exists a a%- { E n f }n, = 1 ..., a subdivision E n f ) a ( E n f< ) 00. of X into a-measurable sets, such that Z n M ( The same type of inequality will then hold for any atf 2 a%. If Ek = E[k - 1 0, then M ( 1; Erik) 2 k - 1 and if k < 0, then M(I f I; Erik) > [ k 1, so that for fixed k>O:
If I;
If
CnM(lf1;
-
Ei&)afEnk)2 (k - 1) Cna(Enk) = (k - l)ff(Ek);
while for k < 0:
cnM(lf 1; Enk)a(Enk) If we take yk' = k
-
2k ly; I
1 I
EnafEnk)
=
I I
1 for k > 0 and ykf = k for k < 0, then 5
cM ( I f
1;
Enk)afEnk)
021
Then L Jxfda exists and by what we have proved above is equal to Y SX fda. We shall show later that if Y Jxfda exists, then f is a-measurable (see VII. 8.11). In case both f ( x ) and a ( x ) are bounded, it is sufficient to limit subdivisions to those consisting of a finite number of sets of X . Let If ( x ) 15 M for all x. Let 0%= { E n }be such that M ( f ; En)a(En)5 fda e.
jx
2
+
02
Since Z,,a(E,J = a(X),it follows that (e > 0, n,, n 2 n,) 2: a(Em) < e / M . Let E,, = Z: Em,and let a, E El, ..., En-l, Then
z,,.
Ic M ( f ,E M E ) - c M ( f , E h f E ) I 0.
Then
c 7
2 M ( f , E)a(E) 5 c M ( f , E)a(E) + 2e 5 jx fda + 3e. dl
sx sx
0
It follows that fda = g.1.b. [ZOM(f;E)a(E) for all finite a of XI. Similarly, - fda = 1.u.b. [ Z m ( f ,E)a(E) for all finite (T of XI.
238
VII. LEBESGUE-STIELTJES INTEGRATION
Since both of these bounds are limits in the a-sense, it follows that if J g f d a exists with f and a both bounded on X , then Jx fda = limu.Zof(x)a(E) where the a are limited to be finite subdivisions of X. For the existence of Y J,fda it also sufficient to limit subdivisions of X to consist entirely of Borel measurable sets. We have: 3.10. A necessary and sufficient condition that Y Jxf d a exist is that Iim,Z,f(x) a (E) exist for all subdivisions consisting entirely of Borel
measurable sets. If Y JAyfdaexists, there exists a a" such that ZuoM((f E)a(E)
1;
If 1;
If
I I;
1;
If I;
If 1;
EXERCISES 1. Suppose a ( x ) is a monotone bounded pure break function on X , so that all subsets E of X are a-measurable, and a ( E ) = ZxinE (a(x 0) a ( x - 0)). Show that 1, f d a exists and is equal to Zx f(x)(a(x 0) a ( x - 0)) if and only if Zxlf ( x ) I (a(x 0 ) - a ( x - 0 ) ) M . Show that L Sxf(m, M ; x ) d a ( x ) = :$ yd,u(y) where p ( y ) = a ( E [ f5 yl).
+
+ +
5.
239
INTEGRAL AS A FUNCTION OF MEASURABLE SETS
4. Properties of the Integrals
Since for a-measurable functions the two definitions of integration we have given in the previous section agree, it is possible to use either definition for the derivation of properties of integrals of such functions. As a consequence, for most of the theorems in this and the following sections, methods of proof alternative to the one used exist. For Riemann integrals, if the integral exists for the basic interval [a, b ] ,then it exists also for every subinterval [c, d] of [a,b]. Correspondingly, we can consider an integral on a measurable subset of X . This can be obtained by considering the function fE(x) = f(x) if x is in E and fE(x) = 0 if x is in CE, or ,fE(x)= f ( x ) x ( E ; x) where x ( E ; x) is the characteristic function of E. For the Y-integral, this is equivalent to considering for any subdivision (T of X which is finer than the subdivision (E, CE) determined by E only the terms of Lof(x,)a(E,), for which Ell are subsets of E. We have:
SX fda exists, then SB fda = SX f'(x)da(x) = Jayf ( x ) x ( E ; x) da(x) exists for every a-measurable subset E of X . If SE fda exists it will be the a-limit of sums Zuf(x,,)a(EE,) with x, in EE,,. If Jxfda exists, then (a) there exists a (T, such that for a 2 u,:ZoM( I; E,Ja(EJ < co. Consequently, if o = ( T ~- (E, CE),
4.1. If
If
then
2 (Wlf 1;
EE,)a(EE,)
+ W ( f1;
(CE)E,)ff((CE)E,))
a
7
GO
I;
and so ZuM ( )f EE,)a(EE,) < co. If Jxfda exists, then (b) g.l.b., C p ( f , E,)a(E,,) = 0. But this implies g.l.b.o Luw (f;EE,) a(EE,) = 0. Since the two conditions (a) and (b) are sufficient for the existence of the integral it follows, that J E ,fda exists, for all a-measurable E. The integral J E fda is then a function of f(x), a(x) and E, which we can bring into focus by writing I ( f , a , E ) = J E fda. We shall treat in succession, I ( f , a, E) as a function of E, o f f and of a. 5. The Integral as a Function of Measurable Sets
Corresponding to the fact that the Riemann integral is an additive function of intervals, we have : 5.1. If Jx-fda, exists, then
is an s-additive function of a-measurable sets. In particular if E = ZlIEn,with En, a-measurable and J E fda
240
VII. LEBESGUE-STELTJES INTEGRATION
sE
sE
disjoint then fda = Z n n fda, where the series on the right is absolutely convergent. Since SE fda exists for each a-measurable En, we have ( e 0, ane,;n 2 O n e ) :1 ~ a n f ( X , i ) a ( E , i) sEnfdff < e/2n, where un are subdivisions of En and consist of the sets ETii.We also have ( e > 0, U s , 0 2 u e ): Za f ( x i ) a ( E i ) - SE fda I < e. If (Te' = ae(Znone) and if a 2 ae', then a 2 a e and a * E n L ane, and the absolute convergence of the series involved yields
I
1
Then the series En JEn fda converges absolutely. Further, for the same a we have
This completes the proof of the theorem. The s-additivity of SE fda leads immediately to the finite additivity:
SAYfda exists and E , and E, are any two a-measurable subsets of X, then
5.2. If
We can also prove the following converse to the s-additivity theorem :
6.
24 1
L-S INTEGRALS AS FUNCTIONS OF INTERVALS
5.3. If E , are disjoint a-measurable sets with E = Zll,ER, if
fda
JE
1
exist for each n, and if LrlJEn f Ida < C O , then JEfda exists a i d is equal to Z nJ E n fda. We shall show later in VII.8.3. that if Jxfda exists, then f Ida exists also. To show that if E = ZTzE,,then fda exists, we need to show first that there exists a subdivision B of E such that ZoM( E)a(E) < co. The existence of fda implies the existence of (f da, so that for e > 0, there exists B,, = { E l l t } of E,, such that 'r5nM ( I f E,,Jff(E,J s S E n I d f f e P " . Then
Jxl
sE
If I;
sEn
sEn
+
If
1;
c' c M f I f 1;
Ell,)ff(E1,J 5
JE n
n
This gives the desired convergence for
B =
"n
(e
0, n, B
~
~
I SE f d a
2~ o,l r Be :) ~
~
n
-
5n
I f I da + e <
CO.
L',,B~,.Further, f(x,fl)a(E,lv)1 < 4 2 " .
Then for any subdivision B 2 u e = S,,B,,~, so that
Then
1
B
E,l = ufl 2 B , , ~ :
SE fda exists and has X,,sEnfda as its value.
We note that in the hypothesis of this theorem, the condition X l 1 SE (f da < co cannot be weakened to Zpl SEnfda < co. For take E,, disjoint a-measurable sets, and f ( x ) the step function f ( x ) = c,, on E71.Select c , ~so that c2,,a(E2,,)= - C ~ , + ~ ( E , ~ and + ~Lvl ) I c, 1 a(E,,) = CO. If EA = E,, + E271+1, then SE, fda = 0 for all n, but if n E = Z7,E; = Zp1Erlthen J E fda does not exist.
I
1
I
6. L-S Integrals as Functions of Intervals
In the Riemann-Stieltjes type of integrals we did not distinguish between open and closed intervals for the basic interval function nor for the interval function determined by the integral. For additive properties, intervals were permitted to be nonoverlapping, without being entirely disjoint. The Lebesgue-Stieltjes integral determines a function
242
VII. LEBESGUE-STIELTJES INTEGRATION
of sets, and when we consider integration over an interval, there may be ambiguity as to whether we are considering the closed interval [a, b ] , or the open interval (a, b ) or the half-open intervals [a, b ) and (a, b ] . For instance, if a and b are points of discontinuity of a ( x ) and if I = [a, b ] and I, = (a, b), then
so that the integrals agree only when the right side vanishes. We note, moreover, that the value of a at the points x = a, or x = b plays no role in the values of the integral, and what is more the value of JI f d a for the closed interval depends on the behavior of a outside of I. Since the additive property of J E f d a as a function of E depends on the disjointedness of the sets involved, to secure a corresponding property for intervals we must impose limitations. A simple way out is to restrict the word interval in this connection to intervals closed on the right and open on the left (or open on the right and closed on the left). If l i s so restricted then for a < c < b, we have fda J: fda = J: f d a , and even
JI
+
Observe that there are two ways of obtaining J: f d a on (a, b ] from J x f d a : (1) Jlfd. = Ssf,da where f,(X) = f ( x ) x ( I ; x ) , or f,W = f ( x ) for a < x 2 b, and f , ( x ) = 0, for x 5 a or x > b ; (2) Jf fda = Jx f d a l , where a,(x) = a ( x ) for a < x < b, a , ( x ) = a ( a fO) for x 5 a, and a , ( x ) = a ( b 0 ) for x 2 b.
+
7. Absolute Continuity
Iff is bounded on X and Js fda exists, then for any a-measurable set E J E fda IS M ( f E ) a ( E ) , so that J E fda 4 0 as a ( E ) 0. This is a continuity property of set functions. We define:
I
I 1;
--f
7.1. Definition. A set function ,4 on a class @ of subsets E of a fun-
damental set X is absolutely continuous with respect to the positive set function a on CF, if limax(E)-tO P(E) = 0. The word " positive " includes positive or zero. We have:
7.
243
ABSOLUTE CONTINUITY
7.2. If J,fda
exists then the set function J E f d a on a-measurable sets is absolutely continuous with respect to the function a(E). For if J,fda exists, then there exists a subdivision 0 of X such that ZoM( If Ek)a(Ek) 0 choose n so that ZEnM(If E k ) a(Ek) < e. Let E' = T:-' Ek and E" = Z:Ek, and let E be any a-measurable set. Then J E . f d a 5 2: M ( If E,)a(Ek) < e, since M ( If E . Ek)5 M ( f Ek) and a(Ek . E ) 5 a(Ek). On the other hand, f(x) is bounded on E', since it is bounded on each Ek with k < n. Consequently lim,(E)-toJ E . E' fda = 0, or (e > 0, d,, a ( E ) 5 d,) : I J E . f d a I 5 e. Consequently, if a(E) 5 d,, then
1;
1;
1;
I
1;
I 1;
If ,B is a set function on a class E of sets, which is absolutely continuous with respect to a positive set function a on Q,then a(E) = 0 implies p ( E ) = 0. This last condition: " a(E) = 0 implies ,B(E) = 0" is sometimes used as the definition of absolute continuity o f set functions, though it is difficult to see that such a condition is a continuity condition. However, under special circumstances it does imply the absolute continuity property as we have defined it. We prove: 7.3. If ($ is an s-algebra of subsets of a set X ; if a(E) and P(E) are
s-additive and positive set functions on E; if p is finite valued; and if a(E) =O implies p(E)=O, then ,B is absolutely continuous relative to a . The proof proceeds contrapositively. If ,B is not absolutely continuous, then there exists e > 0, such that for each of a sequence of constants el, > 0 with C,le,l < co, there exists a set E , with a ( E n ) < e , and ,8(E,) > e. If E m = CZEn, then a(Em)5 ZZa(E,), so that a(nmEm) = lim,a(E'") = 0. On the other hand, ,B(E") I P(E,) > e for all rn. Since ,8 is finite valued and s-additive, it follows that lim P(E") m
=
p(nE m ) > e. m
Then for Eo = nmEm, we have a ( E o ) = 0 but P(E0) > e. It can be shown (see IX.4, Exs. 1 and 2) that the condition that ,B be positive or zero on Q can be dropped. We note also the following:
Q is an s-algebra of sets; if a is an s-additive, and finite valued, positive or zero function on Q, and if ,B is an additive, finite valued set function on Q which is absolutely continuous relative to a , then P is s-additive. 7.4. If
244
VII. LEBESGUE-STIELTJES INTEGRATION
Let En be disjoint sets in G with E = Z7,E,. Denote by E, the set q = (n,, ..., ni, ..., n m ) , any finite set of integers. Then since a is s-additive and finite valued lim,a(E - E,) = lim,(a(E) a(E,)) = 0. Since P is additive P(E - E,) = P(E) - Z,P(EJ. Then the absolute continuity condition gives lim,(P(E) - ZqP(En))= 0, so that by 1.6.2. Znl /3(En) < co, and Z,?P(E,J = P(Z,iE,L) = P(E).
2gn,,where
1
EXERCISE Show that the upper and lower integrals when they exist as g.l.b.u(ZuM(h E,)a(E,)) and l.u.b.u(Zum(f; E,)a(E,)) are completely additive as functions of a-measurable sets. Are they also absolutely continuous with respect to a? 8. Properties of the Integral I f f , a , E) = sEfda as a Function o f f
For a fixed a , and a fixed a-measurable E, the existence of SE,fda determines a class of functions f. We shall limit ourselves to the case when E = X, since
As an immediate consequence of the definition of the Y-integral and the linear properties of limits, we have: 8.1. If J,,. fdu exists and c is any constant, then Jx (cf)da exists and Jx Cfda = c Jx fda. 8.2. If Js,f,dor exists and Jxf2da exists, then SXf&. and ss (A + f * ) d a = J x - f @
Js ( f , +f , ) da exists
+ These two theorems assert that the class of functions integrable
with respect to a fixed function a is linear and that the integral is a linear functional on this class, We have further : 8.3. If
Jx f d a exists then
I
Jx- If da exists and
I Js.fda 15 Jx 1.f
I dor. I f 1;
For the condition : there exists a u,, such that if u 2 uo,then ZuM( E)a(E) < co, is really a condition on If Further, (IJ( f E) 5 E), and so g.l.b.u2uw(f; E ) a ( E ) ) = 0 implies g.l.b.u(2um( E)a(E)) = 0. Then the conditions for the existence of Jx da are fulfilled. The inequality Zuf ( x ) a ( E ) 5 Su f ( x ) a(E) for all 0 induces the corresponding inequality on the integrals.
I.
1
I
I 1;
I
I
If 1;
If 1
8. I f f , a, E)
= J E fda AS A FUNCTION OF
f
245
This theorem asserts that the class of Y-integrable (and so L-integrable) functions has the absolute property. If we adjoin the linear property, we find that with f, the functions f = ( I f f ) / 2 and f - = f - f ) / 2 , that is the positive and negative parts of f are also integrable. This fact is sometimes made the basis for a definition of integral, the integral being first defined for functions f 2 0 on X , and then extended linearly to functions of arbitrary sign. Further, if f l and f , are integrable, then fi V f i = (fi f , f l - f 2 1)/2 and f , A f , = (fl f , -f , /2 are also integrable, that is, the integrable functions form a linear or vector lattice, with f, ( 5 )f , equivalent to f , ( x ) 5 f , ( x ) for all x (or all x except for an a-null set). The fact that any L-integrable function is also absolutely integrable shows that the class of L-integrable functions does not include all improperly Riemann integrable functions, since some of these are not absolutely integrable. For instance, ( l / x ) sin ( l / x ) dx exists as an improper integral since lime+oJ: ( l / x ) sin (l/x) dx exists, but J: (l/x) sin ( l / x ) dx = a.Then (l/x) sin ( l l x ) is not L-integrable on [0, I ] . In the existence of integral after the manner of Lebesgue, we have noted that in addition to a-measurability, a single convergence condition is sufficient to guarantee the existence of the integral. A generalization is the following: +
(1 1
+
If,
I+
+ +I
1)
Jt
I
I
8.4. If f 2 0 on X , and Jx fda exists; if further f, is a-measurable and IS f except for an a-null set, then Jx flda exists and Jxfida 5
If,
I
I
Sxfda. For if for (T 2 ao:ZuM(f;E)a(E) < 03, and (T includes the a-null set Eo for which [ >f, then ZoM(Ifl E)a(E) 5 Z,M(f; E)a(E) < co. If we add the a-measurability off, we conclude that Jxfida exists. The inequality between the integrals is a direct consequence of the definition of integration in terms of o-limits. The class of integrable functions does not have the multiplicative property, that is, if f , and f , are integrable, then f , -f , is also integrable. This is due to the fact that iff, and f , are both unbdunded, the unboundedness off, f 2 might be of higher order than that permissible for integrability. To obtain a theorem, we restrict one of the functions.
If,
fi
1;
8.5. If is a-measurable and almost bounded, and if Jxf2da exists, then Jx f , * f2da exists and f , f,da 5 M Jx f 2 da,, where M =
Sx I
g.1.b. (M’ such that cr(E[[f, > M ’ ] ) = 01.
I I
VII.
246
LEBESGUE-STIELTJES INTEGRATION
For if Jx f,da exists, there exists a a, such that if a 2 a, then ZaM(If , 1; E) a(E ) < 00. Since f , is almost bounded let M , and Eo be such that a(E,) = a ( E [ ( f , > M I ] )= 0. Then if 0 is finer than a,as well as (E,, C E J , we have z U ~ ( *f,!, l f ,E ) ~ ( E < ) ~,~,,M(!f, E ) < co. Further if the a-measurable set E is disjoint from E,, and if f , is bounded on E, then
I
4fl3,; E ) a ( E ) 5 M,w(f,; If then a is finer than
go
1;
E)a(E)
+ WIf,1;
E ) a ( E) u ( f , ; E ) .
as well as (EO,CE,), then:
2w. ( f , * f ,;E ) a ( E )5 M , C (f,;E )a (El +c“WIf,I ;E ) a ( E ) (fi;E) Lu
5 M , ~ a u ( f ,E)cr(E) ;
+ I.u.b.(w(f,;
E ) , E o n a) - K
I I;
where K = ZUM( f , E)a(E). Now the Lebesgue procedure of subdivisions of the Y-axis yields subdivisions a into a-measurable sets such that for e > 0: ~ ( fE;) , < e for all E of a. It follows that g.l.bv0 [zaOJ(f,f,; E ) a ( E ) ] = 0 and f,- f,da exists. Since for any subdivision a 2 (E,, CEO),where a (E,) = a ( E [ f , > M , ] ) = 0 we have
ss
I
I
I I
1 I
it follows that Jx f , f,da S M , Jx f , da, for all M , such that a ( E [ l f , > M,]) = 0. Consequently, the same inequality holds if M is the greatest lower bound of M , for which a ( E [ f , > M , ] ) = 0. In case we know that f , is also a-measurable, then the theorem follows from the fact thatf, * f , is also a-measurable and f,(x)f,(x) 5 M , f , ( x ) except for an a-null set, M , If, being L-integrable, so that f , - f , is also L-integrable. For L-integrable functions, we note :
I
1
I I
I
I
I
I
x except at most an a-null set, and if Jx fda = 0, then f ( x ) = 0 for all x,except at most an a-null set. Assume first that f ( x ) 2 0 for all x , and let E,L= E [f > l/n]. If f,(x) = x(E,; x ) / n , then 0 5 f , , ( x ) 5 f ( x ) for all x,and is a-measurable. Now 0 5 Jxf,da 5 Jx fda = 0, or a(E,,)/n = 0, and so a(E,) = 0 for all n. Since E [ f > 01 = Zr,E,, = Zr L E[ f > l / n ] , it follows that a ( E [ f > 01) = 0, or f vanishes except for an a-null set. In case f 2 0, except for an a-null set, we let f ( x ) = f(x) if f(x) h 0 and 8.6. If f ( x ) is a-measurable; if f ( x ) 2 0 for all
8. I f f , f f , E)
=
SE f d f f
AS A FUNCTION OF
247
f
= 0 if f(x) 5 0. Then l x f d a = Jx f d a = 0, so that f vanishes except for an a-null set, and the same thing holds f o r 5 We take up next the interchange of limit and integration and consider first the case when the sequence of functionsf,(x) is uniformly bounded on X.
f(x)
8.7. If (a) f,,(x) are a-measurable and uniformly bounded on X , so
that there exists an M such that I.fn(x) 15 A4 for all n and x, and if (b) lim,,f,(x) = f ( x ) excepting at most an a-null set, then limn Jx f,,da = fda, as a matter of fact lim,, - f da = 0, and limn J E f,da = J,fda for all a-measurable sets E. If f , are a-measurable and uniformly bounded, then f is a-measurable and almost bounded so that f,, and f are L-integrable on X . Then Jxf,da - Jx.fda IS Js - f da. If we set E n , = E [ l f r ,f > el, then limp(E,J = 0 since lim,,f,(x) =f(x) except for an a-null set and f , are a-measurable. Now
sx
I
sx If,
1
,fI
I
I
For a given e, > 0, we select e so that ea(X) < e,/2, and then ne0 so that for n 2 n, we have 2Ma(E,,) < e,/2. Then
This theorem includes the Osgood theorem for monotonic a as a special case (see 11.15.6). If we examine the proof of this theorem, we note in the first place that we have used the convergence off, to .f only to assure us that lim,la(E,,e) = 0 for all e > 0. It then would have been sufficient to assume that f,, + f ( a ) , or f , , converges to f relative to a. Further the uniform boundedness of the sequence f,, actually leads to the uniform absolute continuity condition: lima(E)+oJ E f,,da = 0 uniformly in n. This leads to the stronger theorem: 8.8. If (a) f,(x) andf(x) are a-measurable; if (b)
Jay f,da exists for all n ; if (b') J l y j d aexists; if (c) f , , +,f(a) ; if (d) J8f,,da are uniformly absolutely continuous relative to a ; then limn Jx If, - f da = 0, and so lim, JEf,da = J,fda for all a-measurable sets E.
1
248
VII. LEBESGUE-STIELTJES INTEGRATION
The proof of 8.7 applies verbatim, if we show that if J E f , { d aare uniformly absolutely continuous, the same is true of J E da. For if (e> 0, d,, a ( E ) < d,, n ) Je.f,,da < e, then if E = En E;, where EZ are the points of E for which f,, 2 0, and E; the points of E for which f,, < 0, then a(E;) < d, and a ( E J < d, and
:I
1,f7, I
+
I
The hypothesis of 8.8. can be weakened further by dropping (b’) the assumption that J,fda exist, because we can show: 8.8.1. The conditions (a), (b), (c), and (d) of 8.8. imply the condition
(b’) that JAl fda exists. Of the various convergence conditions which together with ameasurability give integrability, we can use VII.2.4. to prove that if E,, = E [ I MI and if JEY tia is bounded in M , then Js fda exists. We have
If
If 1
I
J E .w
I f I ch /&
+ [, If, I da 5 .I’RM I f;, J’I da + Jn I f,, 1 da. --
fI
1.1
-
Of the two expressions on the right-hand side of the inequality, the first, by VIT.8.8 converges to zero as n a,for each M , since J E y l f Ida exists. Consequently, our theorem is proved if we can show that the sequence I., Ida is bounded. Now --f
If, J., I J,! I da i,I f , I da + =
.If
I I da, ftl
where E,, is again the set E [ j f 15 M I . Then limIf+r, a(CEl1)= 0, since f(x) is assumed to be finite valued. By the uniform absolute continuity of JE,f,p7aand consequently that of JB Ida, for any k > 0, we can find M = Mk so that Jp,3jf f,, da I k for all n. On the other hand, for this M
1 1
If,
Since the first term on the right-hand side converges to zero, we can for e > 0, find n , such that for n 2 n ,
8.
r(A a ,
E)
=
J E f i a AS
I
A FUNCTION
+
OF^
249
+
Consequently for n 2 n,: If,, da 5 e Ma(EIl) k , where the right-hand side is a constant independent of n. Since there are only a finite number of n < n,, it follows that Js da is a bounded sequence. For the sake of emphasis and completeness we restate the fundamental convergence theorem :
If, I
Jx f,,da exists for all n ; (c) f,, +f ( a ) , that is limnu(EIIf,L- f > el) = 0 for all e>O; and (d) Jflf,,da are uniformly absolutely continuous relative to a, then Jx fda exists, and lim,l Jx If,, - f du = 0, so that also Em,( J E f,da = J e f d a for all a-measurable E. As a corollary, we have the convergence theorem due to Lebesgue: 8.8.2. THEOREM. If (a) f , / ( x )and f ( x ) are u-measurable; (b)
I
I
8.8.3. If (a) f,,(x) and f(x) are a-measurable; (b) limllflL(x)=,f(x) except at most an a-null set; (c) there exists an L-integrable function g ( x ) 2 0 such that fJx) 15 g(x) for all n and x , then limn Js f,,da = Jx fdu. For if f,,(x;) 15 g(x) for all n and x, then Jxf,da exists and J E If,, da 5 J E gda for all a-measurable E, so that J B f , da are uniformly absolutely continuous. For the case when the sequence J;,(x) is monotone in n, we have the following simpler convergence theorem :
I
I
I
n and x;if (b) limnf,,(x) =f(x) so that f is also a-measurable; and if (c) the sequence Jxf,,da is bounded, then Js f d a exists and limn Js.f,da
8.9. If (a) f , ( x ) are a-measurable with ,f,l(x) Zf,,+,(x) for all
Jx fda. It is sufficient to demonstrate this theorem for the case when f , , ( x ) 2 0 for all n and x, the general case results by applying the special case to the functions f , , ( x ) - f,(x). We show that JBY fda is bounded in M , where E,, = EIO 5 f 5 MI. Now limn J B M f,dn = J E M fda since the f f l are uniformly bounded on E l f . Also =
where K = l.u.b.,L(Jxf,,da). Then J E M fda 5 K for all M , and J.y fda exists. Sincef ( x ) 2 f , ( x ) for all IZ and x, Lebesgue’s theorem VII.8.8.3 applies and limn Jxf,da = Jx fda. The condition (a) of the hypothesis of this theorem can be replaced by f,,(x) S J , + ~ ( X )for all n, and all x except for an a-null set.
250
VII. LEBESGUE-STIELTJES INTEGRATION
Moreover the monotonic nondecreasing sequence f, can be replaced by a monotone nonincreasing sequence, whose integrals are bounded below. As a corollary to this theorem, we have: 8.10. Fatou’s Lemma. If f n ( x ) are L-integrable on X with respect to
a, and such that &, Jx f n ( x ) d a > - co, and if there exists an L-integrable function g ( x ) such that f n ( x ) 5 g ( x ) for all n and x , then E,,Jdy,fn(x)da(x) 5 Jx- limnf , ( x ) d a ( x ) . Let gnL(x)= 1.u.b. [ f , ( x ) for n 2 m ] . Then the functions g,(x) are a-measurable and f,(x) 5 g,(x) 5 g ( x ) or 0 5 g ( x ) - g,(x) 5 g ( x ) - f , , ( x ) for all m and x . Since g ( x ) -J;,(x) is L-integrable, g ( x ) - g,(x) will also be L-integrable, and so will g,(x) = g ( x ) ( g ( x ) - g , ( x ) ) for all m. Further, J,g,da1 Sayfnda for n L m, and consequently
Since the g,(x) are monotonic nonincreasing in m, converge to ihrL f , ( x ) , and the integrals JXg7,,(x)da(x)are bounded, it follows from the preceding theorem that Jglim,g, ( x ) d a ( x ) = J x G n f n ( x ) d a ( x ) exists and is equal to lim, Jxg,(x)da(x). Then
The lemma has an obvious counterpart involving the least of the limits. We are now in position to prove that:
.
8.1 1. T HEO RE M The Y-integral definition and the L-integral de-
finition lead to the same class of integrable functions. In view of VII.3.9 it is sufficient to prove that if Y Jx fda exists, then the function f ( x ) is a-measurable. Since the Y Jxf d a exists the upper and lower integrals fda and Js fda exist, are finite and equal to each other and to Y Jxfda. We can then determine serially, a sequence of subdivisions uk such that uk+l2 uk, and
sx
lim 2 M ( f ; E ) a ( E ) k
a .
=
lim k
o
m(f; E)a(E) = J x fda.
8. I f f , a, E)
= J E f d a AS A FUNCTION OF
f
25 1
Let Eknbe the sets in ok with a(Ekn)# 0. Let g k ( x ) = Z n M ( f ;Ek,) X(Ekn;x ) and hk(x) = Z n m ( f ;E,,)x(E,,; x ) , each function being zero on X - zl,lEkll,an a-null set. Then g,(x) and hk(x) are a-measurable, and gk+l(x) 2 gk(x) 2 f ( x ) 2 hk(x) 2 hk+l(x)for all k and all x except an a-null set. Since the g,(x) and hk(x) are step functions which are a-measurable and satisfy the integrability condition for such functions it follows that n
M ( f ; Ek,)a(Ehn)
IX
=
1, gk(x)da(x) 2 s x f ( x ) d a ( x )
hk(x)da(x)
= n
m ( f ; Ekn)a(Ekn)'
Let lim,g,(x) = g ( x ) , and lim,h,(x) = h ( x ) , each except for an a-null set. Then g ( x ) 2 f ( x ) 2 h ( x ) except for an a-null set. Moreover, by VII.8.9,
1, gda
= lim k
sx
gkda = s x f d a
= lim k JX
hkda =
Ix
hda.
Then Jx (h(x) - g ( x ) ) d a ( x ) = 0, with h ( x ) - g ( x ) 2 0, except for an a-null set. Since h ( x ) and g(x) are a-measurable as the limits of a-measurable functions, it follows from VII.8.6 that h ( x ) - g ( x ) = 0 except for an a-null set. Now h ( x ) 5 f ( x ) 5 g ( x ) except for an a-null set, so that f ( x ) differs from the a-measurable functions g ( x ) and h ( x ) by at most an a-null set, and is therefore also a-measurable. Consequently, L J,fda = Y J,fda, the two definitions of integration lead to the same class of integrable functions and are equivalent. Returning to the basic convergence theorem VII.8.8.2, it is natural to inquire whether the uniform absolute continuity of the integrals JEf,da is necessary. We have the following theorem of converse type: 8.12. THEOREM. If for the functions f , ( x ) and f ( x ) on X , the in-
tegrals Jx f , d a and J,fda exist in such a way that for every a-measurable set Ewe have limn J E f n d a = J E fda,then the integrals JE,f,daare uniformly absolutely continuous relative to a. In the proof of this theorem we use the following lemma which plays a role in the space I' of sequences {a,} whose sums are absolutely convergent: znI a , I < co. 8.13. Lemma. Suppose a,,
and alLare real numbers such that (a) Z, [ a m n( < c o for all m, and Zn1 a,, I cco; and (b) limmXoa,,, =
252
VII. LEBESGUE-STIELTJES INTEGRATION
Zoa,,for every set o (finite or infinite) of the integers: u = n,, ..., nk, ...; then Z,$ a,,, converges uniformly in m. Because of the condition (b), it follows that limnznnl,, = a,Lfor each n. Also because of the inequality Z,, a,,, 5 Z,, a,,, - a,, Z,, a, it is sufficient to prove the lemma for the case when a,, = 0 for all n. We proceed by the contrapositive method. We assume that lim, a,,, = 0 for each n, and that Sm a,,, are not uniformly convergent in n, and show that there exists a set of integers o such that lim,Zaa,,, f 0. Then there exists an e > 0 such that for every integer k, there exists nk > k, and m, > k such that Zn"=n,(amk,,I> e. We proceed to establish the existence of a sequence of disjoint finite sets of integers ol, 0 2 ,..., nk, ... and an increasing sequence of integers mk such that
I
I
I
I
I
I
I
I
c
I+ I I
I
a,,,,
... +ak--l
a1 + a 2 +
I
I < e/*
(1)
I c altyj I > e/2
(2)
ak
Suppose the finite sets oI ... ok-lhave been determined. Since lim,a, = 0 for each n, inequality (1) will hold for all m > some mk'. By the nonuniformity condition on Zln am, we can select mk > mk-l and mk' and n, > all the integers in so that a,nk,,> e. Then there exists an integer itk' so that
I
1,
Zz,,I
1
We select ok as that subset of the integers between nk and n,' for which a,,E,,are all of the same sign and such that "ra,amk,, > e/2. It follows that if a = 3riuz, then
I
1
# Zaa,, = 0, and so the same holds for limlnZaum,L. Then lim, Soumkn By using the iterated limits theorem 1.7.4, we can conclude further that under the hypothesis of the lemma, lim, Z,l a,,, - a, = 0, which in the language of linear spaces says that in P, the space of absolutely convergent series, weak convergence implies strong conver-
1
I
8. Z(f, a , E )
= J E f d a AS A FUNCTION OF
f
253
gence. (See S. Banach, " Theorie des Operations Lineaires," Warsaw, 1932, pp. 137-139). Further, the lemma can be strengthened to read:
1
I
8.14. If (a) Z,( alrr,,< co for all m, and if (b) lim,Zoa,,, every set a (finite or infinite) of the integers, then Ztl
I
uniformly in m,lim,nu?n71 = a,, is in
I' (Sll1 a,, I < 00) and
exists for
I converges
For the existence of lim,,,2'oarn,1is equivalent to
Since the double sequence (ml,nz,) has sequential character, the proof for the single sequence case can be adapted to show that Z,, a,,l,,ramZn converges uniformly in m, and m2,that is ( e > 0, k , k 2 k,, m,,m,):c:=~ am,,, - amz,l < e, and consequently
I
I
I
I
If we also take k 2 k ' , so that Z:=_, I a,,,z7l 1 < e, then for k 2 the larger of k , and k,' and every m,: Z:=,< 1 a,l ,I 1 < 2e, which implies the uniform convergence of Z?,I a,,l I in m. The convergence of L',,I a,, I follows then from the iterated limits theorem 1.7.4. Returning to the proof of our theorem, the assumptions that limn f,da = J E f d a for all a-measurable E and the fact that J E f,$a are s-additive functions of E imply that for any sequence {E,} of disjoint a-measurable sets
sE
lim '
c m
f,da
JB
=
lim
m
lT7 f,,da 1, =
-mEm
m m
fda
=
c in
fda. Em
Then for any fixed sequence E,, it follows that for any subset a of the integers m, we have
Applying the lemma, we conclude that : 8.15. If limn J E f,,da =
J E f d a for all a-measurable sets E, then for any sequence Em of disjoint a-measurable sets, the series Zlll J E m f,,da I converge uniformly as to n.
I
254
VII. LEBESGUE-STIELTJES INTEGRATION
We complete the proof of our theorem by showing that if J E f,Lda are not uniformly absolutely continous with respect to a , then for some sequence of disjoint sets Em, the series Em J E f,,da [ do not m converge uniformly as to n. If J E f,da are not uniformly absolutely continuous, and if for e > 0, d,, = 1.u.b. of d such that a ( E ) < d implies J E f,,da < e, then for some e > 0, g.l.b..d,, = 0. Consequently, there exists e > 0, such that N , d imply the existence of I I >~ N ~, and ~ ENd with a(ENd)< d and SEnd f,,, da > e. Take N = d = 1, and find n, and El so that a ( E , ) < 1 and J E f,,, da > e. Let d, be such that a ( E ) < d, implies J E f r da z < e/2, which is possible since JBfR1 da is absolutely continuous. Select n2 > n, and E, with a(E,) < d,/2 so that SB,Jfn,du > e. Determine d, < d J 2 so that a ( E ) < d, implies SE f , da < e/2. At the kth stage we have determined n, < n2 < ... < nk-l and dl ... dk-, with 0 < di < di-,/2. We then find Ek and nk such that a(Ek) < dk-,/2, nk > nk-l,and SEk f,,da > e. Now the sets Ek are not necessarily disjoint. We therefore set
I
I
I
I
I I I
I
I I
I
I
I
I
1
a,
Ek' = Ek - Ek ( C Em) = Ek - Ek". m-k+l
Then the Ek' are disjoint and
c E,,) 5 W
a(Ek") 5 a(
m=k+l
W
W
a(E,) 5 k+l
C
m=k+l
c 112" W
d,-,/2 < dk
= dk.
m-k+I
It follows that
I
I
Consequently, the series Zm J E ' f n kda are not uniformly convergent in k, since for every k : J e k , f,, da I > e/2. Then Z7n m' f , , da is not uniformly convergent in m either, giving us the contradiction we sought.
I
"1
SE
Supplementary remarks: We note that the proofs of the lemma and of the theorem depend on the fact that we are dealing with sequences of elements of the space of absolutely convergent series, and sequences of integrable functions, that is they are of sequential character.
9.
DIRECTED LIMITS AND L-S INTEGRATION
255
Further observe that the statement: there exists e > 0 such that N , d imply the existence of n, > N , and EN, such that “(Ex,) < d and I J B N d f,, da > e, is the negative of the statement: (e > 0, n,, d,, n 2 n,, a ( E ) < d,) :I J E f , d a I 5 e, or lim J E f,da = 0, where the limit is the double limit n + coy a ( E ) -+0. We have consequently shown:
1
I
8.16. If for all disjoint sequences {E7n}of a-measurable sets C7n IJEmf,da
are uniformly convergent in n, then lim n,a(E)+( ~
0
1, f,da
=
0.
)
This might have been expected since it follows from considerations in connection with the iterated limits theorem, that if for a directed set Q and a sequence of functions { f , l ( q ) }such that lim,f,(q) exists for all q and limgf,L(q) exists for all n, then the existence of the double limit: lirnql,,f,,(q) implies and is implied by the uniformity of lim,f,(q) in n. An examination of the proofs shows further that only the properties of SEfda as a function of measurable sets, and not the fact that they are integrals are used. If we use the supplementary statement to the lemma (V11.8.14), it can be verified that the following theorems hold :
p,,(E) are finite valued and s-additive on &, and such that lim,P,(E) exists for all E of &, then Zrn1 p7,(Ern)I is uniformly convergent in n for each sequence of disjoint sets E, in &. Further Crnp,(Ern)are also uniformly convergent in n, so that p(E) is also s-additive on &.
8.17. If & is an s-ring of sets E of a fundamental set X , if
p,l(E) are finite valued and s-additive on the s-ring E of sets E of X ; if lim,~,(E) exists for all E of 6; if a(E) is s-additive and positive or zero on Q; and if P,(E) are absolutely continuous relative to a ( E ) for each n ; then p,,(E) are uniformly absolutely continuous relative to a. For more sophisticated proofs of theorems of this type see S . Saks: On some functionals, Trans. Am. Math. Soc. 35 (1933) 549.
8.18. If
9. Directed Limits and L-S Integration
We discuss briefly, the extension of the convergence theorems of the last section, when the sequence of functions f n ( x ) on X is replaced
256
V11. LEBESGUE-STIELTJES INTEGRATION
by a set of functions f,(x), q on the directed set Q. An examination of the proof of the basic convergence theorem VII.8.8.2 shows that it can be adapted to prove the following: 9.1. Assume that Q is a directed set of elements q. If (a) the functions &(x) and f ( x ) are a-measurable; if (b) Jxf,da exists for each q ; if (c) f , +f ( a ) , that is, lim,a(E[IJ~ - f > el) = 0 for each e > 0;
I
and if (d) the integrals JE.fpdaare ultimately uniformly absolutely continuous in the sense that there exists a qo such that J E J ~ d aare uniformly absolutely continuous on the set of q for which qRq,, then JA-fda exists, lim, Js If, - f 1 du = 0, and so lim, JEf,da = J E f d a for all a-measurable subsets E of X . Note that the ultimate uniform absolute continuity of the integrals J e f q d a implies the same property for J E f , Ida, since
1
where E,+ = E [f , 2 01 and E,- = E [ f , 5 01. Further, that in the demonstration of the existence of fda, we can prove and need only the boundedness of J.\ fqda in the sense that there exists an M and a qo such that if qRq,, then f,da IS M . In addition, a careful examination of the proof of the convergence theorem shows that condition (d) that J E fQdabe ultimately uniformly absolutely continuous can be replaced by (d’) : limq,a(E)+U Jk; f,da = 0, which in case Q is not the sequence of integers N in their natural order may be weaker than (d). We also call attention to the fact that in the proof of this theorem, the a-measure convergence of f, to f , rather than pointwise convergence plays a role. In order to obtain a theorem in which we start with the pointwise convergence of f , ( x ) to f ( x ) , we need to add additional conditions so as to secure also the a-measure convergence. We recall that in the case where Q is the sequence of positive integers pointwise convergence implies a-measure convergence. This suggests :
I
1 I
JAx
9.2. If Q is a directed set of sequential character (where there exist sequences q,, cofinal with Q); if (a) fb(x) are a-measurable; if (b) lim, f , ( x ) = f ( x ) except for an a-null set; if (c) Jxjhda exists for
each q, and if (d) J E f g d aare ultimately uniformly absolutely continuous (or alternatively if limp,or(E)+,, J E f,da = 0); then Jxf d a exists, lim, JA -J da = 0 and lim, J E f,da = J E f d a for all a-measurable E.
If,
I
9.
DIRECTED LIMITS AND L-S INTEGRATION
257
For if Q is a directed set of sequential character, and if a, is on Q to real numbers, then limp, = a if and only if for every monotone sequence (q,,Rq,,-, for all n ) cofinal with Q we have lim,pq, = a (see 1.5.1). Suppose then that {q,,) is a monotone sequence cofinal with Q. Then lirn,fQn(x)= f ( x ) except for an a-null set, so that f is a-measurable. Further the hypotheses for the sequential convergence theorem will hold forfQn(x)so that Jsji-lor exists, and lirn?,Jx- If 4, - f da = 0. Since this is true for every monotone sequence q,, cofinal with Q, it follows that lim, J If, - J I da = 0, and so lim, J E J p d a= J E fda for all a-measurable E. The converse type of theorem in this setting reads:
I
9.3. If Q is of sequential character and if Jx-f,da and J x j d a exist
in such a way that lim, J E f q d a= J,fda for every a-measurable set E, then limq,a(E)+O JE.fpda= 0. For if q,, is any monotonic sequence cofinal with Q , then by the sequential case where Q is the sequence of positive integers, limn,a(E)+O JETg,da = 0. Consequently, if limnLa(Em)= 0, then lim,,ll J E m f g n d = a 0. Now the product set (Q, E) is a directed set of sequential character if we define order by the condition that: (ql, E,)R(q,, E,) is equivalent to q,Rq, and a(E,) 5 a(E,). Then limq,ax(E)+O j E jida = 0. We note that it is not always possible to replace the conclusion by the condition that the integrals J E f g d a be ultimately uniformly absolutely continuous. This indicates that the double limit condition limq,a(E)+,)J E f , d a = 0 is the natural one for these convergence theorems. In the set function setting, the last theorem becomes: 9.4. If Q is a directed set of sequential character; if 6 is an s-ring of
subsets of a fundamental set X , if P,(E) are finite valued and s-additive on 6 and such that lim,P,(E) = P(E) for all E of G, then limp,mCF=m P,(E,J = 0 for all sequences Em of disjoint sets in 6, and p(E) is also s-additive on G. If moreover a ( E ) is s-additive and positive or zero on (3, and if the P,(E) are absolutely continuous relative to 2, then limq,a(El+o P,(E) = 0. The conclusion that limli,"L .Y;P=,, P,(E,) = 0 follows as above from the case when Q is the integers in their natural order. By applying the iterated limits theorem in its alternative form [if lim,.f(p, q) exists for all p of P,and if lim j(p, q ) = g(q) for all q of Q in such
I
I
I
I
258
VII. LEBESGUE-STIELTJES INTEGRATION
a way that 1impJf(p, q) - g ( q ) ) = 0, then lim,lim,f (p, q) = limJimq f(p, q ) ] to the expressions f ( m , q) = P,(E,) and to f ( m , q) = z" ,B,(E,[), then we obtain at once that Zn ,B(E,) converges, and that Z? P(EJ = P(S7 E n ) for each sequence E n of disjoint set in Q. The last part of the conclusion of the theorem is again a direct consequence of the sequential case.
I
fl=1
I
I
I
10. Properties of the Integral I ( f , a , E) = JEfda as a Function of a
In this section we shall consider properties of the integral J E f d a as a varies on the class of monotonic nondecreasing bounded functions on X E - co < x < co. We note that the class of a-measurable sets varies with a , which may prove disturbing. However, all a-measurable sets have in common the Bore1 measurable sets, and these play an important role in the definition of integration. For the present we shall assume that E = X , since the results for other permissible sets can be obtained by considering the functions f E ( x ) = f ( x ) x(E; x)
+
> 0, and if J,fda exists, then J s f d ( c a ) exists also and is equal to c Jn-fda. The condition c > 0 is introduced so that ca is also monotonic nondecreasing. If we wish to include the case c < 0, we may observe that a simple change of sign allows one to go from monotonic nondecreasing to monotonic nonincreasing, and the same kind of theory of integration can be set up for the latter. 10.1. If a ( x ) is monotonic nondecreasing on X , and c
10.2. If a , and a , are monotonic nondecreasing, and such that if I = [a, b ] then a,(I) = a,(b) - a,(a) 2 a,(b) - .,(a) = a,(I) for all I, and if J x f d a , exists, then Jx f d a , exists also. Iff 2 0 for all x, then Jx f d a , L J, fdu,. For under the hypothesis of the theorem a,*(E) 2 a,*(E) for all E, so that by V. 14.1 if E is measurable relative to a1 it is also measurable relative to a,. Consequently, i f f is a,-measurable, it is also a,-measurable. Moreover, if for a given subdivision 0, we have C5M( If I; E)a,(E) < co, then Z5M( I f E)a,(E) < co also. Then if J x fda, exists so will Jay fda,. The inequality between the integrals follows at once by applying limits to the inequalities between the approximating sums.
1;
10.3. If a,(x) and a2(x) are monotonic nondecreasing on
is such that Jx fda, and Jxfda, exist, then Jx fd(a, and is equal to Jxf d a , Jx fda,.
+
+ a,)
X and f ( x ) exists also
10. I(.J a , E )
= J E f d a AS A FUNCTION OF
a
259
We recall that for the existence of the integral by the Y-integral definition it is necessary and sufficient that the limits involved exist when based on subdivisions into Borel measurable sets in X , such sets being measurable relative to all monotonic a (see VII.3.10). The theorem then follows from the additive property of limits. There is a sort of converse involving the last two theorems. 10.4. If a,(x) and a,(x) are monotonic nondecreasing and a ( x ) = a,(x) a2(x) and if Jx f d a exists, then Js f d a , and Js f d a , exist and
+
Jxfda = Jxfda,
+ Jxfda,.
10.4.1. In particular, if a(x) = a,(x)
+
ac(x), where a,(x) is the continuous part of a, and a b ( x ) the function of the breaks, then
+
since J41f d a , = ZXf ( x ) (a(x 0 ) - a ( x - 0)) for any function f(x) for which the right-hand side is absolutely convergent. In connection with the interchange of limit and integration we have : 10.5. If ( a ) f i s bounded on X ; (b) a l l ( x ) and a ( x ) are monotonic
nondecreasing bounded on X , with a,,(O) = a ( 0 ) = 0 for all n ; (c) lim,,an(B) = a ( B ) for every Borel measurable subset B of X ; and if (d) Jd fda?,exists for each n, then Jx fd a exists and lim7LJxfd a IZ = Jx fda. For by V.14.7 if lim7La,t(B)= a(B) for every Borel measurable set B, and if E is measurable relative to all then E is also measurable relative to a and lim,/a,((E) = a(E). If Jxfda,, exists then f is measurable relative to each a,,, every set E [f 5 y ] is measurable relative to all a?,,and consequently relative to a , so that f is a-measurable. Since f is bounded on X , it follows that Jx fda exists. Let p.,,(y) = a,(E[ f 2 y ] ) and p ( y ) = a ( E [ f 5 yl). Then limmpu,(y)= p ( y ) for all y . Moreover, if rn
260
VII. LERESGUE-STIELTJES INTEGRATION
for all subsets E of X ; and if (d) J s f d a q exists for all q, then J, fda exists and lim, JAl fda, = Jzlf d a . For under condition (c) if E is measurable relative to all a , it will be measurable relative to a. The rest follows as in the case of sequences. If we assume that the directed set Q has sequential character, then condition (c) can be replaced by lim,a,(B) = a ( B ) for all Borel measurable sets, and the convergence of Js,fda, follows from the sequential case. In these theorems the integral over X can be replaced by the integral over any set E which is measurable relative to all a,. If the function f ( x ) is unbounded on X , then the question of the convergence of Jzlfilu ,, to J,fda involves an additional iterated limits operation. For if E,i,,l, = E[m 5fS MI then = a*(E)
J., fda =
lim
(7iL, N ) + ( - C o .
+a))
fda = lim lim
JE!,,,.41
( ? % N )n
fda
I,.
SEm,,%4
I
We let E,, = E [ I f 5 M I . The iterated limits theorem 1.7.4 then suggests the following : 10.7. If (a) : a?,(x) and a (x) are monotonic nondecreasing bounded with a,,(O) = a(0) = 0 and lim,,u,,(B) = a ( B ) for every Borel measurable subset B of X ; and if (b): f(x) on X is such that the integrals J.\ fda,, exist uniformly in the sense that lim.,f JCyEnd If 1 da,, = 0 uniformly in YI,then J., fda exists and lim,, J.\,fdu,, = J,l fda. Condition (b) can be replaced either by (b'): limii J E da,,= J E , f da uniformly in M , or (b"): 1imlf,/,J ( y E n l I f da,, = 0. By the iterated limits theorem, the hypotheses of the theorem are sufficient to guarantee the existence of J., da as lim, J E ) r da. Further the same theorem gives us
I
1 W
If
I
If I
If I
If I
Condition (b) can be deduced from the condition: there exists a subdivision u,, = {&} of X such that T o o M ( ~ jE,,Ja,,(E,,J '~; converges uniformly in 17. The sets E,,, are assumed to be measurable relative to all a,(.For then (e > 0, in,, k 2 m,, n) : P'=, M ( )f E,,)a,,(E,)<e. Let M , be the least upper bound of If1 on P i , ETr,,and set Ek = L" n = k E,?,. Then if M > M(, we have (CEII)Ek = CE,,.
1;
1 1. FUNCTIONS
26 I
OF BOUNDED VARIATION
Consequently for M > M , and for all n :
For the case when the sequence U / ~ ( X )is replaced by the set a,(x) with q on directed (2, the theorem reads: 10.8. If (a) a,(x) and a(x) are bounded monotonic nonincreasing
on X;if (b) there exists a q(, such that lim,u,(E) = a ( E ) for all sets E measurable relative to u and all u q for qRq,; if (c) J z l f d a , exists If daq = 0, uniformly in q for qRq,, in the sense that limlI+m with EAll= E [ 5 MI, then J.\ fdu exists and lim, J,fda, = J., f d a . The condition (d) can be replaced by either of the following: (d’) JcE,wfdaq lim, SEA{ (fI da, = JEIW f d a uniformly in M or (d!‘) lim( = 0. Condition (d) is a consequence of the assumption: (d”’) there exists a q, and a subdivision 0, such that for qRq,, the series ZuoM( EnJa,(E,,J converges uniformly in q. This latter assumption implies that the integrals J E I.fIda, are uniformly absolutely continuous on sets measurable relative to all uq for qRq,, in the sense (e > 0, d,, a,(E) 5 d,, qRq,) : fE da, < e. The proof follows the lines we used in VII.7.2 in proving JEfdr*absolutely continuous relative to a. This type of uniform absolutely continuity does not seem to be sufficient to replace condition (d), since the rate at which lim,Jlap(E.,I) approaches zero would depend on q.
If
I
I
1.f I;
If
I
11. Integration with Respect t o Functions of Bounded Variation
The additivity of JA f d a as a function of a suggests that if for a1 and a , monotonic nondecreasing bounded, on X,the integrals J,fda, and Jx fda, exist, we define J,x f d ( a , - a,) = J.r f d a , - JAY fda,. Since a,(x) - a,(x) is at most of bounded variation on X , we are faced with the extension of the integral J<,-fda to the case when a ( x ) is of bounded variation on X.a(x) is of bounded variation on X if JT,” J da is bounded in N , or Zu a ( x , ) - c ~ ( x ? - ~is) bounded in 0 for all subdivisions 0 of X into a finite or denumerable set of nonoverlapping intervals by the points x,.If a ( x ) is of bounded var-
I
I
I
I
262
VII. LEBESGUE-STIELTJES INTEGRATION
iation on X , then there exist monotonic nondecreasing functions p(a, x) and n(a, x) as well as the total variation function v(a, x) withp(a, 0 ) = n(a, 0 ) = v(a, 0 ) = 0, a ( x ) = p(a; x) - n ( a ; x) a(O), v(a; x) = p(a; x) n ( a ; x), and v(a; m) - v(a; - m) = J-: da = v ( a ; X ) . In the case of Riemann-Stieltjes integrals, with a of bounded variation on [a,b ] , we found that J: f d a exists if and only if J: fdv(a) exists, or if and only if J: f d p ( a ) and J: f d n ( a ) both
I
+
I
+
+
exist. (See 11.13.4.) This suggests that we define: 11.1. Definition. If a ( x ) is of bounded variation on X , then Jx f d a exists if and only if Jxfdv(a) exists, and then we set
For if Jx f d v ( a ) exists, then f is measurable relative to v(a). Since for any interval I = [a,b ] , v ( a ; I ) = v ( a ; b ) - v(a; a) 2 p ( a ; b) p(a, a) = p(a, I ) , and also v ( a ; I) 2 n ( a ; I ) , it follows that Jx j d p ( a ) and JAf d n ( a ) exist also. Since a ( x ) = p ( a ; x) - n ( a ; x), the relation J x f d a = Jx-f d p ( a ) - Jx f d n ( a ) is logical. Suppose for two monotone nondecreasing bounded functions a, and 01, the integrals Jx f d a , and JJ f d a , exist. Will Jx f d ( a , - a,) exist also and be equal to Js f d a , - JSyfda,? In the first place Jx fdaw,+ JH f d a , = SLY fd(a, a,) will exist. Moreover, for any I = [a,b ] : (a1 a,) (I) = v(a, a,; I ) 2 v(a, - a,; I ) , Consequently, Jxfdv(a, - a,) exists. If for the difference of the two monotonic functions a1 - a,, a , and a2are not equal to the positive and negative variations p ( a l - a,; x) and n(a, - a,; x), respectively, then because of the minimal character of p ( a , x) for any function a of bounded variation, there exists a monotone nondecreasing function p ( x ) such that al(x) = p(al - a 2 ;x) P(x) and a,(x) = n(al - a,; x) p ( x ) . Since Js f d a , exists, it follows that Jay f d p exists and Jxfda, = sxfdP(al - a,) s,fdP, Sxfda, = J x f d n ( a , - a,) JxfdP. Consequently SxfWa, - "2) = SXfdP("1 - ff2) - Sxfdn(a, - "2) = J x f d a , - Jx fda,. We can then assert:
+
+
+
+
+
+
+
11.2. If a1 and a , are bounded, monotonic nondecreasing on X , and f on X is such that Jx f d a , and Jx fdcr, exist, then Jx f d ( a l - a,) exists and is equal to f d a , - Jx fda,. If E is any v(a)-measurable set, and Jxf d a exists, then JE fda = J E f d p ( a ) - J E fdn(a). Since JBf d p ( a ) and J E fdn(a) are s-addiJAl
tive on v(a)-measurable sets, it follows that
JEfda
is also s-additive
1 1.
263
FUNCTIONS OF BOUNDED VARIATION
on v(a)-measurable sets. In particular if f(x) = 1, we can define a ( E ) = S E d a = JEdp(a) - SEdn(a) = p ( a ; E) - n(a; E), and a(E) will be s-additive on the class of v(a)-measurable sets. If E consists of the point x, then a(E) = p ( a ; E )
-
- n(x - 0)) = p(x = a(x
+ 0) -p(x + 0) - n(x + 0)) (p(x
n( a; E ) =p(x
-
+
0) - (n(x 0) - 0) - n(x - 0))
-
+ 0) - a ( x - 0).
It follows that if E consists of the set of discontinuities of a, and f d a exists then J E f d a = Zzf(x) (a(x 0) - a(x - 0)), where the series converges absolutely. If we decompose a(x) = ab(x) ac(x), .where ac(x) is the continuous part of a and ab(x) is the function of the breaks, then by decomposing ab(x) and ac(x) into their positive and negative variations, it turns out that
+
Sx
+
+
Now the term Zx f(x) (a(x 0) - a(x - 0)) is entirely independent of the value of a ( x ) at the points of discontinuity of a, and the same holds true of a ( E ) . However the monotonic point functions v(a; x), p ( a ; x) and n ( a ; x) may change with a(x), and the existence of the integrals depends to some extent on the absolute convergence of ZXf(x) ( ~ ( ax; 0) - v(a; x - 0)). If a(x) lies between a(x 0) and a(x - 0), then v(a; x 0) - v(a; x - 0) = a(x 0) a(x - 0) but if a(x) is exterior to (a(x - 0), a(x 0)), then v(a; x 0) - v(a; x - 0) = cr(x 0) - a(x) .(X) a(x - 0) > a(x 0) - a(x - 0 ) Consequently,
+
1, + I 1
+
+
I
I.
a(x
I + + I+ I
+
+
-
Sx
may be finite but f d a as we have defined it may not exist. In view of these considerations, it seems sensible either to limit the functions of bounded variation a (x) considered to those which satisfy the regularity condition that a(x) lies between a(x 0) and a(x - 0) or ( a ( x 0) - a(x)) ( a ( x ) - a ( x - 0)) 2 0 for all x, or to determine integrability conditions by a variation function associated with a regularizing equivalent of a ( x ) , for instance by the function a,(x) = a(x 0) for all x, which is right continuous in x and agrees with a(x), where a(x) is continuous. We select the former of these alternatives and define :
+
+
+
264
VII. LEBESGUE-STIELTJES INTEGRATION
11.3. Definition. The function a ( x ) of bounded variation on X is
regular if for all x of X , a ( x ) lies between a ( x (a(x 0 ) - a(x)) (a(x) - a(x - 0)) 2 0. We then have the following:
+
+ 0) and a ( x - 0), or
11.4. THEOREM. If a(x) is a regular function of bounded variation
I
I
on X , then v(u; X ) = 1.u.b. [Xo a(E,) for all subdivisions {E,} of X into v(a)-measurable sets]. For Em a(E7j-z) = Em) Em) I I Em) 4.; EJJ = v(u; X ) .
u =
I C mIP(~;
I
c,(P("; +
On the other hand, suppose for e >: 0,
u
is a subdivision of X into
since a ( x ) is regular. Now the subdivision of X consisting of the points x i and the open intervals (xzp1,xl) is a subdivision of X into v(a)measurable sets and for an open interval we have u(a, b ) = u(b - 0 ) - a(a 0). Then for e > 0, there exists a u of X into v(u)-measurable sets such that v(a; X ) < X I u(EJ e. As a consequence ~ ( a ;X ) = l.u.b.oZ:d a ( E ) Because of the s-additive property of a(E), the sums Za a(E) are monotonic nondecreasing in 0 , so that we have v(a; X ) = l.u.b.uXa a(E) = limaXd a ( E ) In other words, the total variation v(u; X ) is of the nature of an integral of the set function I a ( E ) I and could be written Jx da(E) If E, is any v(a)-measurable set, then
+
I
I
I I
I
v(a, E,)
+ v(u;
I
I.
1+
I
I
I.
I.
CEO)= v(u; X )
= lim a
a
1 a(E) I
since the two terms on the right are monotone and bounded in 0 , so that the limits exist. Now v(a, E') 2 a(E) for all v(cx)measurable E' and all u. Consequently:
I
1 1. FUNCTIONS
265
OF BOUNDED VARIATION
11.5. If E , is a v(a)-measurable set and a ( x ) is regular, then:
v(a; E,)
= lim a
C I a ( E ) I = 1.u.b. [cI a(E) I ,
O./jO
a
for all subdivisions a of E, into v(a)-measurable sets]. In analogy to the Jordan procedure for defining the total variation of a function on an interval, it is possible to define for a(x) regular, p ( a ; X ) and n ( a ; X ) by least upper bounds and as limits as to subdivisions. This leads incidentally to additional information on the functions p(a; E) and n ( a ; E ) . For convenience we shall omit the a in v(a:E), p(a; E ) , and n ( a ; E ) , and assume that v(E), p(E), and n(E) have been determined by a regular function of boundedvariation a(x). Let a be any subdivision of X into v(a)-measurable sets, and let En' be the sets in a, for which a(E,,') 2 0 and Ell'' those for which a(E,") < 0. Define p,(X) = Z,a(E,') = a(Z,E9,'), and n,(X) = - C,a(Efz") = - a(ZoE7?"). Then for all u, we have v(X) 2 p,(X) n,(X) and a ( X ) = p,(X) - n,(X). Let E: = CoE,', and En = C,E," = X - E,+. We show that if a12 a, then
+
u(ET) 5 a ( E z E i ) 5 a(E,:).
For if E is any a-measurable set, such that a ( E ) 2 0, and we apply a subdivision a = {En,} to E yielding (EE7,'} and {EEm"), with a(EE,,') 2 0, and a(EE,'') < 0, then a ( E ) = a(CmEE,,') a(Zn1EEm").But a(ZnIEE9,,")< 0, so that a ( E ) 5 a(EZ,E,'). Applying this to every subset E,?' in E: yields: if al2 u, then a(E:) 5 a(E:* E:). Now E', will also contain sets which are subsets of E;. 1 Consequently, a (ET Em:) 5 a ( E J . We conclude then than p,(X) and n,(X) are monotonic nondecreasing in a, and since they are bounded by v(a; X ) , that lim,p,(X) and lim,n,(X) exist. Since for all a : a ( X ) =p,(X) - n,(X), it follows that a ( X ) = lim,p,(X) - lim,n,(X) and v(X) = lim,p,(X) lim0n,(X). Consequently, limnp,(X) = p(X) and limon,(X) = n ( X ) . We can conclude more. Because of the monotonic character of p,(X) = a(EZ), there exists an increasing sequence of subdivisions a, with o m 2 such that p ( X ) = lim,pam(X) = lim,a(Ebf m ). We show: If X' = limnz E,', then
+
+
~
p(X) = a(X'),
rn
and
n(X)
=
-
a(X - X')
= -
a(X-).
266
VII. LEBESGUE-STIELTJES INTEGRATION
For the inequality a(Elf) 5 (E:Eu:) 5 a(E:) can be at once extended to any finite number of subdivisions ol5 o2 5 ... 5 ok, to read k a(E:) 5 Eo:) 5 a(E;). i= I For instance :
a(n
a(E:) 5 a(E:
*
E.',) 5 a(E;
Now
*
E.', * Eb',) 5 a(E:
m a ,
lim E + - uni
= lim
k=l m=k
rn
E:) 5 a(Eb+a).
i
EL
=
*
(lim
k
Et). m=
a t
Since c1 is the difference between two bounded s-additive set functions on v(a)-measurable sets, it follows that i m=k
a
Since
a(nE L ) . 00
lima(n E+)=
m=k
But lim, a(Eo+) = p ( X ) , and since a is s-additive L
Consequently a(X') = a(1im -m E L ) = p ( X ) . Since a ( X ) = a(X') a ( X - X') = p ( X ) - n ( X ) , and p ( X ) = a(X') it follows that n ( X ) = n ( X - X+) = - a(X-). Further, since p ( X ) = a ( X * ) = p(X+) - n(X'),p(X) L p ( X + ) ,and n(X') L 0, it follows that n(X') = 0, and consequently n(E) = 0 for all v(a)-measurable sets E 5 1'. Similarly p ( X - ) = 0, and p ( E ) = 0 for all v(a)-measurable sets E I X-. If now E is any v(a)-measurable set, then
+
p ( E ) - n(E) = a(E)
= a(EX')
since n ( E X f ) = p ( E X - )
p(E)
= 0.
+- a(EX-)
=p(EX')
- n(EX-),
Then
= p(EX') = a(EX+) = v(EX+)
and
n(E)
= n(EX-) = - a ( E X - ) =
-
v(EX-)
for all v(a)-measurable sets E. Summing up we have shown:
267
11. FUNCTIONS OF BOUNDED VARIATION
11.6. THEOREM. If a ( x ) is of bounded variation on X and regular
+
in the sense that ( a ( x 0) - a ( x ) ) ( a ( x ) - u ( x - 0 ) ) 2 0 for all x , then there exist disjoint v(u)-measurable sets X + and X - , such that X = X f X-,
+
p(X)
= p ( X + ) = a(X')
=
v(X+); p ( x - ) = 0;
and
n(X) = n ( X - )
=
-
a(X-)
= -
v ( X - ) ; n ( X + ) = 0.
For all v(a)-measurable sets E, the sets E f = E X + and E - = E X - , effect the same decomposition for E. An immediate consequence of this theorem is: 11.7. If a ( x ) is of bounded variation on X and regular, then for all v(a)-measurable sets E, we have p ( E ) = 1.u.b. [.(El) for all v(a)-
measurable E, 2 El, and n(E) = - g.1.b. [u(E,)for all v(a)-measurable sets El 5 El. For if El 5 E, and v(a)-measurable, then a ( E l ) = p ( E I X f ) n(E,X-) = p ( E I X ' ) 5 p ( E ) . The least upper bound is attained for El = E X f . Similarly for n(E). We are now in position to prove: 11.8. If a ( x ) is of bounded variation on X and regular in the sense that ( a ( x 0) - a ( x ) ) ( a ( x ) - a ( x - 0)) 2 0 for all x , then Jxf d a exists if and only if: (a) there exists a subdivision a,, of X into v(a)-measurable sets, such that for o 2 o,,, Zuf ( x ) a ( E ) is absolutely convergent for all choices of x in the sets E of a, and (b) limuZ0f ( x ) * a ( E ) exists, Jx-f d a being this limit. The necessity part of the theorem follows at once from the definition of Jxf d a = Jxf d p ( a ) - Js fdn(a), and the Y-definition of integral for the case when a is monotone bounded. To show that the conditions are also sufficient, suppose that Zuf ( x ) a ( E ) converges absolutely for u 2 a,,.Let a. = {E,} and set u = a * [X', X-1, where X', X - , is the decomposition of X induced by a, in accordance with VII.11.6 above. Then a(E,X+) = v(a; EJ') and a(E,,X-) = - v(a; E,X-) and
+
I
This latter series is absolutely convergent, so that z' If(x) v(a; E) for all a 2 al. If next we assume that limuCuf ( x ) a ( E ) exists, then there exists a subdivision a,,'such that for u 1a,,':Zuf(x,)a(E,) is
268
VII. LEBESGLJE-STIELTJES INTEGRATION
absolutely convergent and bounded for all choices of x, on Em. By taking ol' = o,,' * (X', X-),we find that f ( x ) is bounded on any set E of ol' for which v(a; E) f 0, and ZuM(If I; E) v(a; E) < co for any o 2 ol'. Moreover, for the same cr:
where x,' and xm" are any two points of E,. Then the Cauchy condition of convergence gives us that limuZum(f,E) a(E) = 0. But any set of a o 2 ol', is either a subset of X + for which a(E) = v(a; E ) or of X- for which a(E) = - v(a; E), and so a ( E J = v(a; Em) for all m, and limuZow(' E,,)v(a; E,) = 0. But by VZI.3.4.1 this is sufficient to gives us the existence of Jxfdv(a) and consequently that of f d a . As pointed out in the proof of the necessity part of the theorem, we then have limuZuf(x)a(E) = Js fda. The properties of Js f d a when a is of bounded variation and regular are similar to those for the case when a(x) is monotonic nondecreasing bounded on X , and can frequently be deduced from this special case, We consider briefly a few of these properties.
I
I
I
I
Sx
sx
11.9. If fda exists and E is v(a)-measurable, then J E fda exists as the difference of f d p ( a ) and SE fdn(a). If we set P(E) = SE fda then P(E) is absolutely continuous with respect to v(a), P(E) is s-additive on the class of v(a)-measurable sets, and is of bounded variation in the sense that Zu P(E) is bounded in o, the class of subdivisions of X into v(a)-measurable sets. Moreover, v(P; X ) = sx If Ma). The first part of this theorem is obvious. For the last part, suppose that E is any v(a)-measurable set. Then
sE
I
I
I
I P(E)) 15 I j,fdP(W) I + I J E f d d 4 I 5 jE If I dP(ff) + j E I f I dn(a) = j E If I M a ) , consequently v(P; X ) 5 Jx. If I dv(a). On the other hand,
let and E' = E [ f > 01, E" = E[ f 2 01, X + and X - be the decomposition of X into sets such that p ( a ; X ) = a ( X ' ) = v(a; X') and n ( a ; X ) = - a ( X - ) = - v(a; X - ) . Then
1 1.
269
FUNCTIONS O t BOUNDED VARIATION
I
If we set o = (E'X+, E'E-, E"X+, EI'X-), then Zm//3(E) = JAY 1 f 1. dv(a). Consequently v(p; X ) = Js I dv(a). We have incidentally obtained for P(E) a decomposition of X similar to that for a ( E ) , namely: X ' ( p ) = E [ f > 01 X ' ( a ) + E [f 5 01 - X - ( a ) , and X - ( p ) = E [ f > 01 . X - ( a ) E [f 5 01. X - ( a ) . Further
If
+
P(P, X )
=
jB,Jdff
and
n(P; X )
=
-
1^
E '*Y
+ jB"Jda
fda
-
1
EIIX+
fda,
from which the values ofp(P; E ) and n(P; E ) for any v(a)-measurable set can be written down at once. If we observe that for any v(a) measurable set E, it is true that
provided x is on E, and f is bounded on E,, then the s-additivity of ,B(E) gives us at once the 11.10. Approximation Theorem. If a ( x ) is a regular function of bounded variation on X and J.,.fdcx exists, then for every subdivision u into v(a)-measurable sets on each of which f is bounded, we have
The following linearity properties are deducible from the definitions :
SX
Jx flda and f,da exist, then for c, and c2 any two constants, we have the existence of S ~ ( c i f i ~2fz)da= ~1 Js.f,da -I- C, Jxfda,. 11-11. If a ( x ) is of bounded variation on X and
+
11.12. If a,(x) and a 2 ( x ) are of bounded variation on X , and if
for f ( x ) on X , the integrals fda, and JAY fda, exist, then for any two constants cI and c2, we have the existence of Jx fd(c,a, cp2) equal to c, Jx fda, C, Jx fda,. TO apply VII.11.8, some regularity condition on a , and a , would be needed. However, from the definition in VII.ll.l, it follows that if fda exists, and c is any constant, then Js f d ( c a ) exists with value c Jxfdcr. Further if J,fdal and JA1 fda, exist, then since v ( a , a,; X )
+
SX
+
+
270 <
VII. LEBESGUE-STIELTJES INTEGRATION
sx
+
+
v(a,; X ) v(a,; X ) it follows that fdv(a, a,) exists and so Jxfd(a, a,) exists also. If we decompose a,(x) =p,(x) - n,(x) and a,(x) =p,(x) - n,(x) into their positive and negative parts, and apply VII.11.2 to a,(x) .,(x) = ( p , f x ) +p2(x)) - (n,(x) n,(x)), then we find that fd(a, a,) exists with value Jxfda, + Jx fda,. The conclusion of the theorem is then immediate. We observe that J,fda exists for any bounded Borel measurable function f on X relative to any function a of bounded variation on X , since such a function is v(a)-measurable for any such a. If (BB) is the class of bounded Borel measurable functions, and ( B V ) that of functions of bounded variation on X , then by the last two theorems, the integral Jxfda is a bilinear functional on the product class (BB)x (BV.
+
sx + +
+
11.13. If a is of bounded variation on
X and f is such that s,fda
I
exists then f is absolutely integrable in the sense that Jx If dv(a) exists. Iff is v(a)-measurable on A’, and such that there exists a function g(x) 2 0 such that for all x of X except at most a v(a)-null set: If(x) I I g(x), and Jx gdv(a) exists, then Jx fda exists and I Jxfda I I sx gdv(a). In the matter of interchange of integration and limits, we have: 11.14. If (a) a ( x ) is a regular function of bounded variation on X ; (b) f , and f are v(a)-measurable on Xand such that f , +f(v(a)) ; and (c) sE.fr,daare uniformly absolutely continuous relative to ~ ( a ;)then Jxfda exists and limn If, - f I dv(a) = 0, so that limplJ E f,,da = SE fda for all v(a)-measurable sets E. We use the decomposition of X into the sets X + and X- such that p ( X ) = a ( X + ) and n(X) = - a ( X - ) . Since on subsets of X+, we have v(a; E ) = a(E) and on subsets of X-, we have v(a; E ) = - a ( E ) , then for any v(a)-measurable set E
sX-
As a consequence the integrals J E f,dv(a) are absolutely continuous relative to v(a) uniformly in n. The theorem then follows from the corresponding theorem for monotonic bounded a. (See VII.8.8.2.) For a set of functions f , ( x ) on a directed set Q, a corresponding theorem holds. For instance:
11.
27 1
FUNCTIONS OF BOUNDED VARIATION
11.15. If a is a regular function of bounded variation on X ; if f , ( x ) and f ( x ) are v(a)-measurable on X ; if f,--f f ( v ( a ) ) ; and if
sx-If,
I
- f dv(a) = 0, so that lim, then J2-fda exists, and lim, = J E f d a for all v ( a ) measurable E.
JE
fqda
11.16. If a is a function of bounded variation on X ; if f , ( x ) and f ( x ) are v(a)-measurable; if Jx f p d a exists for all q of Q; and if there exists a function g ( x ) such that Jx g dv(a) exists and lim, f , ( x ) = f ( x ) uniformly relative to g in the sense that (e, q, qRq,,x on X ) : f , ( x ) f ( x ) 15 e g ( x ) then Jx f d a exists, and lim, SAX-f q - f dv(a) = 0, so that lim, fqda = J E f d a for all v(a)-measurable sets E. For by the hypothesis, if for given e > 0, q is any such that qRq, then If(x) 2 f , ( x ) e g ( x ) Sincef , ( x ) and g ( x ) are integrable with respect to v(a) it follows from VII.11.13 that Jx fda exists. From the same theorem we conclude that for qRq,:
I I
I
sE
1,
I+ I
I I
I jxfqd*
-
I
I
j x f d a 2 JA.
I
I
I.
If,-f I d v ( a ) 5 e Jx l g I dv(u).
Then lim, j Efqda = SE fda for all v(a)-measurable E. For sequences of functions of bounded variation a,, the use of the definition Jx fda = Jx f d p ( a ) - Jx fdn(a) leads to: 11.17. If a l l ( x ) and a ( x ) are of bounded variation and such that lim a,(B) = a ( B ) and lim,( v(an; B ) = v(a; B ) for every Bore1 measurable subset B of X ; if Jx-fdan exist uniformly in n in the sense that EmM JcEN d v ( a n ) = 0 uniformly in n, where EAw= E [ f 5 MI, then fda exists, and limrl J E f d a n = SE fda for all subsets E of X which are v(a,)-measurable for all n. Other variants of this convergence theorem can be derived from the corresponding theorems, for the case when are monotonic nondecreasing (see VII. 10.7). Parallel to the substitution theorem of 11.1 1.6 for Riemann-Stieltjes integrals, we have a substitution theorem for L-integrals.
sx
If I
I I
11.18. Substitution Theorem. If (a) a ( x ) is a regular function of
+
bounded variation on Xwith ( a ( x 0) - a ( x ) ) ( a ( x ) - a ( x - 0 ) ) 2 0 for all x; if (b) f ( x ) and g ( x ) are v(a)-measurable; if (c) Js f d a
272
VII.
LEBESGUE-STIELTJES
INrEGRATION
exists and we set p(E) = J E f d a ; and if either Jxfgda or Jn-f d p exists, then both integrals exist, and are equal. The integral Jx f d p with respect to the function p(E) on v(a)measurable sets, we define as the lim,Z, f ( x ) P ( E ) , where the subdivisions (T of X consist of v(a)-measurable sets. As in the case of the integral based on the function a ( x ) of bounded variation on X , it is understood that if Js f d p exists, then there exists a subdivision a, of X such that for a 2 a",the sums Sof ( x ) p ( E ) converge absolutely for all x in the corresponding E, and that one can prove that there exists a subdivision (T"'such that for all a 2 go',f ( x ) is bounded on all subsets of a for which vfp; E) # 0, and Z,M(I f E ) v(p; E ) < 00. Suppose that j'(x) is bounded on X , and let a be a subdivision of X such that g ( x ) is bounded on all subsets E of a for which v(a; E ) f 0 . Then if a = {E,,} we have:
1;
wheref ( x ) 5 A4 on X . Since Jx gda exists, limo"rocu(g;E,,)v(a ;E n ) = 0. Then limo 12,f ( x , J g ( x , , ) a ( E , , ) - "r, f ( x n ) p ( E I 1 ) = 0, which leads at once to the conclusion of the theorem. If f ( x ) is not bounded on X , a different type of proof seems to be needed. Suppose first that Js.fdp exists, and let a, be such that if (T I go, then L,M(I f E,Jv(P; E,J < co. If for such a (T = { E l l } we set F J x ) = S,, f(x,,)x(E,; x ) g ( x ) , with x,, on El,, then F J x ) = f ( x , , ) g ( x ) on E,,. F n ( x ) being a convergent sequence of ~ ( a )measurable functions, is then also v(a)-measurable. Moreover since S E ~ F, dvfff) = I Nx,) v(B; E , ) it follows that z',, J E I L F, dv(a) < co, so that by V11.2.3 Js Foda exists, and has as value Zn Foda = z , l f ( x J P ( E , ) . Let ae be any subdivision of X induced via f ( x ) by a subdivision uY of the Y-axis into intervals such that ay < e. Then c u f f ; E ) < e, for any E of a 2 ae. The possibility of determining such a aedepends on the v(a)-measurability off. Consequently, for (T 2 oe * go, we have Fo(x) - , f ( x ) g ( x ) [ 2 e g ( x ) for all x , in other words limoFo(x) = f ( x ) g ( x ) relatively uniformly as to g ( x ) . Since g ( x ) and F J x ) are integrable with respect to v(a), it follows from VII.11.16 that Jxfgda exists and that
I
1;
1 1
I
1 1 SEn
I 1
I
I
I
1 1.
273
FUNCTIONS OF BOUNDED VARIATION
On the other hand, suppose that J L l f g d uexists. Then by using the same functions Fg(x), which are v(u)-measurable and converge t o f ( x ) g ( x ) uniformly relative t o g(x), we find for u 2 cre that Fo(x) 5 f(x)g(x) e I g ( x ) so that by VII.11.13 SAXFn(x)du exists. Obviously Jx- Fo(x)du = ZoJ E f(x,,)g(x)dx = Xof(xn)P(E,,),which will be absolutely convergent. Using again the convergence of Fo(x) to f(x)g(x) uniform relative to g(x) we obtain:
I
1+
I
1,
I
EXERCISES 1. Suppose a ( x ) is monotonic bounded on X . Set up a definition for s , f d ~ as follows: Let u stand for a finite set of disjoint a-measurable sets: El ... E,, such that L',,ETLS X , and order the 0 by the condition that if o1 = Ell ... El,,!, and o2 = E,, ... E,,, then o1 >= 0, means that C T E l , 2 L':E,,, and each E l , is contained entirely in some E,?. Then define SLYf d u = limo2,f ( x J a ( E , ) , if this limit exists. What relation, if any, does this integral bear to the Y-integral, and what properties does such an integral have? 2. Suppose u ( X ) is bounded monotonic on X giving rise to the upper measure u * ( E ) on subsets E of X . Let f ( x ) be bounded on X . Let u y be subdivisions of the range off: (In
3. For u(x) monotonic bounded on X , and for the a-measurable function f(x) define f ( m , M ; x) = , f ( x ) if m M . Show that Sx,f(m, M ; x)du = yd,u(y) if P(Y) = a ( E [ f 5 UI).
s,"
4. The function f ( x ) a ( E ) , x on E, is a multiple valued set function on a-measurable subsets of X,which form an s-algebra. This suggests: Let (3 be an s-algebra of subsets of X . Let p ( E ) be a set function defined on Q. Let u be subdivisions of X into subsets of (3 directed by the usual order o12 0 , if every set of o1 lies in some set of 0 2 .Define S,p(dE) = lirn,L'J(E)
274
VII. LEBESGUE-STIELTJES INTEGRATION
provided this limit exists. What properties parallel to those for integrals of interval functions does the integral Sxp(dE) possess.
5 . Let a ( x ) be a bounded monotone function on X . Let 8 be the class of finite valued functions f on X , which are a-measurable. Show that if for the functions f l and f i of -8, we set
then S(fl, fi) satisfies the metric properties 0 5 S(fl, f i ) = ,a'( f l ) and S(fl, f 3 ) 5 S(fl, f 2 ) S(f2, f 3 ) , but that S(fl, fi) = 0 is equivalent to: f i differs from f 2 on an a-null set. Show that lim,d(f,, f) = 0 is equivalent to the a-convergence of f , , to f, that is f , +f ( a ) . Is it possible to metrize the totality of finite valued functions on X in a similar way so as to secure the equivalence of lim,,8(f,, f ) = 0 to f,+ f ( a * ) ? Show that SZI-~(E,AE2) da, where EIAE, = (El - E,) (E, - E l ) , sets up a similar metric for a-measurable sets, while S(El, E,) = a*(El - E,) a*(E2 - E l ) will metrize the totality of subsets of X .
+
+
+
6. Construct an example of a directed set f , ( x ) of uniformly bounded Lebesgue measurable functions on 0 5 x 5 1, such that lim, f q ( x ) = f ( x ) for all x , but lim, J: f,(x)dx # S,'f(x)dx.
7. What class of functions f ( x ) on X is determined by the condition that f d a exists as an L-S integral for all functions a ( X ) of bounded variation on X ?
Sx
sE
8. Show that if a ( x ) is of bounded variation on X , and fda = 0 for all v(a)-measurable sets, then f ( x ) vanishes except for an v(a)-null set. 9. Show that if a q ( x ) is a directed set of functions of bounded variation on X , such that there exists a function a ( x ) of bounded variation with lim,v(a, - a ; X ) = lim, Ssld(cr, - a ) I = 0, and if f ( x ) is bounded on X such f d a , exists for all q, then f d a exists and lim, Jxf d a , = fda. that Can the boundedness condition on f ( x ) be dropped?
Sx
Sx
sx
12. Integration with Respect to Unbounded Measure Functions
The obvious case of an unbounded measure function is that in which X = - 00 < x < co and a ( x ) = x. For the case of Riemann-Stielt-
+
12.
275
UNBOUNDED MEASURE FUNCTIONS
jes integration we required for the existence of (a) J:f(x)da(x) exist for all - co < a < b <
J-yf(x)da(x)
+
00,
that
and that
I*
lim
( a , b ) + ( - ~ , + ~ )a
fda
exist. For the Lebesgue setting with X = - co < x < + co, and a(x) of bounded variation on every finite subinterval a S x 5 b of X , we define: 12.1. Definition. The function f(x) on X is L-integrable with respect
to a ( x ) , if and only if (a) f(x) is L-integrable on every finite halfopen interval Iahb = a < x ib and (b) lim
(a,b)+(-m,+ m)
j::; If I dv(a)
exists. Then Jxfda = lima,bJ:: fda. It follows that Jxfda exists only if Jx dv(a) exists, so that f is integrable if and only i f f is v(a)-measurable and absolutely integrable with respect to ~ ( a )We . might consequently limit ourselves to the case when a(x) is monotone nondecreasing, but not necessarily bounded on X . The condition of absolute integrability is desirable if we expect to obtain properties of the integrals comparable to those for the case when a ( x ) is of bounded variation. The results obtained apply with obvious alterations to the case where X is the open interval a < x < b, where a is finite or - 00 and b is finite or co.It is then assumed that a(x) is of bounded variation .on every finite closed interval [c, d ] with a < c 5 x 5 d < b. For convenience we shall call such a function a(x) of extended bounded variation on X. An immediate consequence of the definition of L-integrability is:
If I
+
12.2. If a(x) is of extended bounded variation on X , f(x) and g(x) 2 0
I
are v(a)-measurable on X with f(x) 15 g(x) except at most on a v(a)-null set then J,fda exists and Jx fda 15 Jx gdv(a) provided Jx gdv(a) exists. Assuming that a(x) and f(x) satisfy condition (a) of the definition of integrability 12.1 above, we observe that condition (b) is equivalent to the following, where throughout means L-integration over the half-open interval a < x 5 b:
I
Js
276 12.1
V11. LEBESGUE-STIELTJES INTEGRATION
@’)a
s: If I M a )
lirn(,,b)+(+rn,+rn)
lim(a,r,)+(-m,-rn) J:
If I dv(a)
= 0,
and
= 0.
12.1 (b”). For every sequence {a,J, n ranging over the positive and negative integers, such that a,, < a,+1, with limntm a,, = - co and a, = co, the series ZnJ;n+l dv(a) converges. Moreover, JAY f d a = LT1 Jz;;+l f d a .
+
If I
1.f 1;
12.1 (b”’). There exists a subdivision cr of X such that ZOM(
E) v ( a ; E ) < co, where as usual M(I f E) = 1.u.b. f ( x ) x on El, and M(I f E)v(a; E) = 0 if v(cc; E ) = 0. There is no difficulty about showing that (b’) or (b”) can replace (b). If condition (b”’) is fulfilled and we set g ( x ) = M ( f Em)on the sets E, of cr when M ( f Em)< oc, and zero elsewhere, then g ( x ) is v(a)-measurable, Jx gdv(a) exists, since for any finite subinterval (a, b ] : J,“gdv(a)5 ZOM(If Em)v(a; Em). Consequently, since If(x) < g ( x ) except at most for a v(a)-null set, it follows from VII. 12.2 that Jx f d a exists and Js f d a I SOM(If E m ) v ( a ; Em). That the condition (b”’) is also necessary will be shown below. An alternate definition of integrability is contained in the theorem:
1;
1;
[I
I
I 1;
1 1;
1;
I
I
1;
I
12.3. If a ( x ) is of extended bounded variation on X , and regular with
+
( a ( x 0) - a ( x ) ) (a(x) - a ( x - 0)) 2 0 for all x, then a necessary and sufficient condition that f ( x ) be L-integrable is that lim0ZUf(x)a(E) exist, where subdivisions are into denumerable sets of v(a)-measurable subsets for each of which v ( a ; E ) < 03. The set function a ( E ) is defined in terms of the functions p ( E ) and n(E) based on the monotone point functions p ( x ) = (J,” da a ( x ) - a ( 0 ) ) / 2 and n(x) = (Ji I da - ( a ( x ) - u(O)))/2, repectively, by the relation a ( E ) = p ( E ) - n ( E ) . This assigns a definite value to a ( E ) if v ( a ; E ) < co. The condition that cr consist only of sets E for which v ( a ; E ) < co imposes no essential restrictions, since the product of any subdivision and the subdivision determined by the intervals (n, n 13, n ranging over the positive and negative integers will satisfy this condition. Moreover, in asserting that limOCO f( x ) a ( E ) exists, it is understood that S J ( x ) a ( E ) ultimately has meaning, in particular there exists a cro such that for CT 2 cr(,, the sum C0f ( x ) a ( E ) exists as a general sum (see 1.6). This means that S nf ( x , , ) a ( E , ) converges absolutely where cr = { E r L }and x , is on E,. Consequently, the existence of limOluf ( x ) a ( E ) carries with it, not only that there
I
+
1 1+
12.
UNBOUNDED MEASURE FUNCTIONS
277
exists a u 0 such that for (T 2 u,, the expression ZUf ( x ) a ( E ) is absolutely convergent for all x on the corresponding E, but also the boundedness of the same sums for u 2 go. Suppose fda exists. Then L'?,-=I I f dv(a) < co and Jx f d a - v+Co n= ' f d a . For e > 0 let a l l ( and o,, be subdivisions of (n, n 11 such that for ( T ~2~ o,,~:
Jz
I
+
I
J?':
I J;+'fda
-
c f ( x ) a ( E ) IS e/2"
and
n ranges over any finite set of integers. If oe = L ' 7 1 ~ n L , where in L',,, and (T 2 ere, then
xo1
I I a ( E ) 15 xoI f(~)I v(u, E ) < J,r I f I dv(a) + 2e < a.
f(X)
We then have also for cr 2 ( T ~
I
x JIL'
fda
cf ( x ) a ( E ) 15 c [ J " - ' f d a
-
91
U
n
-
C f ( x ) a ( E ) I 5 2e, "7l
so that limolrof(x)a(E) exists and is Jakfda. On the other hand, suppose that linioZof(x)rx(E)exists. Then by the Cauchy theorem of convergence, we have ( e > 0, ue, u1 2 oe, oz 2 ae) Z,,f(x)a(E ) - Z , , f ( x ) a ( E ) 5 e, the sums being absolutely convergent. Let IaI,= (a, b ] , and suppose that ul and uz are finer than the subdivision determined by Inb,X - Iqb. If we keep o1 and u2 fixed and equal to each other outside of the interval rob, so that they differ only inside of this interval, the condition for the existence of J i f d a are satisfied. This will be true for all a and b with - a3 < a < b < co. It follows that f ( x ) is v(a)-measurable on X . Since linioZof(x)a(E) exists, there exists a uo such that the sums Cuf ( x ) u ( E ) are bounded in CJ for u 2 u(,, and in x for x on the corresponding sets E of u. As in the proof of VII. 11.9 we can divide each interval ( 1 1 , n I] into sets E:, and E;, so that f ( x ) 2 0 on E : and f ( x ) < 0 on E;. Further ET can be divided into E i - and E:such that v(u, E ) = u ( E ) for any subset of E;- and v ( u ; E ) = - a( E) for any subset of E;,-, and similarly for E;+ and E;-, all sets being v(a)-measurable. It follows that 0 5 f ( x ) a ( E ) = f ( x ) v ( a ;
:I
I
+
+
278
VII. LEBESGUE-STIELTJES INTEGRATION
E) for any v(a)-measurable subset of Eit or E;-, and 0 2 f(x)a(E) - - f(x)v(a; E) for any v(a)-measurable subset of Ei- or E;+. Let a : = ( E r , Ei-) and a; = (E,', EZ-). For a 2 ao, let a+ stand for the collection of sets :a and a- for the collection a;. Then
+
Cuffx)a(E) = C,,+ffX)QI(E) Cu-ffxI.(E) - C,-f(x)v(.;
=
C,+ffx)v(.;
El
E).
Since Z,, f(x)a(E) is bounded in a for a h ao, by varying subdivisions of a+ and a- independently if follows that Zu+f(x)v(a; E) and ZU-f(x)v(a; E ) are bounded for all redivisions of a+ and a-, Consequently, z1,f(x)v(a; E) = Zaf(x)v(a; E) 2,- f(x)v(a; E ) is bounded in a for a h go. It follows that if a h go, and a = {Em},then f ( x ) is bounded on each set Emfor which v ( a ; Em)> 0. If M(I f Em)= 1.u.b. [(f(x) 1, x on Emsuch that v(a; Em)> 01 then for e > 0 there exists an x, in Em such that M(I f 1; Em)d I f(x,J I e/(2"v(a; Em)).It follows that
+
1;
+
I;
where M(I f E)v(a; E) = 0 if v(a; E) = 0. Then by VII.12.1 (b"') Jxfda exists, and as proved above has for its value limoZuf(x)a(E). The proof of this theorem shows incidentally that if Jxfda exists, then there exists a subdivision a,, such that for a h ao, we have qN( ( f 1; E)v(ff; E) < a. The linearity of Jxfda in f for fixed a, and in a for a fixedf, follow in the usual way from the definition VII.12.1. We also have: 12.4. If a ( x ) is of extended bounded variation on X, Sxfida exists, fz(x) is v(a)-measurable and bounded except for a v(a)-null set, then S x f i fzda exists and I Jx fi f,da I 5 M Jx I f , 1 dv(a), where If2(x) I I M except for a v(a)-null set. For the product ,L(x) f z ( x ) will be dominated by M If,(x) I except for a v(a)-null set. If we set fz(x) = x ( E ; x), where E is a v(a)-measurable set then we find:
12.5. If E is a v(a)-measurable set, and Jxfda exists, then
J E fda
=
SX f(x)X(E; x)da(x) exists. Further if p(E) = SE fda, then P(E) is finitely additive in the sense that if El and E, are v(a)-measurable then p(EJ /3(E,) = /3(E,E,) /?(El E,), and s-additive in the
+
+
+
12.
279
UNBOUNDED MEASURE FUNCTIONS
sense that if {E,} is a sequence of disjoint v(a)-measurable sets, then P(ZE,) = CnP(E,). The additivity follows as usual from the identity: x(E,; x ) + x(E,; x) = x(E, E,; x) x(E,E,; x), and the linear properties of the integral. The s-additivity can be deduced from the fact that if I = (a, b ] then
+
+
uniformly in E. As a converse we have: 12.6. If E,, is a sequence of disjoint v(a)-measurable sets such that
SEn fda
exist and Z,, exists as ZnSEnfda.
SE 1f I dv(a) < co, then
if E
= C,E,,
SE fda
12.6.1. In particular if the function f(x) is a step function on X , and f ( x ) = c, for x on E,, and such that Z I LI c,, v(a, En)< 00, E,, being disjoint v(a)-measurable sets, then Jxfda exists with Znc,a(E,,) as its value.
I
JEfda = p ( E ) is absolutely continuous relative to v ( a ; E). Also if IR,= (a, b ] and Crab= X - lab, then
12.7. The integral
for all v(a)-measurable E. The last part follows from the inequality dv(a).
The first part from the decomposition of E and the absolute continuity on every rob. No new proof is needed for the following:
=E
+
. Iab E Crab, *
12.8. If a(x) is monotonic nondecreasing on X , f ( x ) 2 0 on X , and
SX fda = 0,
then f = 0 except for an a-null set. When it comes to a consideration of the integrals J f,da for a sequence of integrable functions f , L ( x ) ,we note that any theorem on the interchange of limit and integral involves an additional interchange of iterated limits. Additional conditions making possible interchange
280
V11.
LEBESGUE-STIELTJES INTEGRATION
of order in limits are then needed. Since we already have sufficient conditions so that lim,? JX f,,da = Jxfda, when a is of bounded variation on X , which can be applied to the case when X = I,, = (a, b ] , we need only an additional condition permitting the interchange of limit as to N, and limit as to (a, b). The result is the following:
X,(b) f,(x) and f ( x ) on X are such that lim,! f , , +f ( v ( a ) ) , (c) Js f,da exists 12.9. If (a) a ( x ) is of extended bounded variation on
for each yt, (d) J E f,,du are uniformly absolutely continuous relative to v(a), and (e) lim
(a,h)+(-oo.+m)
I*1 f,,I (1
dv(a)
exist uniformly in N (or Jsf,,da exist uniformly in n), then Jx-f d a exists and lim??JAl - f dv(a) = 0 so that Iirn?, J E f,,da = J E f d a for all v(a)-measurable sets including X. Conditions (a), (b), (c), and (d) imply that lim,, Jab 1 f,, - f dv(cr) = 0, for all - 00 < a < b < co. Since
If,
I
I
+
I dv((r) 1 5 ibI f,,- f 1 dv(a), it follows that lim7,J," If,, 1 dv(a) = Jf I f 1 dv(a) for all co < a < b < co. In view of condition (e), the iterated limits theorem applies to give us the existence of Jal I f 1 dv(a) = lim,, Jx If,, 1 dv(a). 1 J: 1 f, 1 dv(a)
-
.f" I f
' 0
0
-
Since
I J I 1 f,,- f I d v f a ) - I f,, f I dv(a) I 5 1 Jx 1 f,, I dv(a) - 1" I f , , I dv(a) I t 1 I f 1 dv(a) j f I dv(a) 1, -
-f bn
JdY
-
/L
it follows that
JI
I
uniformly in rz. Since on the other hand, lim,, f,, - f dv(a) = 0 for all a and b, we can apply the iterated limits theorem to give: lini 71
f, I f ,
.
1 dv(a) = lim lini Ti' I f,,- f I dv(a) = lim lini J': I f,, f 1 dv(a) = 0. -
f
?I
0,h
-
a,C
n
We have the following theorem of' converse type:
12.
28 1
UNBOUNDED MEASURE FUNCTIONS
12.10. If "(x) is of extended bounded variation on X , iff,(x) and f(x) on X are such that f7,daand f d a exist, and if lim, J E f,da =
sx
JAl
1
I
J E f d a for every v(a)-measurable set E, then 2, JEmf,,da converge uniformly in n for each sequence Em of disjoint v(a)-measurable sets, and sEf,da are uniformly absolutely continuous relative to
V(.).
For if we set P,(E) = JEf,,d@,then the P,,(E) are s-additive on the class of v(a)-measurable sets, lirnJn(E) = p(E) = J E f d a , and P,(E) are absolutely continuous relative to ~ ( a ) Then . VII.8.17 and VII.8.18 apply, and yield at once the conclusion of the theorem. In case the function a ( x ) is monotonic nondecreasing, we can state : 12.11. If (a) a(x) is monotonic nondecreasing on X;if (b) f , andf are such that limJ, +f(cr), if (c) Jx f,da exists for each n, and if (d): 2, J E m f,da is uniformly convergent in n for each sequence (Em}of disjoint a-measurable sets, then fda exists, and limn jEf,da = J E f d a for all a-measurable E. In order to prove this we derive first the following
sx
12.12. Lemma. Under the conditions (a), (c), and (d) of the theorem
the series Zm lEm I f , [ da converges uniformly in n for every sequence Em of disjoint measurable sets. The proof of this lemma is by the contrapositive method, and follows the procedure of the proof of VII.8.12. Suppose if possible 2.7nJEm/ f,, Ida do not converge uniformly in n, for some sequence Em of disjoint a-measurable sets. Then there exists e > 0, such that for every N , there exists nN > N and mN > N such that 2;.' J E m fn, Ida > e. Take N = 1 and m, and n, so that 2zl J E m I f n l Ida > e. Then there exists an integer m,' > m, such that
I
Let El' be that subset of 2;;'E, on whichf,, is invariant in sign (that is entirely positive or zero, or negative or zero) and J E 1 ,f.,, da > e/2. Take n2, m, so that n, > m,' and m, > m,' and 2z=m, JEm da > e. Then we can find m,' such that JEm I f n , da > e, while .LYE;+~SE,, 1 f ? L z [ da < e/4. E,' is then the mbset of Z,"z'Em on which
I
.Z:irn
1
Ifnzl
1
282
VII. LEBESGUE-STIELTJES INTEGRATION
I
is invariant in sign and such that JE2,frr, dcx I > e/2. By continuing an increasing the process, we get then a sequence of disjoint sets Ei', sequence of integers ni.such thatfni is of fixed sign on Ei', JEt,fnjdaI > e/2, while n
fn2
I
The last inequality holds because ZF=i+lEi'is contained in the set
It follows that for each i:
Consequently the series ,ZF=l JEk,f7,i da does not converge uniformly in i; which contradicts the assumption (d). To return to the proof of our theorem, we show that the condition that for each sequence of disjoint measurable sets Em, the series ZTflJ E m Iffl I dcx converges uniformly in n, gives the uniform absolute continuity of J E f,,dcx relative to a, and the uniform existence of Jxf,da. The first of these can be inferred from the proof of VII.8.12 if one observes that the finiteness of a ( X ) does not enter into the proof. 11, For the second statement, we note that if we set Em = (m, m for all positive and negative integers in, then ZmJEm If, da converges uniformly in n, which implies that lim Jf If,, I da exists uniformly in n as a + - co and b + co. It follows that the hypotheses of VII.12.9 are fulfilled and we can conclude that limn J E f n d a= J E f d a for all v(a)-measurable E. For the case of a set of functionsfJx) for q on a directed set Q, the following theorem analogous to VII.9.2 can be proved:
I
+
+
12.13. If (a) a ( x ) is of extended bounded variation on are such that Jxf,dcx exists for all q ; if (c)fq -.f(v(a));
lim
X ; if (b)fq(x) if (d)
J fPda = 0,
q7w(a;E)+O
and if Jx If, I dv(a) exist uniformly for qRq,, then Jxfda exists,
12.
UNBOUNDED MEASURE FUNCTIONS
283
1
lim, Jx If, - f dv(a) = 0, and lim, J E f,da = JEfda for every v(a)measurable set E. The following convergence theorem may at times be easier to apply: 12.14. If (a) a ( x ) is of extended bounded variation on X , if (b) f,(x)
Sx
on X and on the directed set Q are such that f,de exists for every q ; if (c) f , converge to f relatively uniformly as to the integrable function g(x) 2 0 in the sense that there exists a v(a)-null set E, such that (e, q,, &,, x on x - E,) :If,(x) - f(x) 5 e * g(x), then Sxfdff exists and lim, Jx If, - f dv(a) = 0. By setting e = l/n, we can select a monotone sequence of q : q , such that limnf,,(x) = f ( x ) except for x on Eo a v(a)-null set. Since J B f,da exists, fJx) is v(a)-measurable, and so f ( x ) is also v(01)measurable, as the limit of a sequence of v(a)-measurable functions excepting for an a-null set. Further, for an e > 0, and a fixed q,Rqe we will have f ( x ) 5 e g(x) f,,(x) except for x on E,, a v(a)-null set.. Consequently by VII.12.2 Jx If dv(a) exists. The linearity of the integral then gives for qRq,:
I
I
I
Jx
and so lim,
jx
+I
I
,fI
1,
I
If, - f I dv(a) 5 e Jx gdv(ff),
-f I dv(a)
= 0.
EXERCISES 1. Show that if a ( x ) is of extended bounded variation and regular on X , then there exist disjoint v(a)-measurable sets X f and X - such that X = X + X - , and for any E with v(a; E ) < m, we have v ( a ; EX+) = a(EX+) and v(a; E X - ) = - a ( E X - ) .
+
SE
2. Show that if for a given v(a)-measurable set E : I f I dv(a) > e then there exists a subset El of E such that f ( x ) is entirely positive or zero, or entirely negative or zero on El and I fda I > e/4.
sE
3. Is it possible to replace condition (a) in theorem VII.12.11 that a ( x ) be monotonic nondecreasing by the condition (a’) that a ( x ) be of extended bounded variation on X , the sets E occurring in the theorem being then
assumed to be v(a)-measurable?
284
VII. LEBESGUE-STIELTJES INTEGRATION
4. Suppose a,(x) and a(x) are of extended bounded variation on X. Find conditions sufficient to guarantee that if f d a , exists for all n, then Jxf d a exists as the lim, Jx-fda,.
sx-
5. Suppose a ( x ) is of extended bounded variation on X, and f ( x ) is such that Jxf d a exists. If J: f d a is the integral on the half-open interval (a, x] in X , is this (indefinite) integral a continuous function of x at all points where a ( x ) is continuous?
CHAPTER Vlll
CLASSES OF MEASURABLE AND INTEGRABLE FUNCTIONS
Any fixed monotonic nondecreasing function a on X = - co < x < + cc defines a class of functions possessing certain measurability and integral properties relative to a. We treat briefly some classes which have proved important and useful especially in connection with the study of linear spaces. Since most of the properties mentioned for a function a(x) of extended bounded variation, that is of bounded variation on every finite subinterval of X , depend on the total variation function which is monotonic, we limit ourselves in this section to the case where a ( x ) is monotone nondecreasing, and bounded on any finite subinterval of X. 1. The Class of a-Measurable Functions
We have already seen in VI.3 that the class of all a-measurable functions f on X is linear and has the absolute property, that it contains if it contains f. Further that a natural mode of convergence in this class is a-convergence or a-measure convergence: f ,+f ( a ) , equivalent to lim,,a(E[)f,, - f > el) = 0 for all e > 0. With the aid of integration, it is possible to set up a metric in this space such that f) + 0 is equivalent to f , , --f f(a), for a bounded. For this purpose we note that the function y = x/(l x) for 0 5 x < co, is a monotone increasing continuous function which maps the positive real axis on the interval [0, 11, and has the property that if x1 > 0, and x2 > 0, then
If
I
I
+
+
285
286
VIII. CLASSES OF INTEGRABLE FUNCTIONS
As a consequence for x, and x2 real we have:
If now S(xl, x,) is a metric on a space X , then the expression S,(x,, x2) = S(x,, x2)/(l S(x ’. x,)) defines a bounded metric on X , such that lim7LS(x7[, x) = 0 is equivalent to lim, S1(x,, x) = 0. We apply this to the space of functions f (x) on X = - co < x < co which are measurable relative to the monotone bounded function a(x) on X .
+
I
1.l.Iff, and f , are a-measurable, then the function fi(x) - f2(x)
(1
+ If,(x)
-f 2 ( x )
1)
I/
is a-measurable and bounded so that
exists and satisfies the metric conditions 0 5 S(fl, f 2 ) = 8(f2, f , ) and d(f,, f 2 ) 5 S(fl, f,) S(f, f 2 ) for all f,, f,, and f,. The conditions S(fl, f 2 ) = 0 is equivalent to the statement that If,(x) - f,(x) I / (1 I fl(x) - f2(x) 1) vanishes excepting for an a-null set, so that f, and f 2 differ at most on an a-null set. The value of B(f,,, f 2 ) is unchanged if we replace each function f , and f, by a corresponding function differing from it by an a-null set. 8(f,, f 2 ) gives us a metric on the a-measurable functions if we consider two functions differing on an a-null set as equal.
+
+
1.2. A necessary and sufficient condition that lim,d(f,, that f , +f ( a ) .
If,
f)
=
0 is
+ If,
- f I /(I - f 1) +Ofa). For suppose f , + f ( a ) , then For if x/(l x) > e, and e < 1, then x > (1 - e)/e, so that the set E [ If , - f I / (1 If,, - f 1) > el is contained in the set E [I f , - f 1 > (1 - e)/e] for e < 1, and in a-measure convergence we need worry only about the case when e is small. By the integral convergence theorem for a uniformly bounded sequence of a-measurable functions with a bounded (VII.8.7), it then follows that
+
+
On the other hand, we note that for any sequence g, of functions integrable with respect to a , the condition lirnyL Jx-I g , I da = 0 together
1.
I
a-MEASURABLE FUNCTIONS
287
with En, = E[I g , > e] implies limn e * a(E,,) = 0, so that g , + O(a). Now E [ l f , - f / (1 I f , - f > el contains the set E [ l f , - f l > el, so that 1im7?S(f,,,,f) = 0 implies f,+f(a). In case the function a ( x ) is not bounded, we alter the procedure slightly. Let I, be the sequence of disjoint half open intervals (m, m + I], m ranging over the positive and negative integers so that Z,I, = X . Let c, be a sequence of positive numbers such that Zc*, a(I,) < co. Then the step function g ( x ) = c, for x on I , will be a-integrable. It follows that for any two a-measurable functions f, and f,, the function
I
+
I)
is integrable with respect to a so that we can set S(fl, f,) = Jx hda. This will define a metric as in the case when a is bounded as it possesses metric properties for each I,n. Since
it follows that the condition lim,S(f,,f)
=0
implies the condition
for every I,, so thatf, +f ( a ; x on I,). From this we conclude that f , + f ( a ; x on E), where E is any bounded subset of X . On the other hand, suppose f , - + f ( a ; x on I,) for all I,. Then
for each k. Moreover,
giving uniform convergence in n for the series on the right. The iterated limits theorem then gives us lim,6(fTL,f ) = 0. It follows then that for our definition of S(f,,f,), the condition lim,S(f,, f ) = 0 is equivalent to the condition: f,, + f ( a , x on I,) for all I,, or f, + f ( a , x on E ) for all bounded a-measurable E.
288
VIII. CLASSES OF INTEGRABLE FUNCTIONS
f,+ f ( a ; x on E) for all bounded a-measurable E, is weaker than f , , + f ( a , x on X ) . 1.3. For unbounded functions a ( x ) , the condition
1.4. Since the space of a-measurable functions is complete under a-measure convergence, it follows that it is also complete under the metric convergence, that is, lim,l,md(f,,,hfl)= 0 implies the existence of a function f such that lim ,L B(f,,, J] = 0. The space is also complete in the more general sense, iff,(x) is a set of a-measurable functions with q on the directed set Q, and limql qzB(f,l, f q 2 )= 0, then there exists an a-measurable function f such that limq6(fq, f) = 0. The last sentence is a result of the following general property of metric spaces : 1.5. Let Y be a sequentially complete metric space of elements y in the sense that lim,,,,,d(ym,y,,) = 0 implies the existence of an element y of Y such that lim7,8(y,,,y ) = 0. Now if y , is a set of elements on directed Q with limql q25(yp1, yq2)= 0, then there exists an element y of Y such that limq8(y,, y ) = 0. The proof follows the procedure for the case when the metric space is that of real numbers (see 1.2.11). 2. The Class of Almost Bounded a-Measurable Functions
For a monotonic nondecreasing function a ( x ) on X
= -co < x <
+ co, the class of almost bounded functions has already been defined
(see VII.2.4.1) as the class of a-measurable functions for which there exists an N > 0 such that a ( E [If . : N ] ) = 0. This class of functions is usually denoted by M or L". It is linear and has the absolute property.
I
2.1. If we define:
llfll
=
g.1.b. ( N such that a ( E [ f > N ] ) = O),
I I f I I is a norm on M , with the following properties: (a) l l f l l 2 0; llfll = 0 if and only i f f = 0 except for an a-null set. 1 4= l a I llfll and II I f 1 II = Ilfll. (b) 1 (c) I1 5 II If, I + I II 5 llfi I1 + 11-
then
Ilfl +fi
If2
Ilf2
(d) I f f belongs to M and g is such that g is a-measurable and
2.
1"8$T)
289
ALMOST BOUNDED a-MEASURABLE FUNCTIONS
1
< f ( x ) except for an a-null set, then g belongs to M and
(I.
Properties (a), (b), and (d) and the first inequality of (c) are obvious. For the rest of (c), with e > 0, let N l and N , be such that N , > Ilf, > N I - e and N? > > N , - e. Then a(E,) = a ( E [ / f , > N , ] ) = 0 and a(E,) = a ( E [ l f , > N , ] ) = 0. If En = El E,, then cc(EO)= 0. On CE,, the complement of En Ifi I If, 5 Nl N,, so that E [ l f , + f , > N , N , ] is a subset of E, and so an a-null set. Then \If, +f,( 1 5 N , N, < 11 2e for all e > 0 and so for e = 0. If we consider functions of M as equivalent when they differ by an a-null set, then the conditions for a norm are satisfied by f We have :
I l f , 11
+
1
I
+
I
+
I
+
11
+f, I If, I + [ I f , 11 + I l f ,
11 I I.
/I
-f = 0 in the class M , is that f,,converges to f uniformly excepting for an a-null set, that is, that there exists a set En for which a(E,) = 0 such that lim,if71(x) = f(x) uniformly for x on CE,. The sufficiency is obvious. For the necessity suppose lim,z If,, - f = 0, so that (l/m, n,, n 2 n?,,):llf,, - f 5 l/m.Let E,,, = E [ ( f l l - f > l/m].Then a(ETI,) = 0 if n 2 n,. Let E, = ZmZ,", Ellm. Then a(E,) = 0. If x belongs to CE,,, then x is in no En, for n 2 n,, that is for n L n, and all x in CE, we have If,(x) - f(x) 5 l/m. This means uniform convergence off,, to f on CEO.
2.2. A necessary and sufficient condition that lim,, Ilf,
I
(1
I
11
I
11
11
2.3. The space M is complete in this norm, that is, limn, f, -f, = 0 implies that there exists an f i n A4 such that 1imT2 -f = 0. Since lirnrl~,If,, - f, = 0. We have (1 /k, nk, n 2 nk, m L nk) : -fntll5 l/k.Set E , ( m k = E[lf,, -A,, > l/kI. Then a(E,,,,k) = 0 if n 2 nk, and m 2 nk. Let E, = Zk PC L'2kEllmk.Then a(E,) = 0,
II
I
l f,
l fTt
11
I
I
and on CE,: lim9t,, If,,(x) - f,,,(x) = 0 uniformly. Then there exists an u-measurable function f defined on CE,, and arbitrary on E, such that lim71fn(x)=f(x) uniformly for x on CEO.Consequently, Iffx) 5 If,,(x) e for some n > n, and x on CEO. Since f , , is almost bounded it follows that f has the same property. Moreover, since limnf,,(x) =f(x) uniformly on CE,, it follows that
I
I+
Ilf,!- f 11 = 0. Since in the space M , convergence depends on a metric, M is also complete relative to the convergence of a set f,(x), q on a directed Q.
290
VIII. CLASSES OF INTEGRABLE FUNCTIONS
2.4. We recall that if a is monotone bounded and f is in M , then
JAYfdaexists. Also iff,, and f are in M and lim, I If, - f I I = 0, then limn JX If, - f I da = 0. These statements need not hold if a! is unbounded. 3. The Space L'
For a given monotone a on X , L1is the class or space of L-integrable functionsf on X,that is such that Jx-f d a and so JX da exists. We have already noted in VII.8.1 and VII.8.2 that L' is linear and possesses the absolute property. We have also shown (VII.12.9) that if (a) f , belongs to L1,(b) f,+ f ( a ) , (c) J E f,da are uniformly absolutely continuous in n relative to a , and (d)
If 1
lim
(a,b)+(-m,
*
[; I f ,
Ida!
I
exist uniformly in n, then f is in L1 and lim, Jx Ifn - f da! = 0. 3.1. If we introduce in L' the notation Ilf
11 = Jx- If I da, then for f
in L1:
I If I I 2 0; I If I I = 0 if and only iff = 0 except for an a-null set. (b) l l c f l l = I c I llfll; II If1 II = IIflI. (c) llfl II 5 Ilf, II + II(d) if g is a-measurable, and f is in L1,and I g(x) 15 If(x) I except for an a-null set then g is in L1 and I I g I I 5 I I f I I. (a)
+f2
Ilf2
Properties (a), (b), and (d) are obvious; (c) follows from
If I
da has properties (a), (b), and (c) it is a norm on L', Since JX. provided functions which differ on an a-null set are considered equivalent.
Ilf 11
I
= Jx If da, that is if f,, is a sequence of functions in L1 such that limm,[\ I f , - f , = 0, then there exists an f in L1 such that
3.2. The space L1 is complete under
11
4. THE For if EmmnJx If, limmneafE[lf,
SPACE
-
L"
f , Ida
WITH
= 0,
0 < p < Go
29 1
then for fixed e
I > el) 5 limm,lSX If, -f, I da = 0,
-fn
so that the sequence of functions f,L satisfies the Cauchy condition for a-measure convergence. Consequently by VI.5.7 there exists a sequence of functions f,,, and a function f such that lirnJ,Jx) = f(x) except for an a-null set and f , &+f ( a ) . From the inequalities
SE If,
Ida 5
SE
-fnf
If,l
and
j E Ifnt
- f , Ida 5
Ida
SX ,fI
+ .IE If, -fn
Ida
Ida
for any a-measurable set E, it follows that (e, n,, m 2 n,, n 2 n,) : J E I f , da 5 e J E If, da. Now J E If, da is absolutely continuous relative to a, so that for a ( E ) 5 d,, we have J E If, dor 5 e. Then also for a ( E ) 5 d, and n 2 n,:JE If, da 5 2e. Since there are only a finite number of n < n,, for each of which J E If, da is absolutely continuous, it follows that there exists a d,' 5 d, such that for a(E) 5 d,' and for all n : J E If, da I e, or absolute continuity uniform in n. In a parallel way we show that the
I
+
I
I
I
I
I
I
lim
(a,b)+(-oo, rn)
Ib If,,I da
exist uniformly in n. We can then conclude that f is in L1 and limn
jxIf,
=
limff
[IflL
-fll
= 0.
4. The Space Lp with 0 < p < 03 4.1. Definition. The space Lp,with 0 01
< p < 00 relative to a monotone
Ip
is defined as the space of all functions f on X such that Jx ( f da
exists. If a is bounded, and p 1 I p 2 , then the space Lpl is contained in the space Lp2. For if a is bounded, admission to Lp depends particularly on the large values of f(x). If If(x) > 1, and p 1 L p 2 , then If(x) 2 If(x) Ipz, so that if Js If lpl da < 03, then Jx If l p z da < 03 also. For a unbounded, no such simple relation exists between the spaces Lp. For a bounded, the class M of almost bounded functions is contained in Lp for all p > 0, and so is sometimes labeled L".
I
Ipl
292
VIII. CLASSES OF INTEGRABLE FUNCTIONS
Lp obviously has the absolute property of containing If I withf. It is also linear. For iff is in Lp, so is cf for any constant c. To obtain the additive property we need the following:
4.2. The space
4.2.1. If a 2 0 and b 2 0, then
+
2?j1(ap bP) 5 (a 4- b ) p S up 4- bP if 0 < p 5 1, and
ap
+ bP5 ( a ib ) P 5 2p-’(ap + bP) if 1 5 p < cc. function (1 + ~)~/(l + xp) on 0 2 x < cc assumes
the For the value 1 at x = 0, approaches 1 as x + co, and has a maximum or minimum at x = 1. If p > 1, the value 2p-1 is maximal, if p < 1, the value 2”-l is minimal. If we replace x by b/a, the inequalities mentioned result. It follows that: 4.2.2. If a and b are any two real (or complex) numbers, then
(a+b[P5(~aI+IbI)P5jaIP+ IbIP if p S l and
I a + b I P 5 2 p - 1 ( I a [ p +I b l P ) if p 2 1 . These two inequalities can be combined into a single statement
Ia+bIPZgfp)
IbIP)
if we define g(p) = 1 for O < p i 1, and g(p) In view of VII.12.2 this inequality yields: 4.2.3. If p
> 0, and f , and f 2 are in Lp, then f l
I
Ip
=
2p-1 if 1 S p < c o
+f 2 is in Lp and
For 0 < p < 1, Jx fi - f 2 da plays the role of a metric on Lp. But Jx da is not a norm, since JAY cf da = c sx If IPda. The spaces Lp with 0 < p < 1 are not normable, but we do not demonstrate this here.
I f Ip
IP
I Ip
I I
Ip
> 1, Jx If da is not a metric, but (JLy If dOr)l/” is not only a metric but a norm on L p for 1 5 p < co. The expression obviously satisfies the conditions (a), (b), and (d) listed for [ 1 f in Section VIII.2. In order to show that the triangle
4.3. For p
II
4. THE SPACE L”
WITH
0 < p < CO
293
inequality (c) also holds, we develope some inequalities which have (c) as consequence and are interesting in themselves. We start from the statement that for x 2 0, and p > 1 : xp2px+ 1 - p
or p x S x p + p -
1.
This follows from the fact that x” for p > 1 is concave upward and so the curve y = x p lies entirely above its tangent at x = 1 . If we set x = a/c, then for a 2 0 and c 2 0: ac“’
If b
=
(I/p)ap
+ (1
-
l/p)cp.
cp-’, then for a 2 0 and b 2 0 ab 2 ( l / p ) a ”
If we let p‘
=p/(p
-
+ (1
l/p)bp/(pl)
~
l ) , so that ( l l p )
ab 5 (1 / p ) up
+ ( l / p ‘ ) = 1, then
+ (1 / p ‘ )bp’,
so that if a and b are any real numbers,
I ab 12 ( l / P ) I a Ip + ( 1 / P / ) I b Ip’* For p
=
2, we get the well known inequality 21ab15a’+b2.
It now follows that if fi is in Lp and f, is in Lp‘, then f,.f, is in L1 and
jxIfi 3,I da 5 ( l / P ) jxIf,I p da + ( l / P r )jsIf, I p’da. If in particular Js If, Ip da 1 and I f i Ip’da 1 (that is f, and =
=
JAY
f,have norm l ) , then
jxIf, -f,I d a S 1. I Ip
If for g, in Lp and g, in Lp’ we set f,= gl/(Js g, da)llp and f2= g, da)l’p’, so that f, and J, each have norm 1 in their space, then we can conclude:
g,/(Js
I Ip’
4.4. If g , is in Lp and g , is in Lp’with (1 / p )
is in L1 and
+ (1 / p ’ ) = 1 , then g,
*
g,
294
VIII. CLASSES OF INTEGRABLE FUNCTIONS
This is known as Hoelder's inequality. For p Schwarz inequality : 4.4.1. If g, and g , are in L2, then g , * g , is in
=
2, it reduces to the
L1and
The Schwarz inequality can also be deduced from the fact that L2 is linear and that
is a positive quadratic form in c. To obtain the triangle inequality for [ I f = (Jx If (pdu)llpwe assume thatf, andf, are in Lp and set g , = or and g , = ( + If2 in Hoelder's inequality. For then g, is in Lp and g , is in Lp', since p ' ( p - 1) = p . Using the additive property of integrals we obtain:
II
Ifi I I f , I
Ifi I
We proceed to show that:
[If I I
Ip
4.5. If = (Jx If da)'lp, then the spaces L p with p > I are complete in terms of this norm. Assume that lirnm,?, - f , = 0, or equivalently limm,n[x -f , du = 0. Then for e > 0, limm,,,e u ( E [ -f , > el) = 0 so that the sequencef,L satisfies the Cauchy condition for a-measure convergence. Consequently, there exists a subsequence f , , and a
Ip
[If,
11
If,
1'
If,
3.
THE SPACE
Lp
WITH
0 < JJ < CO
29 5
function f such that lim, f , , ( x ) = f ( x ) except for an a-null set. Then lim, Ifnk(x) = If(x) except for an a-null set. Since the f , , ( x ) are a-measurable it follows that f , , ( x ) + (x) /"(a). We now show that for any sequence of disjoint a-measurable sets Em,the series Zm J E , Ipda converges uniformly in n. For if p > 1, then
Ip
Ip
I
If
Ip
,fI
IfR
Since Jx f, - f ? ( ,Ipda = 0, it follows that ( e > 0, n,, n 2 n,, n' 2 n,) :ZmJEm lpda 5 2'-' (e Z?,,JET,* Ipda). If n' = n,, we can select me in such a way that for m' 2 rn,:Z,: J E m I f T , lpda 5 e. Then for n 2 n, and m' 2 me we have ZwyJEm lpda 5 2pe, from which the uniform convergence of the series Zm JEm I f , Ipda in n can be deduced in the usual way. Applied to the subsequence f,,,it now lpda follows by VII.12.11 that Jx IfIpda exists and that lim, Jx = Jx Ipda. The same procedure gives us that if g ( x ) is'any - g lpda = Jx - g Ipda. If we set function in Lp, then lim, Jx g ( x ) = f,(x), we have lim, Jx - f, I 'da = Jx I f -f, I pda for all n. Consequently,
+
If,
If,
If,
If
If,
, fI
If
which concludes the proof of the completeness of Lp in the norm. As a result of Hoelder's inequality we can state: 4.6. Iff, and
limn 11 f, -f Jxfgd@. For
f are functions in Lp with 1 < p < co, and such that
11 = 0, then for every function g in Lp':lim, Jxf,gda
=
Since for g in Lp',the integral Jxfgda is linear on Lp,it follows that Jxfgda is a linear continuous functional on the linear class Lp. It can be shown that every linear continuous functional on Lp has this form.
296
VIII. CLASSES OF INTEGRABLE FUNCTIONS
5. Separability 5.1. Definition. A topological space X in which the notion of limiting point is defined, is separable if there exists a sequence or denumerable set of elements {x,,}of the space which is dense in the space. In particular if X is a metric space, than X is separable if there exists a sequence of elements {x,} such that for any x of X and e > 0, there exists an n, such that 6(x, xne)< e.
5.2. If X is a separable metric space, and X , is a subset of X such
that the mutual distances of any two distinct elements x’ and x” of X,, is greater than some fixed positive constant e, then X,, will consist of a denumerable number of elements. For if the sequence x,,is dense in X , and x’is any element of X,,, there will exist an element x , , ~such , that B(x‘, x,,,)< e / 2 and d(x“, x , )~ > e / 2 for all elements x” # x’ of X,, since B(x‘, x”) > e. This sets up a one to one correspondence between a subset of { x , ) and the set X,, so that X,, is denumerable. - 03 < x < 03, and such that there exists a sequence of disjoint a-measurable sets E , such that a ( E J > 0 for all n, and if A4 is the space of almost bounded functions f(x) on X relative to a , normed by the almost least upper bound, then M is not separable. For let q stand for any subset of the integers: q = n,, ... nk, ... and define f,(x) = x(ZqE,;x), the characteristic function of ZgEr1. If q1 f q2 then -fq2 = 1, since the two functions will differ on some subset E,,, which is of positive measure. But the set of q, being the set of all subsets of the integers is nondenumerable, being of the same order as that of the real numbers.
5.3. If a(x) is monotonic nondecreasing on X =
lifqj
(1
5.3.1. If a ( x ) = x on [a, b ] , that is a-measurable sets are Lebesgue
measurable, then the space of Lebesgue measurable almost bounded functions M is not separable. - co < x < co, then the spaces Lp for p > 0 relative to a are separable. We prove this for p = 1, the other cases are entirely analogous. There are various sequences of functions which belong to L1 and are dense in L’. One such sequence consists of all the functions g,(x) which are finite, rational, interval, step functions on X .
5.4. If a ( x ) is monotonic nondecreasing on X =
5.
297
SEPARABILITY
5.4.1. g ( x ) is a finite, rational, interval, step function on X , if there
exists a finite number of disjoint, finite half-open (on the left) intervals I, ... Ik having rational end points, such that g ( x ) has a rational value on each I,, and vanishes on the complement of ZJ,. Since there are at most a denumerable number of such interval sets, and the number of distinct rational valued step functions on any set of intervals is denumerable, the set of step functions g ( x ) so defined is denumerable. Any such g ( x ) is bounded and a-measurable and so in L’. Suppose now that f ( x ) is in L‘. Then f is a-measurable and
f;
. 1.f 1 dcx
lh-n
(a,b)+(-m, a)
I
I
exists. For a given e > 0, there exists an integer N such that Jx If da - JT,” da < e. If we set f N ( x ) = f ( x ) for - N < x 5 N, and f N ( x ) = 0 for x > N , and x 5 - N , then Jx If - fN da < e. For M an integer, set f N A M ( x= ) f N ( x ) for x such that f N ( x ) IS M , and fNAtf(x)= M if f N ( x ) 2 M , and fNIf(x) = - M if f N ( x ) 5 M. Then fNAw(x) differs from f N ( x ) only on EAT.lf = E[IfN,,f > MI. Moreover since JT; f N da exists, lim,la(EN.lf) = 0. Then limM JT; f N - f N M da = 0, and for e > 0 there exists M,, such that J1,” f N - f N , l f o da < e. The function f N A I f u ( xis) bounded ameasurable and by VI.6.4, there exists a sequence of interval step functions h,,(x) on - N < x 5 N such that limJz7t(x)= fN.lfu(x), except for an a-null set. These step functions can be taken of the type of the rational step functions defined above and bounded by M , that is, there exists a sequence of finite rational interval step functions M for - N < x 5 N , and g , ( x ) such that - M S g , ( x ) 5 g,L(x)= 0 for x < N , or x S - N , such that lim,g,,(x) = fNAlfo(x) except for an a-null set. By the integral convergence theorem we then have lim, J_’,” g , - fNMo da = 0, or for e > 0, there exists an n such that I g , - fNAJfodw < e. Consequently,
If
I
I
I
I I
I I
I
+
I
I 1
+
+
I
+
I I
This means that for 3e > 0 there exists one of the rational valued
298
VIII. CLASSES OF INTEGRABLE FUNCTIONS
interval step functions g(x) such that Jx I g - f I da < 3e, so that the denumerable set of rational interval step functions is dense in L’. Obviously the same procedure is usable for the case Lp,with d(fl,f2) = -f2 lpda if 0 < P < 1, and Wl, f 2 ) = (SX- Ifl -fiIpda)l’p if 1 5 p < co. We can also prove in the same way that the space of a-measurable functions on X = - co < x < co is separable when metrized as in VIII.1.2 above by the condition
sX-Ifl
+
where h(x) is an appropriate nonvanishing positive valued a-measurable step function which is integrable with respect to a on X . 5.5. The class C, of continuous functions each of which vanishes on the complement of some bounded interval of X , is dense in L1. We might observe that in the proof of the separability of the space L1, we have incidentally demonstrated that the class of finite valued interval step functions, that is, functions which are constant on each of a finite number of disjoint intervals I, ... Zk and vanish on the complement of 2Ji, is dense in the space L1. If we invoke the theorem that for any a-measurable function f(x), with a ( x ) bounded, there exists a sequence of continuous functions f,(x) such that lim,f,(x) = f(x) except for an a-null set, we can show that the class C, of functions continuous on X and vanishing on the complement of some bounded interval is also dense in L1.We need only apply the theorem to the functions f N J f oon the interval - N - 1 S x 5 N 1, defined in the proof of VIII.5.4. For if the sequence of continuous functions g,(x) is such that lim,g,(x) = fNM,(x) except for an a-null 11, we can assume that g,(x) = 0 for I x I set on [- N - 1, N 2N 1, and I g,,(x) I 5 M , for all n and x. Consequently,
+
+
Since
+
6.
THE SPACE
L'.
ORTHOGONAL FUNCTIONS
299
the density of the class C,, of functions continuous on X and vanishing outside of a bounded closed interval, in the space L1 is immediate. If X is the finite closed interval [a, b ] ,we can use the fact that any continuous function on [a, b ] can be uniformly approximated by polynomials, and any polynomial can be uniformly approximated by polynomials with rational coefficients - a denumerable set - to give us an alternate proof of the separability of L1 on a finite interval. Similar statements hold for Lp with p > 0. 6. The Space L2. Orthogonal Functions. Riesz-Fischer Theorem
Among the spaces Lp, the space L2 for p = 2 has received particular attention because on the one hand it is a generalisation to the continuous variable of Euclidean space in n dimensions with Euclidean distance 6(x, y ) = (Z,"=l(xL-y i ) * ) l l 2replaced by (Jx (f g)'da)'/'. In addition, iff and g are in L' then f . g i s in L', or is integrable. As in Euclidean space this fact allows us to set up an inner product for pairs of functions f and g of L', by the condition (f, g ) = Jx f . gda. This inner product is bilinear, symmetric on L2, and has the property that (f,f ) = f = J z y f ' d a L 0 for all f of L', vanishing only for the functions which are zero except for an a-null set. The Schwarz inequality in this setting can be written
11 11'
(f, g )
*
(g, f
) 5 (f, f)
*
(g, g ) .
We have been considering throughout only real valued functions. If f ( x -~ ) is complex valued on X , and f ( x ) = g ( x ) i h ( x ) , with i = 4 - 1, then f ( x ) is in L' if and only if g and h are in L1 and Sx f d a = Jx gda i Jx hda. f is in L2 if and only if is in L' or Jx I'da exists. For complex valued functions, it is usual to define the inner product (f, g ) of two complex valued functions in L' as f * gda. Then (f, g ) is Hermitian: (f, g ) = (g, f), linear in f , conjugate linear in g and (f,f) = Jx If I'd@ 2 0 for all f, vanishing only iff is zero except for an a-null set. f is then defined by f Jx I'da, and possesses the usual properties. The space L' is an instance of an abstract Hilbert space H in which there is defined an inner product (f, g ) for pairs of elementsf, g of H. The space is a real Hilbert space, if H is real linear, (f, g ) is realvalued symmetric, and real bilinear on H with (f, f) L 0 for all f of H, and (f,f) = 0 if and only f is the zero of H . The space is a
If
+ If 1'
+
sx If
11 1 I
11 II*=
300
VIII. CLASSES OF INTEGRABLE FUNCTIONS
complex Hilbert space if H is complex linear, (f,g ) is 1 Hermitian :I (f, g ) = ( g , f), linear in f and conjugate linear in g , with (f,f) 1 0 f) = 0 if and only iff = 0 of H . for allf, (f, Most of the developments of this section, could be carried out in the abstract setting. We shall, however, stick to integrals of real valued functions on the real axis, for convenience abbreviate f d a by omitting the X and a, which are considered fixed throughout this section.
Sx
Sf,
6.1. Definition. A finite set of functions fl ...f , , in L2 is said to be linearly independent if f (.,A ... c,,fJ' = 0 implies c, = c2 = ... = c, = 0, or if L':c7f, = 0 except for an a-null set implies c, = C ' = ... = c,, = 0.
+ +
An expansion of the integral f (L'?c,f , ) * gives ZTjc,ciJ f,&, which if fi ...f , are linearly independent is a positive definite quadratic form in c1 ... c , ~ Since . the form z",p~ix,xiwith at, = a . is positive definite only if the determinant of a,i (or det a7!) is posiyive (and not zero), it follows from the definition of linear independence that: 6.2. The functions fl ...f,, of L' are linearly independent if and only
if det
(Sf,&.) > 0.
f, ...f , is called the Gramian of these functions and will be denoted by G(fl ...f , ) .
6.2.1. Definition. The det (f f,&.) for the functions
6.3. In view of the fact that for f and g in L2, the integral
J f gda
plays the role of an inner product, we say that f and g are orthogonal if and only if Jf - g = 0. The only self-orthogonal function is equal to zero except for an a-null set, or equivalent to a zero function. 6.4. Iff and g are orthogonal to each other, then they satisfy the Py-
thagorean equality :
/ff+g)2=p2+/g2. I f f is any function in L' then the function normalized.
f/ll
Sf'
1. f 11 = f / ( S f 2 ) ' / ' is
6.5. A function f in L' is normalizedif it is of unit norm, that is
=
6.6. An orthonormal system of functions f,(x) consists of functions
which are mutually orthogonal and each normalized, that is f f q t.fh. = Bqtq,,, where B , ,, is the Kronecker 6, which vanishes if q' # q" ? q and has value unity when q' = q".
6.
THE SPACE
L'.
ORTI-IOGONAL FUNCTIONS
30 1
Iff,, and f,,, are two functions of an orthonormal system, then
that is, the mutual distance between any two distinct functions of an orthonormal system is . Since the space L' relative to a monotonic nondecreasing function (Y on X - - m < x < co is a separable metric space, it follows that:
+
6.7. Any orthonormal system of functions of the space L' determined
by a monotonic nondecreasing function u on X is denumerable. For if {g,} is a sequence of functions dense in L2 and [f,]is an orthonormal system, then for each q, there exists an n such that - glfq < 1/4. If q' f q" then nq, f np,,, otherwise [If,, < 1/2. Hence the system [f,] is denumerable. We note that any orthonormal system consists of linearly independent functions, since if Z'%c,f,= 0, then for 1 < j I n : Z,c, J f , f , = = ci = 0. The question of setting up a system of orthogonal functions is covered by :
llf,
f,. 11
11
f,, ...,f,,, ... is any sequence of functions in L' which are linearly independent, in the sense that any finite subset of the sequence consists of linearly independent functions, then there exists an orthonormal system of functions p, ... p7,... in L2 linearly equivalent to f , ...f,, ... in the sense that p,, = Z,L=,a,,,f, for all n with a,!,, f 0, and f,/= .1.':,=, c,,,,fQ?m with c,,, = l/ann. This means that p,, is a linear combination off, ...fl, for all n, and f , , is a linear combination of p, ... p,,or the linear extension (f,...f,), which consists of all functions of the form Z:=, c , f, is the same as (p, ... p,,),, for each n. It follows that Q,, must be orthogonal to f, ... 6.8. T H E O R E M . If
fl/-,.
The following procedure for setting up an orthonormal system ... P , ~... equivalent to a sequence of linearly independent functions f,...f , ... is due to E. Schmidt. We set p, = cllfl, and determine c,, by the normalizing condition on p,, so that c,, = l/(Jf12)''2. Set y2 = cI2p1 cZ2f 2 . Then the condition Jp2p1= 0 gives cI2= - c 2 2 Jf'P1, so that p l 2 = c 2 , [ f 2- ~sf2Q?,h11. Since f 2 and f,, and sofi and p, are linearly independent it follows thatf, - (Jf2p,)pl # 0, and we can set cZ2' = l / J (fi (J.f,p,)p.,)'. Assuming that p, ... p,
+
-
302
VIII. CLASSES OF INTEGRABLE FUNCTIONS
+
+ +
pn-l have been determined, we set pTl= c, fi cn2f i ... c , , - ~ f,-, c, f,. Then the conditions Jp, p, = 0 for i = 1 ... n - 1 give C1,, = - C n n Sf,P,, SO that p 9 (= C n r L ( f r L - qZll ( f , P,)PJ. Be-
+
cause of the linear independence of f , ...f , it follows that f , , qiil (SfrLpJp, +O, a n d w e c a n t a k e ~ n , ~ =11s ( f n - q~1'( J f r L ~ , ) ~ J 2 It is possible to determine p, in terms of fi ...f, by substituting the values for p,, i = 1, ... n - 1 in terms of fl ...fn-l in prL= c,,(f, (Jf,p,)p,). This can, however, be done directly if we remember that p, must be orthogonal to f,, ...f,-,. If we set p, = Z; b,,f,, and invoke the conditions Sp, *L.= 0, j = 1 ... n - 1, then 0
=
2 b,, sf&,
j = 1 ... n
-
1
2=1
Now the determinant of Sf,& with i, j = 1 ... n - 1 is the Gramian: G ( f , ...,fnJ, which does not vanish, since the functions f , ...fn-l are linearly independent. Then the relation p r l = L'*'b,& ', and elimination of b,,, i = 1 ... n - 1 yields
=o so that
I
...
I
The determinant on the right is a linear combination of f , ...f,, with unity as the coefficient off, and so does not vanish. To determine b,, by the condition Jp: = 1, we note that Sp, f i = 0, for i = 1 ... TI - 1 so that
6.
L'.
THE SPACE
303
ORTHOGONAL FUNCTIONS
Let {pn}be an orthonormal system in L', and f be any function in L'. Then for the square of the distance off from any function in the linear space determined by p, ... y,,, or ( y l ... y J L we have
2 crnpm)2 I f 2 2 c
(f=
=
jf
cm
-
m
jfpm
c ( J f v m )+2 c (c,
-
m
-
m
+ c ,c: m
/fP,J2.
It follows that the minimum distance o f f from (p, ... P ) , ) ~ is attained for c, = fpm, and has the value [f - Z7, fpm)']l1/'. If g = Cm(fPrn)P,,, then Sgy, = SfP?, for m = 1 n, so that ( f - g h , = 0 and f - g is orthogonal to each y T Hconsequently , to all of the functions in (p, ... p J L and in particular to g = C,(S fpm)pm.We then have the Pythagorean equality
S
(S
.-a
S f ' 1(f =
-
g)'
+ j8'
s
*
We observe also that the minimizing distance from f to (vl... p J L is along the vector f - g which is orthogonal to the linear space (p, ... pJL. It follows that if fl ...f , , is any finite set of linearly independent functions of L', then the minimum distance from f to the set (fl, ..., f J L is attained by a function of the form f - Zc, f , which is orthogonal to (A, ...,f J L and consequently to each f,. This can also be verified by applying the differential calculus to the problem of minimizing F(c, ... c,,) = J (f - Zmcmf,)' as a function of c, ... c,. Since 0 5 (f(JfPm)pm)' = (jpm)',
$
/
it follows that L'; (J fp,)
Sf' 2
* 5 Jf
6.9. Bessel's Inequality. If
for all n, and we consequently have :
{ ~ p , ~ } ,is
any orthonormal system in L' and
f is any function in L', then ZTl(J fp,)' converges and Z,! (J fp,)' 5
S f '.
For the case when X = 0 5 x 5 2n, a ( x ) = x, and the orthonorma1 system consists of the functions: l/.\/%, ( l / d n )cos nx, ( l / d7c ) sin nx, the expressions J fp, become ,
I
,
I _
a,
=
1/.\/%j2" f(x)dx;
b,
=
(1 /dT)
and
s:"
a,
=
f ( x ) sin nxdx,
(l/dn)
f ( x ) cos nxdx;
304
VIII. CLASSES OF INTEGRABLE FUNCTIONS
and are called the Fourier coefficients o f f . By analogy, the term Fourier coefficient is sometimes applied to the J fp,l relative to any orthonormal system {pfl}.As a result of Bessel's inequality, we could then say that for any f of L' and any orthonormal system {p,} of L2, the sum of the squares of the Fourier coefficients o f f relative {p7,}is convergent. For any orthonormal system in L2, the expression a,, = J fp, effects, then a transformation or map of L' on the space I' of sequences of convergent square. The following important theorem is a sort of converse : 6.10. Riesz-Fischer Theorem. If (a,,} is any sequence of numbers
in l', that is G,, a% m
so that lim,z,, J (f, - f,)' = 0. Consequently there exists an f in L2 such that J (f,l - f)' = 0. Then since for every g in L':
(Sf/$
-
[fa2= f [
(fPl
I
-.flg)'S f (XI -fl'
it follows that lim,[ J f,,g = Jfg. Now J f,p, quently Jfp, = arn for all m. Further since
= a,
1
g'),
if n > m. Conse-
An obvious additional question is: Under what conditions is the function f in this theorem uniquely determined (up to an a-null set), and the obvious answer is that f is uniquely determined if and only if any f of L' such that J f i , , = 0 for all n is the zero function. This property of any orthonormal system is called completeness. As a
6.
THE SPACE
L',
ORTHOGONAL FUNCTIONS
305
matter of fact there are three equivalent ways of defining completeness of an orthonormal system in L'. 6.11. Definition. An orthonormal system (9,) in L' is complete if
and only if: (1) J fp,
=
0 for all n implies f
=0
except for an a-null set;
or :
(2) .Z, (J fp,)'
=
Jf' for all f in L 2 ;
or :
(3) limn J ( f
- 2; (Jfp,)p,)'
=0
for all f in L2.
We show that (1) implies (2) implies (3) implies (1). (1) implies (2). Let f be any element of L'. Since 2, (J fp,,)' < co it follows from the Riesz-Fischer theorem that there will exist a function g in L' such that lim,, J (g - L'y (Jfp,)p,)' = 0, J gp, = SfP, for all m and J g ' = z,< (SfPJ'. But Sgp, = S f P , or J ( f - g)p, = 0 for all m implies g = f except for an a-null set. Then g 2 = f ' except for an a-null set so that J f ' = Jg' = El, (Jfp,,)'.
(2) implies (3). This is an immediate consequence of the fact that: -2 : (Sfpm)pm)' = Jf' (Sfp,)2. (3) implies (1). For if J fp,, = 0 for all n, then by (3) J f ' = 0 so that f = 0 except for an a-null set.
':
J (f
6.12. In L2 space there exist complete orthonormal systems, which
by VIII.6.7 are at most denumerable. Since L' is a separable space, there exists a sequence of functions f T Lin L2 dense in the space. If we drop from the sequence { f , } any function which is linearly dependent on the functions f , , for m < n, then the resulting subsequence { frl,)> j will consist of linearly independent functions linearly equivalent to { f , , } . Let {p,} be the orthonorma1 system linearly equivalent to { f , , }. Then any function f , of the original sequence will be a linear combination of a finite number of functions p,. If J fp, = 0 for all m,then J f f T = L 0 for all n. Consequently 11,
[f'=
iff
-
J f f l ,= J f ( f - f J
llfll Ilf-f, ( 1
for all n. Since thef, are dense in L2 it follows that J f = 0, so that the sequence (9,) is a complete orthonormal system in L'.
306
VIII. CLASSES OF INTEGRABLE FUNCTIONS
As a consequence of the properties of a complete orthonormal system, the Riesz-Fischer theorem has as corollary : 6.13. Any complete orthonormal system {p,} in L' sets up a one to
one correspondence between functions of L2 and sequences in 12, by the condition J f p r L= a,. This correspondence is isometric in that
Cn(an- b?J2= X,(Jfi, -
gP,)'
=
( f - g)'.
The preceding results throw some light on the relationship between a function f in L2 and its (Fourier) development in terms of an orthonormal system {p,} : L'F (J fv,p)pT,.The sequence 2; (J fp,)p, is a-measure convergent to some function g in L2. Only if the orthonormal system { p, } is complete are we certain that 2: (J fp,)pm -+ f ( a ) . As a consequence, there exists a sequence of integers nk such that lim, 2:lC(J fp,)p, = f, except for an a-null set. The series L': (J pm)pwL need not converge to f at any point of X. The Riesz-Fischer theorem enables us to prove: 6.14. If L(f) is a linear continuous functional on L2, then there exists
a function g in L' such that L ( f ) = J f * g for all f in L'. We recall that L ( f ) is continuous on L2, if for every sequence f , such that limn Ilf, - f = 0, we have lim, L(f,) = L ( f ) . The expression J f - g for g in L2is obviously a linear continuous functional on L' since
11
On the other hand, suppose {p,} is a complete orthonormal system in L'. Then for any f in L' we have lim, f - 2; (J fp,)pm I( = 0. If L(f) is linear and continuous then
I(
Then L'? (J fp,) (L(p,)) converges for everyf. But since the system {p,} via J fp, sets up a one to one correspondence between functions in L2 and sequences a, in ,'Z where 2 [ a , < 00, it follows that zy a,L(p,) converges for every seqeunce {a,} in 12. We now use the following :
6.
THE SPACE
L'.
307
ORTHOGONAL FUNCTIONS
27
6.15. Lemma. A necessary and sufficient condition that a,,b, converges for all sequences {a,,} in 12, is that the sequence { b p 2 be } in l',
I l2
that is Z, b, < co. The sufficiency part of this lemma is an immediate consequence of the (Schwarz) inequality: (LI,a,,blJ2I LY,? a,, ZT1b , To prove the necessity, we proceed contrapositively, and assume that Z, b , is divergent. Now the Abel Dini theorem (see K. Knopp, " Infinite Series," Blackie, London, 1928, Section 39) asserts that if d, > 0, and Z7?d,, is divergent then Z,,(d,,/(Z:d7Jpis divergent if p I 1 , and convergent if p > I. Consequently s,<( b,, '/ (2: b, ') is divergent but Zn(1b, l'/(L': by, is convergent. If then we set a, = Ib,, I sgn b,l/(LYy b, then C,,a,,b,lis divergent, but z'?, a, is convergent. This contradicts the hypothesis of the lemma and completes its proof. Returning to the proof of Theorem 6.14, since Z,a,LL(p,) converges for all {a,,} in L2, it follows that 2 , ( L ( q , ~ ~converges. ,))~ Consequently by the Riesz-Fischer theorem, there exists a function g in L2 such that
I 12.
I I
I I
I 1')
I 12),
n
JYq,J?JP, - g
Then
1 1'
I I I 1'
11 = 0.
or
This gives us the desired representation.
EXERClSES 1. Show that if a ( x ) = x,that is u-measurability is Lebesgue measurability then on X = - c o < x < c o
XfL f J
=
J"
-00
exp (
-
x')
Ifi(X)
-
.fdx) I d X l ( 1
+I
f X X ) -f 2 X )
II
can serve as a metric for L-measurable functions, where lim,d(fT1,f) = 0 is equivalent to f,+ f (in measure) on every bounded measurable set E. 2. Give a proof of the statement 1.3 above that if cy is monotonic unbounded on X - (- co, a),then f, --f f ( a ; x on E ) for all bounded a-measurable sets E is weaker than f , +f ( a , x on A').
+
308
VIII. CLASSES OF INTEGRABLE FUNCTIONS
3. Show that if {p,} is a complete orthonormal system in L2, then (a) for we have 2,1 f p , s g p , = S f .g ; (b) for any two functions f and g in L2, any two measurable sets E, and E2: 2, sE,p,l SE,pn = a(E,E2).
s
4. Suppose that { f,} is a sequence of functions in L2 such that if n < m, then f, - f,,is orthogonal to f,,. Show that if the sequence f,' is bounded then f , converges to a function f in the sense limn ( f , -f)2 = 0.
s
s
5. Suppose that I f t l } is a sequence of linearly independent functions in
L2. Show that a necessary and sufficient condition that there exist a function f i n L2 which satisfies the infinite system of integral equations f f , = a,,, is that the sequence j 0 a, ...
s
be bounded in n. Hint: Find a solution of the equations l f .f i = a,, i = 1 ... n such that f = Ly c , f L ,and show that the result is a solution of minimal norm for this finite system. 6. Suppose X = 0 I x I 1, and consider the class 5 of functions f ( x ) on X such that Z z ( f ( x ) ) 2converges. Show that iff and g are two functions of the class 5 , then Z z f ( x ) g ( x ) converges and defines an inner product ( f , g ) = Jzf ( x ) g ( x ) on 5.Show that there exist orthonormal systems in 5 relative to this inner product which are not denumerable. 7. Show that if g is a-measurable and L2,then g is in L2.
sf. g exists for all functions f in
8. Show that if L,,al1bnconverges' for all sequences {a,} in I", that is such that 2, I a , 12, < co, with p > 1, then b, is in P', or 2, I b , Ip' < co, where
P'
= P/(P -
1).
9. In an abstract Hilbert space H based on a Hermitian form (f, g ) orthogonality is defined by the condition (f. g ) = 0. Show that in such a space H , it is possible to determine orthonormal systems of elements [p,,], whose linear closed extension is the space H . If [p,,] is a complete orthonormal system and f is in H , then (f, pp) vanishes except for a denumerable set of pp and f = 2,(f,pp)pp.Also if L ( f ) is a linear continuous functional on H, then there exists a g in H such that L ( f ) = (g, f ) .
CHAPTER IX
O T H E R M E T H O D S O F DEFINING T H E C L A S S O F LEBESGUE INTEGRABLE F U N C T I O N S . A B S T R A C T INTEGRALS
In the preceding sections, we have defined and developed properties of Lebesgue-Stieltjes integrals of functions f(x) on the linear interval X relative to functions cc(x), either of bounded variation on X , or of bounded variation on every bounded subinterval of X . Here X is assumed to be the interval - m < .x < co, but the theory can easily be adapted to the case where X is the bounded closed interval [a,b ] or the bounded open interval (a,b), respectively. The integral definitions were of two types, one by an analog of the Rieman-Stieltjes definition using subdivisions into measurable sets instead of intervals and the other, due to Lebesgue using properties of measurable functions depending on the sets E [ , f 2v]. In either case, it was necessary to develop first a theory of measure and of measurable functions. Both procedures lead to the same class L' of Lebesgue integrable functions with respect to a or Lebesgue-Stieltjes (L-S) integrable functions at least for the case when a(.) is of bounded variation on X , which is either - co < x < cx) or a 5 x 5 b. It is natural to pose the question whether the class L' and the integrals of functions of L1 can be obtained by other types of procedure, avoiding perhaps the development of a theory of measurable sets and measurable functions. In this connection, we observe that the class L1for X a bounded interval [a,b ] and ~(x)of bounded variation on [a,b ] includes the class of Rieniann-Stieltjes integrable functions, with the same values for the integrals, so that L1 is an extension of the class of R-S integrable functions, and consequently also of the class of continuous functions on [a,61. In taking our cue from the development of the real number system from the rational numbers, we recall that one extension procedure
+
+
309
310
IX. OTHER DEFINITIONS FOR LEBESGUE INTEGRALS
is to assume that any Cauchy sequence of rational numbers {a,} for which lim,,, I a, - a, I = 0, can be considered a new number and yields in effect the real number system. Similarly if X is a metric space which is not complete in the sense that there exist sequences of elements { x , } of X which are Cauchy sequences satisfying the condition lim,,7nd(x,z,xTn)= 0 but for which no x in Xexists such that lim,d(x,,x) = 0, we can extend the space by adding all Cauchy sequences in X as additional elements. Two Cauchy sequences { x , ) and { y , ) define the same element in the extended space if lim, 6(xYL, y,) = 0. The sequence E = { x , } for which x , = x for all n corresponds to the element x of X , so that a sequence {y,} corresponds to the element x of X if lim, 6(y,, x) = 0. The extended space becomes a metric space if we define the distance between two elements E = { x , ~ and } 7 = {y,} by condition d ( E , 7) = lim, 8(x,, y n ) , and this distance remains unchanged if we replace the sequences {x,} and {y,) by equivalent sequences. If the space of sequences E is metrized in this way, it turns out to be complete relative to Cauchy sequences. 1. The Space L' as the Completion of a Metric Space by Cauchy Sequences
+
If X = (-- co < x < co), and a ( x ) is of bounded variation on every bounded interval, we start out with the space C, of functions g ( x ) continuous on X , each vanishing on the complement of some bounded closed interval. The class C, is linear. If g ( x ) = 0 for x 5 a and x 2 b, then g ( x ) d a ( x ) exists and can be written Jx gda. We introduce a metric or norm in C,, not by the condition that [I g I I = max (1 g ( x ) 1, x on X ) but by the integral norm : I I g I = Jx I g I dv(a), where as usual v(a, x ) = J: Ida I. The norm gives rise to the metric 6(g,, 8 , ) = Jx g , - g , I d v ( a ) . The class C, is not complete under this norm. We show:
Jt
I
I
1.1. T H E O R E M . If a ( x ) is of bounded variation on every bounded
subinterval of X and C,, the class of functions g ( x ) continuous on X , each vanishing outside a bounded closed interval is normed by setting 1 I g I I = SX I g I d v ( a ) , then the extension of C, by Cauchy sequences yields the class L' of functions L-S integrable with respect to v ( a ) . If { g , } is a sequence of functions of C, '(determining " the function f i n L', then Jx f d a = limn Jx g,da.
1.
SPACE
L'
31 1
AS COMPLETION OF A METRIC SPACE
(a) Any function in the completed extension of C, is in L'. This is almost obvious. For since L1 is complete relative to = Jay dv(a), and contains the class C,, any sequence {g,(x)} of C, such that limn,mJx g,, - g , dv(a) = 0 determines a function in L1 up to a v(a)-null set, where (g,x) +f(x) ( ~ ( a ) ) and , there exists a subsequence g,,ksuch that lim,gnk (x) =f(x) except for a v(cr)-null set. Two equivalent sequences determine functions in L' which differ at most by a v(w)-null set and are consequently equivalent in L'. Iff is determined by {g,,), then J,fdct = limli JAY g,da and J, dv(a) = limn Jx- g,, dv(a) = The bounded v(w)-measurable subsets of X are determined by the condition that the corresponding characteristic functions x ( E ; x) are in the completed extension of C,, from which the class of v(a)-measurable sets is determined in the usual way.
I(fll
If I
I
I
If I
I I
llfll.
(b) Any functionf(x) in L' is determined by a Cauchy sequence in C,. In VIII.5.5 we proved that the class C, is dense in L1 in the sense that for any function f in L', there exists a sequence {g?z(x)} in C, such that limvLJAY g,, dv(a) = 0. The sequence {g,,} is obviously a Cauchy sequence in C, which determines f An alternative way of obtaining the space L' as the completion of a class of functions is based on the observation made in the proof of VIII.5.5 that the class S of finite valued step functions is dense in L' under the norm g = JAl g dv(cr). The type of step function g(x) which proves effective here is defined as follows: Let 3 consist of a finite number of disjoint open intervals (xtr,x L r r )i, = 1 ... n and a finite number of points x,,j = 1 ... k disjoint from the intervals. Then g ( x ) is in S, if g(x) = c Lfor x on (x,', x,Ir) and g(x,) = d, and vanishes if x is not in 3. By taking individual points as end points of the intervals, this type includes step functions defined to be constant on sets of closed or half-open intervals. We now define
If-
1 I 11
I
I I
for any such g(x) and a(x) of bounded variation on every bounded interval. This is essentially the L-S integral rather than the R-S integral, since the latter may not exist if g and CI have common discontinui ties.
312
IX. OTHER DEFINITIONS FOR LEBESGUE INTEGRALS
If we set
then the class S of functions g(x) completed under this norm will also yield L1 and a value for the integrals SAY fda for f in L’. If we have at our disposal the v(a)-measurable subsets of X , then L’ can also be obtained as the completion by Cauchy sequences of the class of measurable step functions which are constant on each of the sets of a subdivision of X into v(a)-measurable subsets, and such that g dv(a) = Z7, c , v(a,E7J < co, where g(x) = c , on E,. The use of Cauchy sequences as a basis for obtaining the space L1 of L-S integrable functions is due to N. Dunford: Trans. Am. Math. Soc. 37 (1935) 441-453. He developed a theory in a more general setting which can be adapted to the space of functions of one variable as indicated in the preceding paragraphs.
SLYI I
1 1
2. Construction of the Space L’ by the Use of Osgood’s Theorem
An approach to the L-S integral closely related to the preceding one, at least for the case when X is the finite interval [a, b ] is suggested by the Osgood theorem of convergence (11.15.14) which asserts that if a(x) is of bounded variation on [a,b ] and f,(x) is a sequence of functions uniformly bounded on [a, b ] for each of which R-S J$f,da exists, and such that lim, f,(x) exists except at most on a v(a)-null set, then lim, $f f,da exists. If C is the class of continuous functions g(x) on [a,b ] and if a(x) is of bounded variation on [a,b ] , then we know that $$gda is defined for all g in C. We extend the class C by adding the class of functions f(x) for which there exists a uniformly bounded sequence of continuous functions g,(x) such that lim,g,(x) =f(x) except for a v(a)-null set, and define $1fda = lim, $fg,da, since the limit on the right exists. Because of the Osgood theorem this value will be independent of the particular uniformly bounded sequence of continuous functions used to approach f(x) as indicated. It is obvious that the resulting class of functions is contained in the class of almost bounded functions measurable relative to v(a). On the other hand for any almost bounded v(a)-measurable function f(x) we can find a sequence {g,(x) } of continuous functions which we can assume to be bounded
2.
CONSTRUCTION OF THE SPACE
L1
313
by the almost bound of f(x) such that limngn(x) = f(x) except for a vfa)-null set. Moreover, lim R-S J b gnda = lim L-S J b g,da n
a
n
a
= L-S
j bfda. a
Consequently : 2.1. If we extend the class C of continuous functions on [a, b ] by
adding the class of functions which are limits of uniformly bounded sequences of continuous functions excepting for a v(a)-null set, we obtain exactly the class of almost bounded functions in L1 and the L-S integral of these. When it comes to the matter of obtaining the unbounded functions on L' by this type of extension process, it is obvious that unbounded sequences of functions will be needed. Since there exist sequences of continuous functions {g,(x)} such that limngn(x) = 0 for all x of [a, b ] for which the limit of J:g,,(x)da(x) does not exist, it is necessary to impose some restriction on the sequence {gr1(x)}. One cue to the type of condition which might serve is contained in the basic convergence theorem for L-S integrals on a finite interval [a, b ] , namely, the uniform absolute continuity of the integrals of the sequence of functions involved (see VII.8.8.2). If we start from the R-S J%gda, which determines an interval function (as contrasted with a function on measurable sets), it is necessary to adapt the notion of absolute continuity to interval functions. For this purpose assume a monotonic nondecreasing function a (x), which gives rise to the interval function a ( I ) = a(d) - a (c), for I = [c, d], and a function of intervals F(I). Each of these defines a function on finite sets of nonoverlapping intervals: 3 = (I, ... In) by the condition a ( 3 ) = 2: a(Ik) and F ( 3 ) = 2; F(Ik). Then we define : 2.2. An interval function F(I) is absolutely continuous relative to the
monotonic function a, if for e > 0, there exists d, such that if
It follows at once that: 2.3. If R-S J: fda exists for a of bounded variation, then the interval function F(I) = J,"fda, with I = [c, d ] , is absolutely, continuous relative to ~ ( a ) .
314
IX. OTHER DEFINITIONS FOR LEBESGUE INTEGRALS
For if J s f d a exists, then there exists a finite number of subintervals I, ... Ik and an M > 0, such that a ( x ) is constant on the closure of the intervals complementary to 2: Ii and I f ( x ) I M on each I<.Consequently, J f f d a I M ( v ( a , d ) - v(a, c ) ) for every interval [c, d ] We can then demonstrate an extension of Osgood's theorem in the form :
I
I
I
2.4. Lemma. If a ( x ) is monotone nondecreasing on
[a, b ] ;g,(x) are
such that g , ( x ) 2 0 for all n and x;lim,g,,(x) = 0 except for an a-null set; g,da exists for each n and the indefinite integrals Jf g,da are uniformly absolutely continuous relative to a , then limn g,da = 0. The proof follows that of 11.15.7. If limn J; g,da f 0, there exists an M > 0, and a subsequence n, of the integers such that gnkda > M for all k . We can assume that {g,,} is the original sequence. Let c > 0 be such that c(a(b) - .(a)) < M/2. Then for 0 < e < Jsg,da - M , there exists a subdivision ( r T 2 such that Zbnw(gn;I ) a ( I ) < e. Let I ' denote the intervals I in ( r l l such that g,(x) 2 c for all x in the 1', and I" those intervals of G~ for which there exists an x in I" such that g,(x) < c. Then
Js
Js
Js
since
I C J,
g,da
-
2 gn(xi)a(Ii")
4 g , ; Ii")a(Ii').
15
j
3
j
It follows that for each n, there exists a finite set of nonoverlapping intervals such that JInk
g,da 2
Jb
g,da
-
c(a(b)
-
.(a))
-
e > MJ2.
a
Now the integrals JI g,da are uniformly absolutely continous, SO that there exists a K such that if Zi .(Ii) < K, then ZiJ I z, g , da < M/2. It follows that for every n, the intervals I,, determined above satisfy the condition Zk" ( I s k ) 2 K, a fmed number independent of n. The use of Arzela's lemma 11.15.8 as in 11.15.7 gives us that the set E =
2.
CONSTRUCTION OF THE SPACE
L1
315
1%. 2, I,, on which g,,(x) does not converge to zero is not an a-null set. It follows from this lemma, that: 2.5. If {g,(x)} is a sequence of continuous functions such that lim,g,(x)
I I
exists except for a v(a)-null set and the integrals J g, dv(a) are uniformly absolutely continuous relative to v(a), then lim, Jf g,da exists. For
I
and the double sequence g,, - g, ma since
I satisfies the conditions of our lem-
The extension procedure is now immediate in that we add to the space C of functions continuous on [a, b ] the space of all sequences of continuous functions {g,(x)} such that limptg7L(x) exists for all x excepting a vla)-null set and J I g,, dv(a) are uniformly absolutely continuous relative to ~ ( a ) . To show that the functions f(x) so determined are in L', we can apply the basic convergence theorem for L-S integrals, provided uniform absolute continuity of JI g, dv(a) in the interval sense implies uniform absolute continuity of J E g , dv(a) in the measurable set sense, that is
I I
I I
e > 0, d,, v(a; E ) < d,, n :
I I
jE I g, I dv(a) I < e.
This is proved seriatim. First in the definition of uniform absolute continuity, the intervals I can be replaced by open intervals Zo, where v(a; 1') = v(a; d - 0) - v(a; c 0) if I' = (c, d). For if Z; ... I:n are a finite number of disjoint open intervals such that 2% v(a; I : ) < d,, then there exist intervals I?, in I : at whose end points a is continuous, and such that lim, v(a; I l k ) = v(a; I:).Since Zzv(a; Itk) < 2% v(a; Z:) < d,, we have for all n and k : 2% J I ( g , dv(a) < e, zk and so also 2% J I o I g , dv(a) < e, for all n. Similar steps can be * taken to replace the sum of a finite number of open intervals by open sets, and then in turn open sets by v(a)-measurable sets E to give the uniform absolute continuity defined above.
+
I
I
316
IX. OTHER DEFINITIONS FOR LEBESGUE INTEGRALS
The basic convergence theorem for L-S integrals VII.8.8 now gives us the fact that functionsffx) determined by the sequences of continuous functions are in L1. Conversely, iff is in L1, then there exists a sequence of continuous functions { g , ( x ) } such that lim,,g,(x) = f ( x ) except for a v(a)-null set, and such that lim?zL-S I g , - f dv(a) = 0. Consequently,
Js
lim L-S n,m
Jb
a
I
I g,, - g , I dv(a) = lim R-S J: I g , n,m
-
g,
I dv(a) = 0,
If I, ... Ik is any set of nonoverlapping intervals, then
e > 0, d,,
C a ( I i ) < d,, m
=
n,:
c1 (g, 1
dv(a) < e,
I<
i
i
then for
From this one can deduce in the usual way that the integrals fI g n (dv(a) are uniformly absolutely continuous. Consequently the sequence ( g , } determines the function f. In this extension procedure, the class C of continuous functions on [a, b ] can be replaced by the class S of finite valued step functions, constant on intervals. If g ( x ) = c, on the open intervals ( x z ' , x ~ " ) , i = 1 ... k, disjoint, and g(x,) = di, j = 1 ... m,for x = x j not on the (xt', xz"), and g ( x ) zero elsewhere, then as above we define
I
Sb" gda c c,(a(x," 0) + c g(xj)(.fxj + 0 ) =
-
-
a(x,'
+ 0))
z
-
4 X j
-
O)),
J
or the value of J g d a as an L-S integral. The space L1 is then the extension of S by sequences of step functions converging excepting for a v(a)-null set, for which the integral J I g, dv(a) are uniformly absolutely continuous relative to v(a). It is clear that this method of extension is almost identical with
I I
3.
317
L-S INTEGRATION
and equivalent to that by Cauchy sequences of continuous functions based on the norm Jf g dv(u). Both of these extensions are illustrations of the fact that new spaces may be obtained as extensions of the space of continuous functions by regarding as new elements the class of sequences of continuous functions which have a certain property in common.
I I
3. L-S Integration Based on Monotonic Sequences of Semicontinuous Functions
We recall that the real number system in addition to being the extension of the rational number system by Cauchy sequences is also the extension of the rational number system by bounded monotonic sequences, with a proper definition of equivalence. We try to adapt this idea to the definition of L-S integrals. We again assume X to be the finite interval [a, b ] but limit u ( x ) to be a monotonic nondecreasing function on [a, b]. C is again the class of continuous functions on X . If {g,,(x)} is a monotonic nondecreasing sequence of continuous functions bounded for each x, then lim,,g,(x) = h ( x ) exists finite valued for each x, and by VI.1.5 h ( x ) is a lower semicontinuous function (1.s.c.) on X . Since u ( x ) is monotonic nondecreasing, the sequence of integrals Jf g,,du is monotone in the same way and converges to a finite number or co. If h ( x ) is continuous, thenthe g,,(x) are uniformly bounded and lim?,J,”g,,du = J$hdu. Consequently, it is sensible to define Jf hdu = limn J,”g,,du. This value will be independent of the particular sequence of continuous functions which is monotonic nondecreasing and for which lim~,gll(x) = h ( x ) for x on X . While this can be seen by using Fatou’s lemma (V11.8.10), that is the theory of Lebesgue integration, it can be proved directly. For suppose ,f(x) is any function in C such that h ( x ) 2 f ( x ) for all x. We show that Jf hdu 2 J$f d u . Let f , , ( x ) = g,,(x) A f ( x ) , the lesser of g,, andf. Then limrZ f , , ( x ) = f ( x ) for all x, Jf g,,du 2 Jf.f,,du for all n, and so
+
j b hdu = lim 1’g,,du 2 lim J‘’f,du n
a
a
n
=Jb
n
fdu.
a
If now f,,(x) is any monotonic nondecreasing sequence in C, such that limnf,(x) = h ( x ) for all x on X , then
Sb hdu n
=
lim n
1’ g,du 2 n
Jb
a
f,du
318
IX. OTHER DEFINITIONS FOR LEBESGUE INTEGRALS
for all n so that lim 1L
ibg,da 2 lim
[”f,da.
R
Since the sequences { f,,} and {g,,) are interchangeable, it follows that lim7zJ”fg,,da= Iim?{J”:f,da. In VI.1.6 we have shown that for any finite valued 1.s.c. function h ( x ) there exists a monotone nondecreasing sequence of continuous functions { g > ( ( x )such } that limv,gl,(x)= h ( x ) for all x . As a consequence we can extend J: gda on C to Jf hda, h in 1.s.c. the class of finite valued lower semicontinuous functions on X , and Jf hda is finite valued or co. In the same way, by using monotone nonincreasing sequences of continuous functions we can obtain an integral for every finite valued upper semicontinuous function (u.s.c.) k ( x ) : kda as a finite value or - co. We now follow the basic idea involved in the Darboux integrals for the case when a ( x ) = x , by defining an upper integral fda and a lower integral J,” fda for any finite valued functionf. We define J”,” fda as the greatesclower bound of Jf hda for all lower semicontinuous functions h ( x ) 2 f ( x ) except for an a-null set, and as co if no such 1.s.c. function exists. Similarly, J: fda is the least upper bound of Jf kda for all upper semicontinuois functions k ( x ) 5 f(x) except for an a-null set, and is - co if no such U.S.C. function exists. Then :
+
J”:
Tc:
+
3.1. Definition. f ( x ) is integrable if and only if
7:
s:
fda and - fda are finite and equal. In order to show that any function integrable by this procedure belongs to the class L’, we use results demonstrated in the theory of L-S integrals. By Fatou’s lemma (VII.8.10) any 1.s.c. or U.S.C. function whose integral as defined above is finite, is in L1and has the same value for its integral. If fda is finite, then there exists a sequence of 1.s.c. functions i 7 , ( x )1f ( x ) except for an a-null set, such that Jf h,da -b J”, fda < l/n. If for n > 1, h,,(x) is the lesser of $,L(x)and h,-l(x), then the h,(x) will also be l.s.c., h,,(x) 2 h,+l(x) and limn h,da will be finite and have fda as its value. If h ( x ) = limnh,(x), then h- will be in L’, be finite valued except for an a-null set, and J: hda = $,” fda. Moreover, h(x) 2 f ( x ) except for an a-null set. Similarly, if $1 - fda is finite, there exists a monotone nondecreasing sequence of
7;
7:
J”:
3.
319
L-S INTEGRATION
U.S.C. functions {k,(x)} with k,,(x) 5 f ( x ) except for an a-null set, with limn J: k,da = J: kda = J$fda so that k ( x ) = limnk,,(x) is in L1. If fda = J$f d a , then J : i h - k)da = 0, with h ( x ) 2 k ( x ) except for an a-null set. Then h = k except for an a-null set. Since h ( x ) 2 f(x) 2 k(x) except for an a-null set, it follows that h ( x ) = f(x) except for an a-null set, that is f is in L'. On the other hand, suppose f is in L'. Then there exists a sequence of continuous functions {g,,(x)} such that lim,,glI(x) = f(x) except for an a-null set and liml, J: g,, - f da = 0. By using, if need be, a subsequence, we can select this sequence g,, so that it satisfies the additional condition that for some sequence of positive numbers { e l l }such that ZlIe,, < co, we have J$ g,i+l- g,, Ida < e,,. For m > n, let g,,,(x) = the largest ofg,,(x) ... g,(x). Then g,l,n(x) is continuous, for fixed n, the sequence g,l,rl(x) is monotonic nondecreasing in m, so that lim,,g,lm(x)= h , l ( x ) is a lower semicontinuous function for all n. The sequence {h,l(x)} of 1.s.c. functions is monotonic nonincreasing in n for each x. Moreover,
I
I
I
lim,,h,l(x)
= lirn,~im,g,,,(x)
= E l t g l t ( x )= lim,,g?,(x) = f(x)
;
except for an a-null set, with h , , ( x ) 2 ~ f ( x except ) for an a-null set. Similarly, if g*,,,(x) = smallest of (g,(x) ... g,,(x)) with m > n, then g*,,,(x) for fixed n converges monotonically to an upper semicontinuous function k,,(x), the sequence { k , l ( x ) }is monotonic nondecreasing in n, with k,<(x)5 f(x) and liml,k,,(x) = f(x) except for an a-null set. We proceed to show that lirn,/ J: h,, da and limn J: k,,da are finite and equal to fda. We recall that if a, ... a, is any finite set of real numbers, then
Jt
The same is true of
Then
I (least of a, ... a,J
-
a,
I. Consequently,
320
IX. OTHER DEFINITIONS FOR LEBESGUE INTEGRALS
provided n 2 n, and rn 2 rn, for which Z : et < e and J: I g , -f I da < e. This means that limmTiJ:g n m da = J: fda. Then J: g,,da are bounded in rn for n 2 n,, so that Jfi h , da = lim, Jfi g,tmda exists. We can therefore conclude that Jfi fdu = lim7n)t J: g,,da = limn J: h,$a. Similarly, we show that lim,, J: k,da = Jfi fda. Since
it follows that the upper and lower integrals off agree, and are finite and f is integrable according to this procedure. It follows that the space of functions for which the upper and lower integrals are finite and equal is identical with the space L1, and gives the same value for Jfi fda. For the case of ordinary Lebesgue integration, where a ( x ) = x on [a, b ] the above extension procedure is developed in detail by L. C. Young in his book on “ The Theory of Integration ” (Cambridge Univ. Press, London, 1927). The steps in this method of arriving at an extended integral have been abstracted to obtain a general integral by P. J. Daniel1 in the paper “A general form of integral” [Ann. Math. (2) 19 (1917) 279-2941. 4. Lebesgue-Stieltjes Integrals on an Abstract Set
The theory of Lebesgue-Stieltjes integrals developed in the preceding pages has limited itself to functions of one variable, or functions on the one-dimensional interval. However, many theorems and proofs do not really involve the domain of the function and are valid as well if the linear interval is replaced by a Euclidean space of two or higher dimensions, As a matter of fact, an examination of the theory reveals that it is possible to set up a theory in which the dimension or the character of the domain does not play any particular role. We briefly sketch the salient points in such a theory. We postulate a basic space X of elements x. Among the subsets of X,we single out a class & of sets which we might call “ measurable.” The class & of such sets E of Xis assumed to be an s-ring (or sometimes an s-algebra), i.e., closed under s-addition and differences. In case & is a ring, we note that the totality of sets of & which are subsets of fixed set E of &, form an s-algebra on E, that is, if we replace X by E as the basic space. Suppose then & is an s-ring.
4.
32 1
L-S INTEGRALS ON A N ABSTRACT SET
On Q we assume defined a finite valued function a(E) which is s-additive in the sense that if { E n }is a sequence of disjoint sets of Q (so that Z?[E,belongs to Q also), then a(ZnE,) = Zna(En). Since a denumerable collection of disjoint sets, does not presuppose any order, it follows that if a is s-additive, and { E n }is any sequence of disjoint sets in Q, then Z1,a(En)is convergent in any order (unconditionally convergent) and so Zll a(E,) < co. We also hav,e:
1
I
Q, then a (E) is bounded on Q. If not, then for every M > 0, there exists a set EAwsuch that I a(E,,) I > M . We determine a sequence of disjoint sets El,as follows: Suppose El ... En have been-defined as disjoint sets such that I a(EJ > 1 for i = 1 ... n. Let En+,be such that
4.1. If a(E) is finite valued and s-additive on the s-ring
I
-
If
=
E,,,
- Zz==lEk,
then E,?,, is disjoint to E, ... Eft and
For this sequence {E,} of disjoint sets, L'lza(En)will not converge since a(E,) > 1 for all n.
I
I
Q and given function a(E) on Q, the total variation on E,: v ( a ; E,) is defined as follows: Let CT be any subdivision of E, consisting of the disjoint sets E,/ in Q such that Z7zEn= E,. Then v(a; E,) = l.u.b.u @(En) where the right-hand side may be co.
4.2. For any set E, of
+
I
1,
4.3. If a ( E ) is finite valued and s-additive on 6, then v(a; E ) is a
positive, finite valued, s-additive function on Q. In the first place v(01, E ) is finite valued. For if (T = {E,} is any subdivision of E, and E,: are the sets of the sequence for which @(El,)2 0 while E l are the sets for which a(E,) < 0, then
coI c o n ) I where E,'
= Cu.(E;)
= Z7B;
a(E)
+ I Cua(E,) I
= a(E:)
+ I 4 E J 1,
and En= Z',LE;, sets in Q. But for all
= CUa(E:)-
C u a ( E , ) = a(E;)
-
CT,
~i(Ei).
322
IX. OTHER DEFINITIONS FOR LEBESGUE INTEGRALS r
1
If Xu a(E,J I were unbounded in u, then both a(E:) and a(E;) would be unbounded in u which violates the boundedness of a on Q. Then v(a, E ) is finite valued on &. We observe next that if a ( E ) is s-additive on &, then the function f(o) = ZUI a(E,) I is monotonic nondecreasing as a function of the subdivisions o, directed by successive refinements. It follows that
The s-additive property of v(a,E ) as a function of E is now immediate. Since v(a, E ) 2 a ( E ) for all E of &, we can define the positive and negative variation of a in the usual way:
I
p(a, E)
=
(v(a, E )
I
+ a ( E ) ) / 2 and n(a, E )
= (v(a, E )
-
a(E))12
which will be positive and finite valued for all E of O., as well as s-additive. As in the case of functions of bounded variation on the line, p ( a , E ) is also the 1.u.b. of a(E,) for all E, of Q which are subsets of E and n(a, E ) is the negative of the g.1.b. of a(E,) for E,, of Q in E. As in VII.11.6 this leads, for every E of &, to the existence of two sets E+ and E- in Q, such that E+ and E- are disjoint, E = E f - Eand p ( a , E ) = a ( E + ) , n(a, E ) = - a(E-). If E, is a subset of the set E in &, then E I E f and EIE- will serve as a similar decomposition for El. Since any s-additive function on 6 is expressible as the difference of two positive s-additive functions on Q, it is sufficient to develope an integral relative to positive s-additive functions on Q. Such s-additive positive functions will be called measures. The integral relative to any s-additive function a is then dependent on the existence of the integrals relative to p ( a ; E ) and n(a, E ) , the positive and negative variations of a , and expressible as their difference. Assume then that CI is a positive s-additive function or a measure on Q, E is a set in Q, andf(x) is a finite valued function on E. Then it is natural to define:
f on E is integrable with respect to a , if limUZG f ( x , ) a ( E J exists, where u = {E,} are subdivisions of E in Q, x, is any element in E N ,and the limit is taken in the sense of successive subdivisions or refinements. As in the case when X is a linear interval, we see that: 4.4. Definition. The function
4.
323
L-S INTEGRALS ON AN ABSTRACT SET
4.4.1 If SE fda exists, then there exists a subdivision a. of E such that f is bounded on each set of a,,for which a ( E ) f 0, and .Z M ( If 1, E,)a(E,,) < 00, where M ( I f 1, Ell) is the 1.u.b. of If on El(.
I
4.4.2. If
SE fdu
exists, so does
SE
I
If da and
I SE fda
00
15
SE If I da.
4.4.3. Iff, and f z are functions on E of Q, and such that fE f,da and f,da each exist, and if c, and c2 are any two constants then (c,fi c2f J d a also exists and has as its value c, f,da c2 f,da.
SE
+
SE
SE
+ SE
In particular, if fE fda exists, then SE f fda and fE f - d a exist, and SE fda = SB f ‘da SB f-da. Here f ’ ( x ) = f ( x ) for f ( x ) 2 0, and zero elsewhere while , f - ( x ) = f ( x ) for f ( x ) 5 0, and zero elsewhere.
+
4.4.4. If
SE fda exists and E, is a subset of E in &, then SE f d u exists,
and the function of sets: g(E,) = SE, fda is a s-additive function on subsets El of E which belong to C. Iff is defined on all of X and J E f d u exists for all E of &, then g(E ) = $,Jda is a finite valued s-additive set function on Q and v ( g ; E ) = fe If da for all E of Q. This last holds if X belongs to Q (that is, C in an s-algebra) and ,fdu exists. In some of the theorems on integrals, the notion of measurable functions plays a role. As usual we say that a function f ( x ) on a set E of & is measurable on E, if for all real numbers c, the subsets E [f > c ] belong to Q. The condition E [ f > c] belongs to (f for all c, can be replaced by E [f 2 c ] , E [f < c ] , or €[ f 2 c ] , belongs to (f for all c where all of these sets are subsets of the original E, relative to which Q is a s-algebra. The measurability condition has the consequence that any function measurable on E, is the uniform limit of a sequence of measurable step functions f , , ( x ) = 2-L :;m cmn~(E,,,;x ) , where for instance
I
For functions measurable on E one can show the equivalence of the Lebesgue procedure for defining an integral, to that used above. Theorems on measurable functions which do not involve the measure function a, only the s-ring 6, are available in this more general setting. However difficulties arise in connection with statements which contain the phrase: “except for an a-null set.” For instance, it is not certain that iff is measurable on Q, and g differs from f by an a-null set, that is, a(,?[\ f - g > 01) = 0, then g is measurable. For the sets
I
324
IX. OTHER DEFINITIONS FOR LEBESGUE INTEGRALS
E [f > c ] and E [ g > c ] will differ on the set E [f > c]dE[g> c ] ,which is a subset of the set E [ l f - g I > 01, but may not belong to Q. This is due to the fact that the subclass Q, of a-null sets of Q, may lack the hereditary property: if E, belongs to 6, and E, is any subset of X contained in E,, then E, is in &,, also, or if a(,?,) = 0, and El 5 E,, then El is in Q also and a(E,) = 0. For instance the class of Bore1 measurable subsets of a linear interval [a,b ] with a ( E ) the Lebesgue measure, lacks the hereditary property for its sets of zero measure.
Q is such that the class Q,,of a-null sets does not have the hereditary property, then one can find an extension of @ to a class @
4.5. If
which does have the property. We set up the class @, consisting of all sets E of X contained in some E,, of Q,, adjoin Eo to Q, and extend the resulting class to be an s-ring. It can be demonstrated that the resulting class 6 consists of all sets of the form E E,, - Eo2,where E belongs to the original Q, E,, and Eo2belong to &, E,, being disjoint from E and E,, being contained in E. The function CI on Q is extended to in the obvious way, that is, if E is in 8, and E = E E,, - Eo2,then a ( E ) = a(E). If the class Q is such that the class Q, of a-null sets has the hereditary property, then theorems concerned with convergence of integrals of sequences of measurable functions are valid as in the onedimensional case. For instance we have the basic convergence theorem : If the a-null subclass of Q has the hereditary property; iff, and f are measurable functions on E of Q ; if lim. a ( E [ - f > e) = 0 for all e > 0; if J E f , ,da exist and the functions JB,frrdaon subsets El of E in Q, are uniformly absolutely continuous relative to a ; then SE fda exists and lim,, SE, f,,dor = SE fda for all sets E,, subsets of E in &. The condition lim,t a(E[If,, - f > el) = 0 for all e > 0 can be replaced by the condition limI,flL(x)= f(x) except for an a-null set. Further if J E fda exists, on E measurable, then we can show that f is necessarily measurable. For we can show that there exist measurable functions g ( x ) 2 f(x) 1 h(x) except for an a-null set of E such that SE gda = J E fda = SE hda so that g ( x ) = f(x) = h(x) except for an a-null set. If the measure function a ( E ) on Q, for which a ( E ) 2 0 for all E of Q, is allowed to assume the value co, then we have a generalization of the measurable sets on the interval - co < x < co, where
+
+
If,
I
I
+
+
4.
L-S INTEGRALS ON AN ABSTRACT SET
325
a function f on a set E of Q integrable only if we can determine J E fda as a finite number, then there exist nontrivial integrable functions on a set E of Q only if there exists a subdivision u, of E : oo = {E,}, such that a(E,) is finite valued for each Ell of uo. a ( x ) is monotone but unbounded. If we consider
4.6. The s-additive function a ( E ) is said to be a-finite on a set E of G, if there exists a subdivision a, = {E,!}of E into sets of Q such that
a(E,) is finite valued for all n. In case a is a-finite on a set E of Q , we can proceed as usual and say that J E f d a exists if and only if limoof &Zo f(x,)a(E,) exists as a finite number. Integrability then requires among other things that there exist a subdivision u of E such that f is bounded on every subset El, of u for which 0 < a(E,) < 03, and ZoM ( f E,)a(E,) < co. It is an illuminating exercise to carry through the details indicated above and determine what results valid for Lebesgue-Stieltjes integrals on the linear interval can be extended to abstract sets of elements as outlined above. Consideration of an abstract integral as sketched in this paragraph was initiated by M. FrCchet : Sur l'integral d'une fonctionelle itendue a une ensemble abstrait, Bull. soc. math. France, 43 (1915) 249-267.
I 1,
EXERCISES 1. Carry through the proof of the statement made after Theorem IX.4.3: If a ( E ) is finite valued and s-additive on the s-ring 6, then for any E of Q there exist disjoint sets E+ and E- in Q such that E = Et E-, p ( a , E ) = a(E') and n(a, E ) = - a ( E - ) so that v(a; E) = u ( E ) - a ( E - ) .
+
2. Show that if Q is an s-ring, and a ( E ) and ,B(E) are s-additive functions on Q, with a ( E ) 2 0 for all E ; and if a ( E ) = 0 implies P(E) = 0, then p is absolutely continuous relative to a (see VII.7.3).
This Page Intentionally Left Blank
CHAPTER X
PRODUCT MEASURES. ITERATED INTEGRALS. FUBlNl THEOREM
1. Product Measures
In V.15 we considered briefly the problem of measurable sets and measure functions determined by point functions a ( x , x, .. . xJ on spaces of two or higher dimension. Thus, for n = 2, the point function a ( x , y ) on X x Y , where X and Y are finite or infinite intervals, gives rise to the interval function a ( I ) =a( [a,b]x [c,d])=a(b,d) -a(b,c) -a(a,d) +a(a,c)
=Az,a(x,y).
If a(1) 2 0 for all a 5 b, c 5 d, then the positive interval function so obtained can be used to determine a class of measurable sets and a measure function a on these measurable sets. If a ( x , y ) is bounded on X x Y, the class of measurable sets is an s-algebra, and the set function a(E) on this class is s-additive. For any such a, open sets are measurable sets, and are basic in the definition of the upper measure and of measurability. It is to be noted that if I' is a closed interval: [a 5 x 5 b, c 5 y 5 d ] , then as a member of the class of a-measurable sets, we have a ( I c ) = a(b
+ 0, d + 0 ) a ( b + 0, c + a ( a - 0 , c-0), -
a ( a - 0, d + 0)
-
0)
-
a(b, c )
-
a(a, d)
which may be different from a(I)
=
Azya(x,y )
=
a(b, d)
-
+ a(a, c)
on which the measure is based. Of special interest is the case when a ( x , y ) = p(x)y(y), where p ( x ) and y ( y ) are monotonic nondecreasing in x and y , respectively. The measure function a ( E ) determined by such an a ( x , y ) depends
327
328
X. PRODUCT MEASURES. ITERATED INTEGRALS
in a sense on the product of the two measure functions P(E(x)) and y(E(y)) and is called a product measure. It is then desirable to explore the relations between the measure function a(E) on the measurable subsets of X x Y and the measure functions and sets defined on X by p and on Y by y, as well as the relation between the LebesgueStieltjes double integral: Ssx1- f ( x , y)da(x, y ) and the iterated integrals sx dPfx) SYf(X, Y ) d Y f Y ) and SY dY(Y) Sxf(X, Y)dP(X). While we treat only the case when X and Y are each one-dimensional, so that X x Y is two-dimensional, it is possible to obtain similar results for the case when X is m-dimensional and Y is n-dimensional, so that X x Y is ( m n)-dimensional, and
+
fffx, Y )
=4x1
*-.x,,
Y1
- * a
Y,)
=
*..xm) Y f Y * *..Y,).
Pfx,
+
Let X be either the open interval - co < x < co, the closed finite interval a 5 x 5 b, or the half-open interval - co < x I b, or a d x < co, and similarly for Y. Let E denote subsets of X x Y, E ( x ) subsets of X and E(y) subsets of Y. Then any subset E of X x Y determines for each yo of Y, the one-dimensional set' E J x ) , the set of x for which ( x , yo) belongs to E ; and for each x, of X , the one-dimensional set E J y ) , the set of y for which (xo, y ) belongs to E. If for each y; the set E J x ) is either the empty set or a fixed set E ( x ) , then for each x , E,(y) will be either the empty set or a fixed set E ( y ) , and E = E ( x ) x E(y). For instance, if E ( x ) is the fixed closed interval [a, b ] , and E ( y ) the fixed closed interval [c, d ] , then E = E ( x ) x E(y) is the closed rectangle [a, b ] x [c, d]. By analogy, sets of the form E = E(x) x E(y) are sometimes called rectangular sets. Let p ( x ) be a monotonic nondecreasing bounded function on X , and y(y) a monotonic nondecreasing bounded function on Y. Then a(x, y ) = p(x)y(y) will be a bounded monotonic nondecreasing function on X x Y. a ( x , y ) will give rise to a class of measurable subsets E of X x Y, which will include the open: (a, b) x (c, d ) , the closed: [a, b ] x [c, d ] , and the half (right) open [a, b) x [c, d ) rectangles. If I o = (a, b ) x (c, d), I' = [a, b ] x [c, d ] and I h = [a, b) x [c, d), then
+
a(]') = (P(b - 0)
ff(IE= ) fP(b and
+ 0)
a ( I h ) = (p(b - 0 )
+ 0 ) ) (y(d
- 0) - y(c
-
P(a
-
Pfa - 0)) f r f d
- P(a - 0))
+ O)),
+ 0) - Y f C - 0))
( y ( d - 0) - y(c - 0)).
1.
329
PRODUCT MEASURES
We note that for each of these three cases we can write:
where the integrals are Lebesgue integrals. Our aim is to extend this result to any set E measurable relative to a , and prove: 1.1. THEOREM. If P(x) and y ( y ) are bounded monotonic nondecreasing functions and E is measurable relative to a ( x , y ) = P(x)y(y), then E y ( x ) is measurable relative to P except for a set of y which is a y-null set, E J y ) is measurable relative to y except for a set of x which is a p-null set, and
where the integrals are Lebesgue integrals. We demonstrate this theorem first for the case when E is an open subset of X x Y. On the one-dimensional interval, any open set can be expressed as the sum of a denumerable number of disjoint open intervals, but for the plane this is not always possible. Instead we make the following observation : 1.2. Any open set of X x Y can be expressed as the sum of a denu-
merable number of disjoint half (right) open intervals Zh [defined above as the points ( x , y ) such that a 5 x < b, c 5 y < d, for some [a, b ; c, dl. For suppose we divide X x Y into a network by means of the lines x = m/2', y = n/2,, k = a positive integer, m and n ranging over all positive and negative integers. We select serially collections of squares S,. S, consists of all half-open squares m / 2 2 x < ( m 1)/2, n / 2 2 y < (n 1 ) / 2 which are contained in G. In general, S, consists of all squares of the form m/2k 5 x < ( m 1) / 2 k ; 5 y < ( n 1) /2,, which have no points in common with any of the half-open squares in S , ... S,-], and which are contained in G. Then each S, consists of a denumerable number of squares and every point of G belongs to some S,, so that G = ZkS,. Obviously the same type of reasoning is applicable in any finitedimensional space. In contrast to the case of an open set on a linear interval, where G determines uniquely the open intervals of which it is constituted, any G can be expressed as the sum of a denumerable number of disjoint half (right) open intervals in many different ways.
+
+
+
+
330
X . PRODUCT MEASURES. ITERATED INTEGRALS
As noted for any half-open interval I"
=
[a, b ) x [c, d), we have
since the step function y(I,h(y)) = y(d - 0 ) - y ( c - 0) for x on [a, b) and zero elsewhere; and similarly for (I,"(x)). If now G is any open set and G = Z,),", where the I: are disjoint, then since the intervals I," are +measurable, and CI is s-additive, we have:
Now the set G,(Y) = z J r r z ( ~ ) .Then Y(G,(Y)) = z n ~ ( I : z ( ~ ) ) , which as a function of x is the sum of a convergent sequence of positive step functions all of which are measurable relative to P. Since
for all m and x , we have by the basic convergence theorem for L-S integrals (VII.8.8.3) :
=
J
Y(G,(Y))~P(~).
In the same way, one shows that
It is to be noted that the functions y(G,(y)) in x and P(G,(x)) in y are measurable relative to P and y , respectively. The same type of formula holds for any closed set F. For if G = X x Y - F, then G is open and a(F)
=
~ (XxY )
=
jxf Y f V
-
-
a(G) = P ( x ) y ( Y )
YfG,fY)))dP(X) =
-
1, y(G,fy))dP(x)
jb V(F,(Y))dP(X)*
For the general case where E is any set of X x Y which is measurable
1.
331
PRODUCT MEASURES
relative to a , we find open sets G I ,2 E and closed sets F f lS E such that a ( G , - F J < l / n so that a(G,, - F J
=
/ x [Y(G,,(Y))
-
~ ( F , , f y ) ) I d P ( x ) < 1/n-
Then lim?zJA. YfG,,(Y)
-
Ffl,(Y))dP(X)
= 0.
But this has as consequence that y(G,,,(y) - Fll,(y)) + O(P). Consequently, there exists a subsequence n,, of the integers n, such that
except for a set E,,, of x which is a P-null set. But since G,,,(y) 2 E J y ) 2 F,,,(y) for all n and x, this means that the sets in y , E,(y), are measurable relative to y except for x in E,,,, and Y(E,(Y))
= lim,
Y(G,mx(Y)).
If y(E,(y)) is defined arbitrarily for the x for which E,(y) is not measurable relative to y , for instance, as the upper measure y(E,(y)), then y(E,(y)) as the limit except for x on a P-null set, of a sequence of P-measurable functions, is measurable relative to P. Moreover, since y(GT,7,LX(y)) 5 y ( Y ) for all x , the sequence of functions y(G,7n,) is uniformly bounded in m and x. Since lim,a(G, m ) = a ( E ) it follows that a(E)
= lim ?n
Jx Y(GII,,,,(Y))CIP(X)= J KY(E,(Y))dP(X).
Similarly the sets in x : E J x ) are measurable relative to /3 except for a set of y which is a y-null set, and a ( E ) = J y P ( E y ( x ) ) d y ( y ) . As a special case of this theorem, we note that if G ' ( x ) and G " ( y ) are open sets on X and Y , respectively, and G = G ' ( x ) x G"(y), the set of all ( x , y ) for which x belongs to G ' ( x ) and y to G"(y) simultaneously, then a(G) = Jr.P(G,(x))dy(y) = P ( C ' ( x ) ) y(G"(y)), since G J x ) = G ' ( x ) for y on G " ( y ) and the null set for y on the complement of G"(y). Similarly, if F ' ( x ) and F " ( y ) are closed sets on X and Y , respectively, then for F = F ' ( x ) x F"(y) we have a(F) = P ( F ' ( x ) ) y ( F " ( y ) ) . More generally we can show that:
P, and E " ( y ) relative to y , x E " ( y ) is measurable relative to
1.3. If E ' ( x ) is measurable relative to
then E
= E'(x)
a(x, Y ) = P(X)Y(Y),
and
a(E)
=
P(E'"E''(Y)).
332
X. PRODUCT MEASURES. ITERATED INTEGRALS
Then E = E ' ( x ) x E"(y) is measurable and
a f E ) = PfE'fX))Y(E"(Y)). 1.4. Suppose now that either of the monotone functions P(x) or y ( y )
is unbounded on X = - q < x < + o o or Y = c coy<+^, but bounded for every finite subinterval of Xor Y. Let E be measurable relative to a ( x , y ) = P(x) y ( y ) , with a ( E ) < co. Let I , be the open rectangle: (- n, n) x (- n, i z ) and set En = E * I,. Then we can apply Theorem X.l.l to subsets En of In and obtain
4 E n )=
jdY(E,,(Y))dP(X)
=J
y
I I
P(E7,y(x))dY(Y)7
1 1
since y ( E J y ) ) = 0 for x > n, and P(E,,(x)) = 0 for y > n. The sets E,,,(y) are measurable relative to y for all x except at most a P-null set, so that y(E,Jy) is defined as a function of x except for a P-null set for each n. If this set is Ello(x),and E,(x) = ZILE,o(x), then Eo(x) will be a P-null set. Since y(E,Jy)) is monotonic in n except for x in E o ( x ) ,limn y(E,,,(y)) will exist (finite or co) as a function of x defined except for a P-null set. Similarly P(E,Jx)) is monotonic in n except for at most a y-null set in y and limn P(E,,,(x)) exists except for a y-null set. Since limn a(E,) = a ( E ) the Fatou convergence theorem VII.8.10 gives us :
+
a(E)
=
lim Jx Y(E,,,(Y)ldPfX) = Jx lim YfE,,(Y))dP(X)
= lim J y
P(E,,WdY(Y)
=Jy
lim P(E,,(X))dY(Y).
Now E,,(y) for x not in E,, is a sequence of sets of Y measurable
2.
L-S INTEGRALS AS THE MEASURE OF A PLANE SET
333
relative to y, with E,(y) = lim,tErJy). Then 1imny(E,,,(y)) = y ( E x ( y ) ) except for x on E o ( x ) or a P-null set, a result of the additive properties of measurable sets and measure. Similarly lim ,P(E,,(x)) = P(E,(x)) except for y on E,(y) a y-null set. Then
f f ( E )= Jx Y(E,fY))dP(X)
=
jy P(E,(X))dY(Y).
Here y(E,(y)) is not defined for x on a B-null set E,(x), and may be + co on a P-null set since Jx- y(Ex(y))d/3(x) exists. Similarly for P(E,(W' In case a(E) is not finite, the same type of reasoning applies, E,(y) is measurable relative to y ( y ) except for x in a P-null set, and E,(x) is measurable relative to P(x) except for y in a y-null set, but the equality on the integrals holds only if we allow CQ as a value for the integral.
+
2. The Lebesgue-Stieltjes Integrals as the Measure of a Plane Set
The theorem just proved can be made the basis for another method of defining a: Lebesgue-Stieltjes integral, at least for positive finite valued functions. Any function f ( x ) I 0 on X determines the plane set E = Z, [x, 0 5 y 5 f ( x ) ] . Let P(x) be any monotonic nondecreasing function on X . Then the function a ( x , y ) = P(x) y defines a planar measure such that for any rectangular set E = E ( x ) x E(y), where E ( x ) is measurable relative to P and E ( y ) is Lebesgue measurable, we have a ( E ) =P(E(x)) y ( E ( y ) ) , where y ( E ( y ) ) is the Lebesgue measure of E ( y ) . We then have the following theorem: 2.1. THEOREM. If f ( x ) 2 0 on X , and P(x) is monotonic non-
decreasing, then a necessary and sufficient condition that Jxf(x)dP(x) exists is that the set E = Cx[x, 0 5 y 5 f ( x ) ] be of finite measure relative to Q ( X ,y ) = P(x) y , and then a ( E ) = Js f ( x ) d P ( x ) . For the necessity, if Jx f ( x ) d P ( x ) exists as a finite number, then the upper and lower integrals fda and -Jx fda are finite and equal. As a consequence there exists a sequence of subdivisions u,rL= {Enm(x)}of X such that 2 ur2,f is bounded on each E , , and
ry
C
' i ~ M ( f ; Enm(x))P(Enm(x)) *n
=
lip
C m(f; E ~ L ~ ( x ) ) P ( E ~ ~ ( x ) ) ~ on
where as usual M ( f ; E ) = 1.u.b. f on E and m ( f ; E ) = g.1.b. f on E.
334
X. P R O D U C T MEASURES. ITERATED INTEGRALS
Define the step functions p P 1 ( x= ) M ( f , ETl,) for x on E n ,and y , ( x ) = m(f, E,,)for x on Let Ell be the plane set Z, [x, 0 5 y 5 p l l ( x ) ]and E , the plane set 2%[x, 0 5 y 5 y , ( x ) ] . Then since by X.1.3 the rectangular sets E , l , l ( ~ )x [0 5 y 5 M ( f ; E J ] are plane measurable relative to a ( x , y ) = P(x) * y and E, is the sum as to m of these disjoint sets, it follows that E, are measurable for each n relative to a. Similarly, E , are measurable relative to a. Moreover, =
and
c M ( f ; E,,,)P(E,l,) On
4%) = 2 m(f ;
E,,)P(E,,).
=?I
NOW
Fn2 E Z
E n and
Then: cx(limlIEn)- a(lim,En) = 0, the set E is measurable relative to a , and a(E) = Jx fdp. For the sufficiency, suppose that E is a-measurable and a ( E ) < a. Then a(E)
= J1,
P(E,(X))dY
=
J,
Y(E,(Y)dP(X).
Now the set E,(y) = [0 5 y 5 f ( x ) ] is measurable in y for each x, with measure f ( x ) , so that
a(E)
=
is Y(E,(Y))dP(X)
=ilf(X)dP(X).
We might note that our theorem also tells us that the set E,(x) is measurable relative to P except for a set of y in a y-null set, that is, a set of Lebesgue measure zero. Now the set E y ( x ) for E = Z,[x, 0 5 y 5 f ( x ) ] is the set of x : E [ f ( x ) 2 y ] . Consequently, the set E [f ( x ) 2 y ] is P-measurable except for a set of y of Lebesgue measure zero, that is for a set of y which is dense on Y. It follows from VI.2.4 that E [f ( x ) Z y ) is measurable for all y , and f ( x ) is measurable relative to P. If we apply integration by parts to J y P(E,(x))dy, then
3.
335
FUBINI THEOREM
Now if f ( x ) is L-S integrable with respect to fact that E,(x) = E [ f ( x ) 2 y ] , we have by VII.2.5. Consequently,
P, then in view of the yP(E,(x))
=
0
if p ( y ) = P(E[f ( x ) < y ] ) ,which connects with the Lebesgue method of defining the L-S integral (see V11.2.3). In case f ( x ) is allowed to take on both positive and negative values, we write f ( x ) = f ' ( x ) - f - ( x ) , where f + ( x ) is the greater of f ( x ) and 0, and f - ( x ) the greater of -f ( x ) and 0. If both sets EC = Z Z [ x ,0 2 y 5 f ' ( x ) ] and E- = Zr[05 y 5 f - ( x ) ] are measurable relative to a ( x , y ) = P(x) y , with a(E +) and cr(E-) both finite, then f is L-S integrable relative to P and Jx fdP = a ( E f ) - cr(E-). 3. Fubini Theorem on Double and Iterated Integrals
As we have seen, if a ( x , y ) = P(x)y(y) on X x Y, and E is measurable relative to a, with a ( E ) < co, then afE) =
j x x y x f E ; x , Y ) d f f ( X ,Y ) = j y P(E,fX))dYfY) = jxY(E*(Y))P(X).
If for a fixed y , the set E J x ) is measurable relative P, then sx x(E;x , y)dp(x) = P (E y(x)). Since E,(X) is P-measurable except possibly for a set of y which is a y-null set, we have a f E ) = J x x y x ( E ; x7 =
Y W ( X 7
Y)
=
j y P(E,(X))dY(Y)
j y M Y ) jxx(E ; X,Y)dP(X)
where the inner integral exists except for a y-null set. Similarly, a(E)
=
j X X Yx(E ; x, Y)dcX(X,Y )
=
dP(x) J y x ( E ; x , Y ) M Y ) .
This is a special case of the following theorem relating double integrals to iterated integrals : 3.1. Fubini Theorem. If on X x Y , a ( x , y ) = P( x ) y ( y ) , with P(x) and y(y) monotonic nondecreasing on X and Y , respectively, and if
336
X. P R O D U C T MEASURES. lTERATED INTEGRALS
f ( x , y ) on X x Y is finite valued and such that J s x y f ( x , y)da(x, y ) exists, then J s f ( x , y)dP(x) exists except for a y-null set in y , and J y f ( x , y)dy(y) exists except for a #?-nullset in x , and
where the values of Jxf ( x , y)d#?(x)and J y f ( x , y)dy(y) are arbitrary for the y and x in the null sets where these integrals do not exist. Since the integration process is linear, we see at once that the preliminary observations above, on the theorem, lead to a proof of its validity for any step function f ( x , y ) , which is measurable and assumes only a finite number of values, that is, one expressible in the form f ( x , y ) = Z:t=l c,, x(E,; x, y ) , where Enlare disjoint sets, measurable relative to a, and ZmEm= X x Y. In order to extend this to the case where f ( x , y ) is a general integrable step function, that is, f ( x , y ) = Zc, x(E,; x, y ) , with Znt c, a(E,) < 00 and ZmEm= X x Y , we note that it is sufficient to make the demonstration for the case where c,, I 0 for all m, as any integrable step function is obviously the difference of two such positive functions. If we set f , ( x , y ) = 2:-1c, x(E,; x, y ) ; then f n ( x , y ) will converge monotonically in n to f ( x , y ) , the sequence J x x yf n ( x , y)da(x, y ) will be bounded in n and so
I
lim n
j
XXY
I
f,,(x,Y)dOr(X, Y )
=Jxxyf(X,
y)da(x, Y ) .
Now
where the inner integral exists excepting for a #?-nullset in x. If E J x ) is the set of x such that J y f,2(x,y)dy(y) does not exist for some n, then P(E,(x)) = 0. If x is not in E,(x) then the sequence J y f,(x, y ) dy(y) will be nondecreasing, and such that Jx dP(x) J y f i L ( x ,y ) d y ( y ) are bounded in n. Then by the Fatou convergence theorem of VII.8.10, we have
It follows that limn J y f n ( x , v)dy(y) exists as a finite number, except-
3.
FUBINI THEOREM
337
ing at most a P-null set and because of the monotoneity of f,(x,y) in n, that
lip J Y f ,Y)dY(Y) k = J y li?f,(X,
Y)dY(Y) = J y f ( x , Y)dY(Y).
Then we have shown that J y f ( x , y ) d y ( y ) exists except for a P-null set, the sum of the sets in x for which E , J y ) are not measurable relative to y ( ~ ) ,and the set on which lim, SY f,(x, y ) d y ( y ) might be infinite. By combining these considerations we have
where the inner integral exists except for a @-null set in x, and is arbitrary at the points of this set. We take up next the case of any finite valued function integrable with respect to a ( x , y ) , and because integrability implies absolute integrability, restrict ourselves to the case when f ( x , y ) 2 0 on X X Y. Then there exists a subdivision u,, of X x Y , such that f ( x , y ) is bounded on every subset E of u,,, and if u 2 o,,,then ZuM ( f ; E J a ( E J
2 M ( . E ) f f ( E )= J x x y f ( x ,y)da(x, Y ) = l.u.b.u 2 m ( f ; E)a(E). U
U
For any subdivlsion c we define the measurable step function f,(x, y ) = M ( f ; E ) if ( x , y ) is in the set E of u. Then
338
X. PRODUCT MEASURES. ITERATED INTEGRALS
This holds for every
CT
2 g o . Similarly, we show that
for every u 2 uo. Taking the greatest lower bound as to CJ in the first inequality and the least upper bound as to u in the second, we obtain -
-
.iXXY f(x9 Y ) d f f ( X ,u) 2 Js dP(x) j y f k
Y)dY(Y)
Then equality holds throughout and because of the properties of integrals we can even assert
= JA.
-
dP(x)
jym7Y)dY(Y) = -IxdP(x) -l y f ( X 7Y)dY(Y).
From these equalities it follows that f y f ( x , y ) d y ( y ) and .fyf(x,y ) dy(y) are each integrable with respect to ,B and that
Consequently, the positive or zero function T y f ( x , y ) d y ( y ) y ) d y ( y ) is measurable, and vanishes except on a ,hull set. This means that J y f ( x , y ) d y ( y ) exists except for a P-null set. Moreover, we can conclude that
JYf(x,
3.
339
FUBINI THEOREM
where the inner integral exists except for a P-null set, where its value can be assigned arbitrarily. Since X and Y and P and y are of a parity, we also have
where the inner integral exists except on a y-null set. As we have indicated, we can deduce the validity of the Fubini theorem for any function f ( x , y ) integrable with respect to a from the above case where f ( x , y ) 2 0, by decomposingf(x, y ) = f +(x,y ) - f -(x, y ) into its positive and negative parts. The existence of Jxxy fda cannot in general be deduced from the existence of the iterated integrals JAY dp(x) J y f ( x , y ) d y ( y ) and J y dy(y) Jxf ( x , y)dp(x) and their equality. For a counterexample see : W. Sierpinski : Sur une probleme concernant les ensembles mesurables superficiellement, Fundarneizta Math. 1 (1920) 140-150. However, in case f(x, y ) is measurable relative to a ( x , y ) , we have: 3.2. THEOREM. If p(x) and y ( y ) are monotonic nondecreasing on X and Y, respectively; and ~ ( xy ), = p ( x ) y ( y ) ; if f ( x , y ) is measurable relative to a(x, y ) ; if Jx dP(x) Jz7 f ( x , y ) dy(y) exists, ( J y f ( x , y) dy(y) need exist only for a set which differs from X by a P-null set), then S X X V f f x , y ) d a f x , Y ) and Sk. M Y ) S X f ( X , Y)dP(X) exist and are both equal to J s d P ( x ) JY,f(x,y ) d y ( y ) . Since the hypotheses of the theorem involve only absolute integrability of f ( x , y ) , it is sufficient to prove this theorem for f ( x , y ) 2 0 on X x Y. Let X,, = [- N, n ] and Y,L= [- n, n ] . Further, set & ( x , Y ) = f ( x , Y ) iff(x, y ) 5 M a n d j i d x , y ) = M iff(x, Y ) > M. Then since f ( x , y ) is measurable relative to a, the functions /3 and y are bounded on X , and Y,,, respectively, and Xll(x, y ) is bounded, we know that
I
I
and
all exist and are equal. Successive applications of the Fatou monotone convergence theorem (VII.8.10) then complete the proof of the theorem.
340
X. PRODUCT MEASURES. ITERATED INTEGRALS
The Fubini theorem can be extended to the case where for a(x, y ) P(x)y ( y ) , P(x) and y ( y ) are each functions of bounded variation on every finite subinterval of X = - co < x < co and Y = - co < y < co, respectively. To this end recall that if a function P(x) is of bounded variation on every finite subinterval of X , then there exist monotonic functions p ( P ; x ) and n(P; x ) such that for any finite interval [a, b ] =
+
+
It follows that the positive and negative variation functions p(a; x, y ) and n ( a ; x, y ) associated with a(x, y ) can be written
and for any rectangle R A x y [ p ( P ;x)P(Y; Y )
and
+ n(P; x ) n ( r ; Y ) ; RI 2 A x , [ P ( P ; x)P(Y;Y ) ; RI 2 Axy
b ( P ; x ) n ( r ;Y ) ; RI
.it follows from VII.10.4 that the existence of the integral J x x yf ( x ,y ) I .
dp(a; x,y ) carries with it the existence of the integrals J x x yf ( x , y )
3.
34 1
FUBINI THEOREM
d ( p ( P ; X ) P ( Y ; Y ) and S , r x Y f ( x , Y M n f P ; x ) n ( y ; Y ) . Similarly from the existence of SSxE. f ( x , y ) dn(a; x, y ) we derive the existence of J x x Y m , Y ) d ( P ( P ; x ) n ( y ; Y ) ) and S S X Y . f ( X , y ) d ( n f P ;x ) p ( y ; y ) ) . To each of these integrals we can apply the Fubini theorem for monotone functions. As a consequence, we have that Jyf ( x , y)dp(y; y ) and SY f ( x , y)dn(y; y ) each exists except for a p(P)-null set. Then S Y f ( x ,Y ) ~ ( P ( Y Y; ) - 4 ~Y ) );= J y f ( x , Y ) ~ Y ( Y will ) exist except for a p(P)-null set and
j,,,f(&
y ) d f n ( P ; x ) p ( y ;Y ) ) =
sx
-
dn(P; x )
jXXYf(X,
y)d(n(P; x ) n ( r ; Y )
jy f ( x , Y)dY(Y)
SY
and the integral f ( x , y)dy(y) will exist except for a n(p)-null set. Consequently, SY f ( x , y)dy(y) will exist excepting for a v(P) null set and
Obviously the roles of X and Y can be interchanged. We have then shown : 3.3. THEOREM. If X = - a < ~ < + a , Y = - - < x < + c o ; P(x) and y ( y ) are of bounded variation on every finite subinterval of X and Y, respectively; a ( x , y ) = P ( x ) y ( y ) ; f ( x , y ) on X x Y is such that J x x y f ( x , y)da(x, y ) exists; then f ( x , y ) d y ( y ) exists except possibly for a v(p)-null set and Jx f ( x , y)dP(x) exists except
SY
possibly for a v(y)-null set, and
in which the inner integrals may fail to exist on v(p)-null and v(y)null sets, respectively.
342
X. PRODUCT MEASURES. ITERATED INTEGRALS
EXERClS E The integration by parts formula for Riemann-Stieltjes integrals can be written :
where E is the triangle a s x s y s b. Suppose that p ( x ) and y ( y ) are of bounded variation on a s x 5 b and a 5 y s 6 , respectively. What does the Fubini theorem yield concerning the relation between the LebesgueStieltjes integrals $: P ( y ) d y ( y ) and y(x)d/3(x)?
st:
CHAPTER XI
DERIVATIVES A N D INTEGRALS
1. Riemann Integrals and Derivatives
In elementary calculus, we find a close relationship between integrals and derivatives for functions f(x) on a finite interval: a 5 x 5 b. This relationship is embodied in the two fundamental theorems of the integral calculus : 1.1. T H E O R E M A. If f ( x ) is Riemannintegrable on [a,b ] andg(x)
Jy f ( x ) d x , then the derivative g ' ( x ) exists for all points x
=
for which
f ( x ) i s continuous and g ' ( x ) = f ( x ) at these points. 1.2. T H E O R E M B. If f(x) on [a, b ] is such that f ' ( x ) exists for all x of [a, b ] and is Riemann integrable, then f ' ( x ) d x = f ( b ) f(a), or f ' ( x ) d x = f(x) - f(a) for a 5 x 5 b. The proof of Theorem A is well known and quite simple. For
Js
Jy
If f ( x ) is continuous at x,, then e > 0, d,,., f ( x , ) < e. If d x < d,,., then
I
1
1
343
1 x - x0 I <
'ezo:
If(X)
-
344 or
XI. DERIVATIVES AND INTEGRALS
I
k(X,
+ Ax) - g(x,))/Ax
-f(x,)
I < e.
This means that g’(x,) =f(x,). An immediate consequence of Theorem A is: 1.2.1. Corollary. If f(x) is continuous on
[a, b ] , then g’(x)
=f(x)
for all x of [a, b].That is for every functionf(x) continuous on [a, b ] , there exists a function g(x) = J: f(x)dx such that g‘(x) = f(x) for all x of [a, b ] , or every continuous function is the inverse derivative of a (continuous) function. We note further that since any Riemann integrable function is continuous except for a set of Lebesgue measure zero, Theorem A asserts that d/dx f(x)dx exists and is equal to f(x) except for a set of x of Lebesgue measure zero. Theorem B tells us that the indefinite Riemann integral recovers the function f(x) from its derivative if this latter is R-integrable. The usual proof is as follows: Let a be any subdivision of [a, b ] . Then by the mean value theorem of the differential calculus there exists points xi‘ on [xi-l, xi] such that
Jl
But the last expression is one of the approximating sums for Jif ’(x)dx. Consequently, f ( b ) - f ( a ) = f ‘(x)dx, -and this holds also if b is replaced by any point x in [a, b]. This proof uses the mean value theorem of the differential calculus. It is natural to ask whether it is possible to prove this theorem using only the existence of the derivative f ‘(x) for each x of [a, b ] and the R-integrability of this function. This has been done for the RiemannStieltjes integral relative to a strictly monotone function g(x) in 11. 16.3, so that the proof is also valid for the special case g(x) = x. The main object of this chapter is to obtain the extension of these two fundamental theorems to the case when Lebesgue integration is involved. Since the proofs above depend on the properties of Riemann integration, different procedures will be necessary for Lebesgue integrable functions. Instead of taking up the L-S integral with respect to a function of bounded variation, we shall limit ourselves to Lebesgue integrals with a(x) = x on a finite interval [a, b ] . Measurable sets are then Lebesgue measurable or simply measurable, and derivatives are with respect to x. We shall denote by p ( E ) the Lebesgue measure
Jt
2.
345
ON DERIVATIVES
of a measurable set E, and by p * ( E ) the Lebesgue upper measure of any set E. We note in passing that the two fundamental theorems of the integral calculus give a partial solution of what might be called the inverse derivative problem: Given a finite valued function f(x) on [a, b ]; under what conditions does there exist a (continuous) function g(x) such that g'(x) = f(x) for all x of [a, b ] and how can one construct such a function g(x) when it exists.
EXERCISES 1. Show that if f ( x ) is Riemann integrable, then g ( x ) = J: f ( x ) d x has a right-hand derivative at all points where f(x) is continuous on the right.
2. Prove the following interval lemma : If to every point x of a I x
+
3. Is the following theorem true: If a(x) is of bounded variation on [a, b ] , if f ( x ) is continous at x, and such that J: f ( x ) d a ( x ) exists in either norm or s-sense, and if g(x) = f ( x ) d a ( x ) , then
JI
where Ax is limited to values for which a ( x o
+ Ax) # a(xo).
2. O n Derivatives
Before attacking the problem of derivatives as related to Lebesgue integration, it seems desirable to review briefly some of the properties of derivatives of finite valued functions f(x) on a finite interval [a, b ] . If, as usual, we say that f(x) on [a, b ] has a derivative at x, if lirn,,,, (f(x, Ax) - f(x,)) /Ox exists finite or 00 or - 00, then we have :
+
+
2.1. THEOREM. If f(x) on [a, b ] has a derivative f '(xJ at xn with
a < xo < by then if h > 0, k > 0, we have
346
XI. DERIVATIVES AND INTEGRALS
0 < k < dexo:
If f ' ( x , ) is finite, then e > 0, deX0,0 < h < d,,, -
he
-
ke
+ h) - f(x,)
'(x,) < he
- hf
and -f ( x , - k )
-
kf '(x,) < ke.
Consequently : -
(h
+ k)e < f ( x , + h ) -ffx,
-k) -
(h
+ k l f '(x,) < (h + k)e.
This leads at once to the conclusion of the theorem when f '(x,) is finite. I f f'(x,) = 00, then M > 0, dAwx0, 0 < h < dAMX0, 0 k M f( x , h) -f( X J > hM, and f( ~ 0 -
+
+
so that f ( x , -
+ h ) -f( x ,
-
k ) > M(h
+ k ) . Similarly for f '(x,)
=
w.
We note that since f ' ( x J exists, we could assume h 2 0, k 2 0, (h, k ) f (0, 0 ) in the double limit. Conversely we have : 2.2. T H E O R E M .
I f f ( x ) on [a, b ] is such that
with either h 2 0, k 2 0, (h, k ) f (0, 0), or h > 0, k > 0 with f ( x ) continuous at x,, then f ' ( x , ) exists with f '(x,) = A. We prove this theorem only for the case when A is finite, the case where A = co or - co follows similarly. The first of the two alternatives in the hypothesis leads to the existence of the derivative by assuming h = 0, k > 0 and h > 0, k = 0. In the case of the second alternative, we note first :
+
e>O, dexo,O
I
+ hh) +- f kb o
-k)
-Al<e.
If in the final inequality we allow k 0, or h + 0, then because of the continuity of f ( x ) at xo, we have --f
e>O, d,,,, O < h < d e X o :I f f f x , + h ) - f ( x o ) ) / h - A e > 0, d,,,, 0 < k < d,,,:
I (ffx,) - f ( x ,
These two statements together yield
f '(x,)
-k ) ) / k -A =
A.
I<e
I < e.
2.
347
ON DERIVATIVES
These theorems are useful in connection with the study of nondifferentiable continuous functions, since it is sometimes easier to deal with secants straddling a point than with those going through a given point. We note in passing that if we define relative to the function f ( x ) the interval function F ( I ) =f ( d ) -f(c), where I = [c, d ] , then (f(x Ax) - f ( x ) ) / A x = F(I)/l(I), where I = [ x Ax, x ] for Ax < 0 and I = [ x , x Ax] for A x > 0 and l(I) = Ax is the length of I. f ( x ) then possesses a derivative at x if liml(I)-toF(I)/l(I) exists, I being limited to the intervals which have x as an end point. Theorem 2.2 points up the fact that a continuous function f ( x ) possess a derivative at x when the class of intervals allowed for F ( I ) / l ( I ) is extended to those containing x either as an end point or an interior point. There is here the germ of a derivative notion of one function of intervals with respect to another such function, leading possibly to a derivative in higher dimensional space. If f ( x ) does not have a derivative at x,, that is,
+
+
lim 1L+O ( f ( x ,
I
+ I
+ Ax) - f ( x , ) ) /Jx
may fail to exist, then we have values approached by this ratio, which are called derivates. The greatest of such values or greatest of the limits is the upper derivate at x , : Df(x,) and the least, the lower derivate at x,: _Df(x,). On a linear interval we can also distinguish between the left (Ax < 0) and the right ( A x > 0) approaches giving unilateral derivatives: f ’ ( x , - 0), f ’ ( x , - 0), as well as unilateral extreme derivates : D+f(x,), D J(x,), D-f(x,), D - f ( x , ) . Obviously B f ( x , ) is the larger of D+f(x,) and D-f(x,) and D f ( x J is the smaller of D + f ( x , ) and D-f ( x , ) . Note that at a given point the “tangent” line representing 0 - f (x,) is below that representing D-f ( x , ) . 2.3. If 5 is any fixed upper derivate and D the corresponding lower derivate, then for two functions f and g we have by 1.4.6, Df
+ Q g 5 g(f + h ) 5 of + B g 5 B ( f + g) 5 @ + Dg, +
provided no expressions of the form 00 - 00 occur. In particular, if f has a finite unilateral or bilateral derivative ( D f = D f and finite), then B ( f + g ) = f ’ + & and @ ( f + g ) = f ’ + g g and
and D - are distributive in this case.
348
XI. DERIVATIVES AND INTEGRALS
The following theorems are easily proved : 2.4. If all the extreme derivates of
f(x)
at a point are finite then
f(x) is continuous at this point. 2.5. Iff(x) is continuous on [a, b ] then at any point
x, of [a,b] the values approached by (f(x, Ax) - f(x,))/Ax as Ax + 0, that is the values of the derivates fill up the closed interval [Df(x,),_D(f(x,) 1.
+
2.6. Necessary condition that
xg be a relative maximum of f(x) [that is, f(x) Sf(x,) for some vicinity of x,] is that Dff(xo) 5 0 and D-f(x,) L 0. We state without proving the: 2.7. Mean Value Theorem of the Differential Calculus. Iff(x) is con-
tinuous on a 5 x 5 b and f '(x) exists for a < x < b, then there exists a n x, with a < x, < b such that (f(b) - f(a))/(b - a) = f '(x,). Following are consequences of the mean value theorem: 2.8. If f(x) on [a,b ] is such that f '(x) = 0 for all x, then f(x) is constant on [a,b]. The conclusion also holds if f ' ( x ) = 0 except for a finite number of points, provided f(x) is continuous.
I
I
2.9. Iff(x) is continuous on [a,b],f'(x) exists for 0 < x - x, < d with a < xo < b, and lim,,,o f '(x) exists, then f '(xJ exists and limT+{f'(x) =f'(x,). Changes in the statement of the theorem when x, = a, or xo = b are obvious. The mean ,value theorem applies to f(x) on the intervals [xo; xo d] and [x, - d, x,], so that if 0 < I x - x, I < d, then (f(x) f(x,))/(x - x,) = f '(x'), where x' is between x and x,,. Under the hypotheses of the theorem we then find lim2+l (f(x)-f(x,))/(x - x,) = limL+Tf' ( x ) .
+
2.10. If
f(x) is continuous on [a,b ] and f '(x) exists as a finite value for all x of [a,b], then f '(x) takes on all values between any two of its values. Let a 5 c < d S b. On [c, d] consider the functions g,(x) g,(x)
(f(d) - f(x))/(d = (f(x) - f(c))l(x =
x) for x f d ; = f '(d) for x = d; - c) for x f c ; = f '(c) for x = c.
-
Then g,(x) and g,(x) are continuous on {c,d], and so g,(x) takes on all values between (f(d) - f(c))/(d - c) and f '(d), and g,(x)
2.
349
ON DERIVATIVES
takes on all values between f ‘(c) and ( f ( d ) - f(c))/(d - c). If y o lies between f ’(c) and f ‘ ( d ) , then either g , ( x ) or g,(x) takes on the value y o in [c, d ] . Suppose it is g , ( x ) . Then there exists x1 with c < x1 < d, such that yo = g , ( x , ) = ( f ( d ) - f ( x , ) ) / ( d - x,). The mean value theorem applied to f ( x ) now gives a point x2 between x1 and d such that
f
’(XJ
=
(f(d) - . N x , M d
-
XI) = g,(x,) = Yo.
The same method of proof can be applied even iff ’ ( c ) and f ’ ( d ) are not finite, so that the theorem holds provided f ( x ) is continuous and f ’(x) exists for all x of [a, b ] . The theorem is of importance in connection with the inverse derivative problem, in that it says that a function g ( x ) can be the derivative of a continuous function f(x) for all x of [a, b ] only if for all a 5 c < d 5 b, g ( x ) takes on all values between g ( c ) and g ( d ) on [c, d ] . For instance, the function g(x) = 0 for 0 5 x < 1 ; = 1 for x = 1, is not the derivative of any continuous function on [0, 1 1 . A function f ( x ) on [a,b ] having the property that for every a 5 c < d 5 b, f ( x J assumes all of the values between f ( c ) and f ( d ) on [c, d ] is sometimes called Darboux contimious. Continuous functions and derivative functions of continuous functions then have this property. H. Lebesgue (“ LeCons sur l’htegration,” Gauthier-Villars, Paris, 1928, p . 97) has given an example of a Darboux continuous function on [0, 1 1 which is not continuous at any point. 2.11. THEOREM. If f , , ( x ) are defined on [a, b ] such that the derivatives f , , ‘ ( x ) exist as finite numbers for all n and x, and if f,’(x) converge uniformly on [a, b ] , then f , , ( x ) - f , [ ( a ) converges uniformly to a continuous function f ( x ) such that f ’(x) = limnf,’(x). This theorem is usually proved by using the fundamental theorems A and B of the integral calculus, but this requires in addition that the derivative functions be Riemann integrable. The following proof uses mainly the mean value theorem of the differential calculus. Since the f , , ’ ( x ) are finite, f , , ( x ) are continuous on [a, b ] . We then apply the mean value theorem to the functions gm,,(x) = r , ( x ) - f,(x) to give
(fm ( x ) = (.m(x) = (fm’
-
J;,( a )) - (f ( x ) 71
-fn(x))
(Xm
Ti)
-
f
,I‘
-
(fm(a)
-
f ( a )) 7,
-fT,(a)
(xm 7 1 ) 1 ( X - a ) >
350
XI. DERIVATIVES A N D INTEGRALS
where a < x,, < x. Since lim,, (f,'(x) [a, b ] , we have
I
e>O,n,,m,n>n, : (f,( X I -fa
( a ) ) -f,(x)
-f , ' ( x ) ) = 0
-f,(a))
I<
e (x-1
uniformly on
<e(b--a).
Then the sequence f,(x) - f,(a) converges uniformly to a necessarily continuous function f ( x ) . If we replace x by x , Ax, and a by x,, then
+
Taking limits as to m gives:
On the other hand, if we set lim,f,'(x) of [a, bl e > 0, n,', n > n,' : lfn'(xo)
= g(x), - g(x,)
then for any x ,
I < e.
Further, since f,'(x,) exists:
If now we select an no > n, and n,' and take d,
I
I
I
e > 0, d,,O < Ax < d, : (f(x,
= den0, then
+ Ax) - f ( x , ) ) / A x
- g(x,)
I < 3e.
This means that f ' ( x , ) = g(x,) = limnfn'(xo) for all xo of [a, bl. With the aid of this theorem it is possible to show that for any continous function f ( x ) on [a, b ] there exists a continuous function g ( x ) so that g'(x) = f ( x ) on [a, b]. For by the Weierstrass polynomial approximation theorem there exists a sequence of polynomials P,(x) converging uniformly to f ( x ) on [a, b]. For polynomials P,(x) we have formulas giving us polynomials Q,(x) such that Qn'(x)= P,(x). Then Q,'(x) converge uniformly to f ( x ) and SO
2.
ON DERIVATIVES
35 1
Q,(x) - Q,(a) converges uniformly to a function g(x) such that g’(x) = f(x). The proof of the Weierstrass approximation theorem in 11.18.3. was made by the use of Bernstein polynomials and does not involve integration. It could then be used to give us a formula for the inverse derivative of a given continuous function f ( x ) on [a,b] without integration. Note that whether we use indefinite integrals or the Weierstrass approximation theorem, the desired inverse derivative depends on the convergence of some sequence of functions (see H. Lebesgue : “ Legons sur l’Integration,” Gauthier -Villars, Paris, 1928, p. 95). For the case when f(x) is continuous but the derivative does not exist at all points of [a,b], we have:
[a,b]; if Df(x) is one of the four extreme unilateral derivate functions of f(x) : DYf(x), D,f(x), D-f(x) or D -f(x) ;and if Df(x) 2 A for all x of [a,b], then ( f ( d ) - f(c))/(d - c) 2 A for all [c, d ] such that a 5 c < d 5 b. It is sufficient to prove this theorem for c = a and d = b. We take Df= D+J Then for e > 0 and x = a, we can find a point x, to the right of a such that f(x,) -f(a)2 ( A - e ) (xl- a). Then there exists a point x2to the right of x, such that f(x,) - f(x,) 2 ( A - e) (x2- x,) and consequently 2.12. THEOREM. If f(x) is continuous on
And so on. Let x‘ be the least upper bound of points such that if x < x‘ there exists an x, with x < x, < x’ and such that f(x,) f(a) 2 ( A - e) (x, - a). Then because of the continuity of f(x) we also have f(x‘) - f(a) 2 (A - e ) (x‘ - a). Either x’ = b or x’< b. But in the latter case there exist points x” > x’ such that f(x”) f(x‘) 2 (A - e) (x” - x‘) and so f(x”) - f(a) 2 (A - e ) (x” - a). Consequently, x” = b and for all e > 0 :f(b) - f(a) 2 (A - e) (b - a). Then f(b) -f(a) 2 A(b - a). The same procedure works for the other derivates excepting that for D-fand D-f we start from b and go to the left, If the condition Df(x) 1A for all x of [a,b] is replaced by Df(x) 5 A, then the conclusion becomes (f(d) - f(c))/(d - c ) 5 A(d - c) for all a 5 c < d 5 b. Note that the proof depends on the unilateral character of the derivates involved. This theorem leads at once to the following:
352
XI. DERIVATIVES A N D INTEGRALS
2.13. T H E O R E M . A necessary and sufficient condition that the continuous function f(x) be monotonic nondecreasing on [a, b ] is that one of the four extreme unilateral derivatives satisfies the condition Df(x) 2 0 for all x of [a, b].
f(x) is continuous on [a, b ] and has a zero right (left) derivate at each point of a 5 x < b (a < x 5 b), then f(x) is constant on [a, b]. For then D+f(x) 2 0 for all x so that,f(x) - f ( a ) 2 0 for a 5 x 5 b. Similarly D,f(x) 5 0 for all x so that f(x) - f(a) 5 0 for all a 5 x 5 b. Then f(x) = f ( a ) for x on [a, b]. The proof for the " left " case is the same. 2.14. T H E O R E M . If
f(x) is continuous on [a, b ] and M and m are the least upper bound and greatest lower bound, respectively, of a fixed extreme unilateral derivate Df(x) on [a,b ] then m(d - c) 5 f(d) - f(c) 5 M ( d - c ) for all [c, d ] with a 5 c < d 5 b. This theorem is a sort of generalisation of the mean value theorem of the differential calculus. It has a number of interesting consequences. 2.15. T H E O R E M . If
2.16. T H E O R E M . If f(x) is continuous on [a, b ] ,then all the extreme
unilateral derivates have the same least upper and greatest lower bounds on all subintervals of [a, b ] . For suppose D,f(x) has M I and D,f(x) has M, as least upper bound on [c, d ] and suppose Ml > M,. Then there exists an x1 of [c, d ] such that M , 2 D , f(x,) > N : > M,, and consequently a point x, in [c, d ] in a vicinity of x1 such that (f(x,) - f(x,)) /(x, - xl) 2 N. But since D,f(x) 5 M , for all x on [a, b ] it would follow that
(f(x,)
-
f(x,)) I(x,
--
XI)
s M , < N.
For the bilateral upper derivate D f ( x ) and lower derivate Df(x) we can assert only that the 1.u.b. of Df(x) is the same as that for Dff(x) and D-f(x), while the g.1.b. of D f(x) is the same as that for D,f(x) and D-f(x). 2.17. T H E O R E M . If f(x) is continuous on [a, b ] and one of the extreme unilateral derivates is Riemann integrable, then all of these derivates are, and the integrals on [a, b ] have f(b) - f ( a ) as value.
2.
353
ON DERIVATIVES
Moreover, the bilateral upper and lower derivates are also integrable to the same value. All of the extreme derivates are equal except for a set of measure zero andf(x) has a derivative except for a set of measure zero on [a, b]. For if for any subdivision 0 of [a, b ] into intervals we set up the upper and lower sums ZoM l ( x l - x t P l ) and Zom , ( x , - x t p 1 ) ,then these sums will be the same for all extreme unilateral derivates since all have the same 1.u.b. and g.1.b. on each of the intervals [xZpl,x J . This means that if one of the unilateral extreme derivates is integrable, then all are and have the same value for their integrals. Since on [X,-l'
x,I: m,(x, - x,-J S f ( x , ) - f ( X , - J 5 MJX,
-
q-J,
it follows that when J,"Df(x)dx exists, then its value is f ( b ) - f(a). Since the 1.u.b. of Df(x)on any interval agrees wit.h that of D+f(x) and D-f(x), it follows that -
s" @(x)dx
=
a
j h" D"f(x)dx =f(b)
-
f(a).
Similarly since the g.1.b. of D f ( x ) on any interval agrees with that of D , f and 0-f, we have
J;- g f w d x =
Jb
Then since -a
f(x)dx 4
-a
sh
Df ( x ) d x 5
,L
and Jb
D , f ( x ) d x = f ( b ) -.f(a).
f(x)dx 4
s"
-4
sb
Bf(x)dx
a
-
Df(x)dx S
j b Bf(x)dx, a
conclude that D f ( x ) and Of ( x ) are both integrable with S$ gf (x)dx o f ( x ) d x . Then J: ( z f ( x ) - D f ( x ) ) d x = 0, which since i?f(x) .D f ( x ) 2 0 for all x , implies that z f ( x ) = D f ( x ) except at most at their points of discontinuity, a set of measure zero, where we use 11.15, Ex. 6. This means also that f ( x ) has a derivative except for a set of measure zero. (For these results compare H. Lebesgue: "LeCons sur l'htegration," Gauthier-Villars, Paris, 1928, Chapter v.)
s:
354
XI. DERIVATIVES AND INTEGRALS
EXERCISES 1. Show that if f(x) is continuous on [a, b ] and if Df(x) 5 A for all x of [a,b ] , then for all a 5 c < d S b, we have f(d) - f(c) 5 A ( d - c); and if D f(x) 2 A, then f ( d ) - f(c) 2 A ( d - c). Will these statements still be valid if in the first Df(x) 5 A is replaced by D f(x) 5 A, and in the second D f(x) 2 A by Df(x) 2 A? 2. Is the following theorem true: If f(x) is continuous on [a, b ] and has
a zero derivate for all x of [a,b ] then f ( x )
= f(a)
for all x?
3. In the proof of theorem XI.2.17. deduce the existence of a derivative for f(x) except for a set of measure zero from the fact that f(x) - f(a) = Df(x)dx, where Df(x) is any of the four extremal unilateral derivates.
JI
3. The Vitali Theorem
The Vitali theorem is of the nature of a covering theorem in that it gives a means of reducing a system of intervals to a finite or denumerable number of intervals. It is particularly concerned with the class of intervals arising in connection with derivatives and proves to be a powerful tool in the extension of the fundamental theorems of the integral calculus to Lebesgue integrals, that is, in determining the relation between indefinite integrals and their derivatives.
A point x of the finite interval [a, b ] is said to be Vitali covered by a system of intervals 3 = [Z], if there exists a sequence of intervals I,, in 3, each containing x either as an interior or end point, and such that limn Z(Z,,J = 0. 3.1. Definitions.
3.1 .l.A set E of [a, b ] is Vitali covered by a set of intervals 3, if every
point x of E is Vitali covered by 3. For instance, if f ( x ) is a function on [a, b ] and E consists of the set of points x of [a, b ] for which f ( x ) has A as a derivate at x, and if 3 consists of the intervals [x, x A,x] such that
+
then E is Vitali covered by 3. The basic theorem is:
3.
355
THE VITAL1 THEOREM
3.2. Vitali Theorem. If E is any subset of the finite closed interval [a, b ] and 3 is a Vitali covering for E, then for e > 0, there exists a sequence of disjoint intervals I , from 3, such that
Consequently, for e > 0, there exists a finite set Z, intervals in 3, such that
2 l(I,) < p*(E) ie, and 1
... ZPt of
p*(E- E t IJ
disjoint
< e.
1
For any e > 0, there exists an open set G 2 E such that p ( G ) < p*(E) e. If we delete from the set 3 all intervals which are not contained in G, we will still have a Vitali covering 3’ for E, and also for any sequence {I,) of disjoint intervals chosen from 3’, we shall have C, l(1,) < p*E e. We can then assume that our original set of intervals 3 lies entirely in G. Let c, = 1.u.b. [l(I) for I in 31. Select as I , any I of 3 such that l(Z,) > 4 2 . Delete from 3 any interval which intersects I, and call 3, the remaining set. 3, will be a Vitali covering for E - IIE. Let c, = 1.u.b. [l(I) for I in 3,], and take Z, in 3, so that l(I,) > c,/2. Delete from 3, all intervals intersecting I2 and call 3, the remaining intervals which will be a Vitali covering for E - (I, I J E . Continuing in this way, at the nth stage we get a set of intervals 3,, no one of which intersects I, ... I,1,we let c , + ~= 1.u.b. [l(I) for I in 3,], and I,/+, an interval of 3, such that l(I,+,) > c,+J2; and 3,+, the intervals of 3 , not intersecting In+,. Since the intervals I,, are disjoint and subintervals of [a, b ] it follows that Z,l(Z,) < co, so that lim,c, = 0, with c , + ~S c,. Let I: be the interval of length 51(I,), obtained by extending Z,? on each side by double its length. Then we show that for every n, every point of E belongs to one of the sequence of intervals I,, ..., I,, * In+,, * .... The only points which cause trouble are those which In+,, do not belong to any I,. Since x is Vitali covered by the interval set 3,L,there exists an interval I, in 3,, with length l(IJ 5 c ? ( +containing ~ x. Now I, is eventually deleted. Suppose that c,+, < l(Z,) 5 c,. Then I, has been deleted by the time the mth stage has been reached, that is, Z, intersects one of the intervals I,+, ... I,, say Ik, with n 15 k 5 m. Now l(I,) > ck/2.Then I, lies entirely in the extended interval Zz, since I(!,,) 5 ck, as a simple diagram will show. Then :I contains x . Since l(I,*) = 51(In),and L’,,l(In) < co, it follows that lim, 2: l(I:)
+
+
+
+
356
XI. DERIVATIVES AND INTEGRALS
= 0. For d > 0, take k so that Zzl l ( I f ) < d. Then since E is contained in Z;,IX,it follows that k 1
2 l(1:) < d,
ZTI,,
m
30
I,) 5
p*(E - E
-E
and p ( E
-
k+l
E
I,)
= 0.
I
A useful form of this Vitali theorem is 3.3. If E is any bounded set of the real line, and 3 is any Vitali covering for E, then for every e > 0, there exists a finite set of disjoint intervals: I, ... I , of 5, such that
For :
if the I, ... I , have been chosen in such a way that p * ( E - E 2: I,) < e. Further, since the intervals I, are disjoint and measurable, p* is by V.10.5 additive on the sets E I,, k = 1 ... N, so that
The remaining inequalities are obvious. The Vitali theorem gives us a generalization of Theorems XI.2.8 and XI.2.14 under which the vanishing of derivates of a function guarantee that the function is a constant on an interval: 3.4. T H E O R E M . If f ( x ) is absolutely continuous on [a,b] andf'(x)
except for a set of zero measure, then f ( x ) is constant on [a, b ] . Absolute continuity can be taken in the interval sense that for an interval function F ( I ) , the expression 2: F(I,) for I, disjoint converges to zero as 2; l(Im) approaches zero. A point function f ( x ) is then absolutely continuous if the corresponding interval function f ( I ) =f(d) - f ( c ) for I = [c, d] is absolutely continuous. An absolutely continuous point function is continuous and also of bounded variation. Suppose e > 0, d,, 2; l(1,) < d,: L':f (I,) < e. If E is the set for which f ' ( x ) = 0, then for e > 0, there exist for every x of E, =0
I
I
intervals I,, such that
4.
DERIVATIVES OF MONOTONIC FUNCTIONS
=
[x, x
+ d , ] , or I,,,
I f f x + d,) - f f x )
=
[x
+ d,,
357
x ] with limrLdTL =0
I I I d,, I = If(I,d I I K,) < e.
The set of intervals I,,, for all yt and x of E, form a Vitali covering for E. By the Vitali theorem we then obtain a finite number of these intervals I , ... I , such that p ( E ) - 2; I(&) < d,. If li' denote inIk, then Zi l(Ii') < d,. Then tervals complementary to
Consequently,
Since this is true for all e, it follows that f ( b ) = f ( a ) . Also, since the the hypothesis of the theorem is valid for all intervals [a, c ] with a 5 c 5 b, we have f ( x ) = f ( a ) for all c, that is f ( x ) is constant on [a, b l . 3.5. Corollary. If f ( x ) and g ( x ) have the same finite derivative on
a set E of [a, b ] whose complement CE is of zero measure, and if f ( x ) - g ( x ) is absolutely continuous on [a, b ] , then f ( x ) differs from g ( x ) by a constant. It is to be noted that the condition that f ( x ) be absolutely continuous in the hypothesis of the theorem is necessary. For example the Cantor function [defined as follows: if x = C,p,,/3n and all a, are 0 or 2, then f ( x ) = (27~a$7'):if x = a,/3 ... ~ , / 3 ~... and a, = 0 or 2 for ic k with ak = 1, then f ( x ) = 6 (aJ2 ... (ak 1)/2')] is continuous and constant on the intervals complementary to the Cantor nondense perfect set. Consequently, f ' ( x ) = 0 except on the Cantor set which is of content and so measure zero. Obviously the Cantor function is not absolutely continuous since it maps the Cantor set on the unit interval.
fr
+ + +
+ +
+
4. Derivatives of Monotonic Functions and of Functions of Bounded Variation
If f ( x ) is finite valued and monotonic nondecreasing on [a, b ] then all of its derivates are positive or zero and may be a.
+
358
XI. DERIVATIVES A N D INTEGRALS
4.1. THEOREM. If f ( x ) is finite valued and monotonic nondecreasing on [a, b ] and E is the set of values for which zf(x) = co,
+
then p( E) = 0. If M is any fixed large positive number, then for any x of E, there exist intervals I,, of the form ( x , x+ d,) or ( x d,, x ) with lim,d, = 0 such that
+
If(x or
+ dn) - f ( x ) I I I dn I = f ( I n J f(In,)
I '(In,) > M
> M '(InJ
Then the set of intervals I,, form a Vitali covering for the set E. Consequently, by the Vitali theorem XI.3.2 for e > 0, there exists a finite number of these intervals Il ... I , such that p*(E) - e < q l ( I k ) . Since for all intervals I, we have f ( I ) h 0, it follows that
Then for all e > 0, P*(E) < e
+
-f(a))lM.
Since e and M are arbitrary, it follows that p * ( E ) = 0. The derivates of a monotonic nondecreasing functions are then finite excepting possibly at a set of x of zero measure. It is possible to say more. 4.2. THEOREM. If f ( x ) is finite valued and monotonic nondecreasing on [a, b ] , then with the exception of a set of points of zero measure,
the derivative f ' ( x ) exists as a finite number. If x is any point for which Df(x)> D f ( x ) 2 0, then there exist rational numbers rl and r2 such that B f ( x ) > rl > r2 > Q f ( x ) . So that
Now the sets E,,,, as rl and r2 range over the rationals is a denumerable collection of sets. If then we prove that for any two rationals rl > r2 we have p(ETl,,) = 0, then ,u(E[Df(x) > D f ( x ) ] ) = 0 also. Let D f ( x ) ] . If D f ( x ) < r2, then then E = E,1,2 = E [D f(x) > r1 > r2 > there exists a sequence of intervals I,, with x as one end point such
4.
DERIVATIVES OF MONOTONIC FUNCTIONS
359
that f ( I T 7 J< r21(I,J with limIil(I?ix)= 0. Such intervals form a Vitali covering for E. Then by the Vitali theorem XI.3.2 for e > 0, there exists a finite number of disjoint intervals I, ... I, such that
and for which
If we assume that the intervals I, are open, which we can do without changing the right-hand side of these inequalities, then for every point x of Ik . E, we have o f (x) > r1 so that there exists a sequence of intervals Iknxinterior to Ik having x as one end point and such that
These in turn yield a finite set for each k : Ikn,n
=
1 ... nk, such that
?I = ?I y:
Then since f (x) is monotone nondecreasing :
and so
But m
2 pu*(Ik E ) > p * E - e *
1
Then
and
2 f ( I k ) < r,(p*E + e ) . 1
3 60
XI. DERIVATIVES AND INTEGRALS
Since any function of bounded variation on [a,b] is the difference of two monotonic nondecreasing functions, it follows at once : 4.3. T H E O R E M . Iff(x) is of bounded variation on [a,b] thenf(x)
has a finite derivative f'(x) except at most for a set of a zero measure. We apply this theorem to the functionf(x) = p * ( E [ a , x]), where E is any subset of [a,b].This function is monotonic nondecreasing in x and consequently has a finite derivative except for a set of zero measure. Now since intervals are measurable sets by V.10.5 the interval function p * ( E * I ) is additive relative to Z, so that for any interval [c, d] we have:
Consequently, if I, is any interval with x as end point, we have f(Zx)/l(Iz)5 1 , so that any derivates or derivatives of p*(E[a, x]) have values between 0 and 1, inclusive. We can say more: 4.4. T H E O R E M . The derivative of f ( x ) = p*(E[a, x]) is unity at
all points of E excepting at most a set of zero measure on E. For 0 < c < 1, denote by E, the set x of E such that D p * ( E [ a , x]) < c. Then we determine a Vitali covering for E, by taking for all x of E,, the intervals I, having x as one end point such that p * ( E I,) < c l(Iz). Let e > 0 and I, ... I , be those intervals determined by the Vitali theorem for which
p*(E,)
-
e<
m
2I p U * ( E cI,) *
c [(I,) m
5
P*(E,)
e.
1
Then since Ec is contained in E :
(1
-
c)p*(E,) < e(l
+ c)
or p*(E,) < e(l
+ c ) / ( l - c).
Since this holds for all e > 0, it follows that p*(E,) = 0 for all 0 <
5. c
< 1. If
stance c ,
DERIVATIVES OF INDEFINITE LEBESGUE INTEGRALS
361
{c,,} is a monotone sequence such that limlLc,z = 1, for in=
1
-
l / n and E, is the subset of E for which Dp*(E[a, X I )
< 1,
then Eo = Z71Ee,,and consequently p*(E,) = 0. The derivative of p*E[a,x ] ) at any point x is called the metric density of E at x , so that the theorem says that the metric density of a set E is 1 at points of E except at most for a subset of E of zero measure. If E is measurable then
Then at any point of E where (d/dx)pE[a,X I ) = 1, we also have (d/dx)p(CE[a,X I ) = 0. Similarly if (d/dx)p(CE[a,X I ) = 1, then (d/dx)p(E[a, X I ) = 0. Consequently: 4.5. THEOREM. If E any measurable subset of [a, b ] , then except for
a set of zero measure on [a, b ] , the derivative of p(E[a,X I ) is 1 for x on E and 0 for x on CE. 5. Derivatives of Indefinite Lebesgue Integrals
If E is measurable, then p(E[a, X I ) = J: % ( E ;x)dx. The last theorem (XI.4.5) can then be restated :
x) of a measurable set E of [a, b ] , the derivative of the indefinite integral J,"x ( E ; x ) d x exists and is x(E;x ) except at most for a set of zero measure on [a, b ] . 5.1. For the characteristic function x ( E ;
This extends at once to:
f(x) is a measurable step function assuming at most a finite number of values, that is, f ( x ) = 2; c, x ( E m ;x ) , where the measurable sets E, are disjoint and 2; Em = [a, b ] , then (dldx) J: f ( x ) d x exists and is equal to f ( x ) except for a set of zero measure. This follows at once from the linearity of the integration and the derivative processes. In order to extend this theorem to the case when f ( x ) is any integrable function, we need additional information on the interchange of 5.2. THEOREM. If
362
XI. DERIVATIVES AND INTEGRALS
derivatives and limits. The following theorem due to G. Fubini [Atti Accad. Naz. Lincei Rend. Classe sci. 3 s . mat. e nut. (5) 24 (1915) 2042061 serves this purpose: 5.3. THEOREM. If f , , ( x ) are monotonic nondecreasing in x on [a, b ]
and monotone in n in a strong sense that for every interval [c, d ] of [a, bl we havef,,(d) - f,,W S f , , + &-fn+1(C), tJ andif lim,,f,,(x) = f ( x ) , a finite valued function on [a, b ] , then lim,, f , l ( x ) = f ' ( x ) except for a set of zero measure. The monotoneity conditions of the theorem are fulfilled in case h n ( x ) is a sequence of monotonic nondecreasing functions and f , , ( x ) = 2; h,(x). Also if k , , ( x ) are integrable and satisfy the conditions: k,,(x) 2 0 and k12-l(x)5 k , ( x ) in n and x on [a, b ] , then the indefinite integrals f , , ( x ) = J: k , , ( x ) dx satisfy the monotoneity conditions. Since a monotonic function has a finite derivative except at most for a set of zero measure, let E, be the set of x such that if Eo = CE, then p ( E o ) = 0 and f , l ( x ) and f ' ( x ) exist as finite values for all x of E,. If we set g , ( x ) =f ( x ) - f , ( x ) , then the sequence of functions g , , ( x ) has the following properties: (1) g , ( x ) 2 0 for all n and x ; (2) lim,gll(x) = 0 for all x ; (3) g,,(x) are monotone nondecreasing in x for each n ; (4)for every [c, d ] of [a, b ] and every n :
g,,(d) - gn(C) 2 gn+l(d) - gn+l(C); and (5) g,'(x) 2 g?;+,(x)2 0, for all n and x in El. The first two of these properties are obvious. As for (3) let us denote by Ag the expression g ( d ) - g ( c ) for c < d in [a, b ] . Then
Ag,
=
A(f
-
f,,)
=
'(fim,f,
-
f,)
- f,)
= fim,'(fin
2 0.
For (4) we have:
'g,
-
'gn+1 =
-fn)
= '!,+I-
'In
2 0.
(5) is an immediate consequence of (3) and (4). We show that lim,,gA(x) = 0 except for a set of zero measure on Eland so on [a, b ] , from which the conclusion of the theorem is immediate. For e > 0, and fixed n, let E,,, be the set of x on El for which gA(x) > e. Then in the usual way, we can use the Vitali theorem to produce for a given d > 0, a finite number of disjoint intervals I, ..,I,.such that P*(E,,) - d < '(I,) p*(E,e) -t d
9
5.
DERIVATIVES OF INDEFINITE LEBESGUE INTEGRALS
and gn(Ik)> e l(Ik) for
Then
m
2
gn(b) - g,(a)
1
k
=
363
1 ... m.
c ‘(I,) ’ m
>
gyL(’k)
e(p*(E,e)
- d)*
This holds for any d > 0 so that for fixed e > 0
@+YE,,)
(gm)
- g,(a))le.
Since lim,(g,(b) - g,(a)) = 0, it follows that limvlp*(E,,) = 0. This means that the sequence g7:(x) converges to zero relative to the measure function p*(E) and therefore by V1.5.5 there exists a subsequence gAm(x) such that limmg7;,x) = 0, except for a set of measure zero. Since the g,:(x) are monotone nonincreasing in n on El, we conclude that lim,g7:(x) = 0, except for a set of measure zero on El and so on [a, b]. In the statement of the theorem, the increasing monotoneity condition in n of the functionsf,(x) can be replaced by a corresponding decreasing condition, that is, f,+,(d) - f,+l(c) If,(d) - f,(c) for all [c, d] of [a, b] and n. We now prove: 5.4. THEOREM. If f(x) is any measurable step function which is integrable on [a, b ] , then (dldx) J,”f(x)dx exists and has f(x) as value except for a set of measure zero. The measurability condition is redundant, for iff (x) is an integrable step function, then there exists a sequence of disjoint measurable sets E n such that ZnE,= [a, b] and
f(x)
=2?cn
xm,; x)
Z?I c, I dE,)
with
<
If c: are the positive cn and c; the negative ones and if the theorem holds for .Znc’, x(E,; x) and - Znc; x(E,; x), that is, for positive valued step functions, then it holds for any step function. Suppose then c , 2 0 for all n. Then the sequence of functions fm(x) = c , x(E,; x)dx satisfies the conditions of the Fubini convergence theorem (XI.5.3) and we can conclude that if f(x) = fim,fm(x), then except for a set of measure zero,
Jz
dldx Jx f(x)dx a
=
d/dx
J” 5 c , x(E,;x)dx “
1
cc
2 m
=
lim dldx m
0
2 c, 1
x(E,,; x)dx
c, x ( E ; x) =f(x).
= 1
3 64
XI. DERIVATIVES A N D INTEGRALS
A second application of the Fubini convergence theorem gives us the result we seek relative to the derivative of the indefinite integral: 5.5. THEOREM. If f(x) is integrable on [a, b ] then (dldx) J: f(x)dx exists and is equal to f(x) except at most for a set of zero measure. It is sufficient to prove this for the case when f(x) 2 0 on [a, b ] since any integrable function is expressible as the difference of two positive integrable functions, the positive and the negative part of f(x). By the definition of integral: Jf f(x)dx = limoXom(' E,)p(E,) where o = {En}is a subdivision of [a, b ] into measurable sets, and m ( f , En)= g.1.b. of f(x) on EPI. Let ok be a sequence of subdivisions of [a, b ] such that ok 2 ok+land lim, L'nkm(f,E,lk)p(E?,k) = Jf f(x)dx.
'
If we set gk(x) = z,lm(f,E,lk) X(E,,; for all k and x, and
x), then .f(x) gk+l(x)
gk(x)
In the proof of Theorem VII.8.11 we showed that we also have limkgk(x)=f(x) except for a set of zero measure. Consequently, since for all k and x : gk(x) 5 f(x), and f(x) is integrable it follows that lim k
/' gk(x)dx /' lirn gk(x)dx =
u
(1
k
=
/'f(x)dx a
for all x. The sequence of functions hk(x) = J:gk(x)dx satisfy the hypotheses of the Fubini convergence theorem. Moreover, gk(x) are step functions so that h;(x) = gk(x) except for a set of zero measure. Then except for a set of zero measure:
f(x)
= lim k
gk(x)
=
lim d/dx k
/"" gk(x)dx
= d/dx lim k
/'gk(x)dx (I
= d/dx /'f(x)dx.
"
This theorem is the generalization of the fundamental theorem A of the integral calculus to Lebesgue integrable functions. In one way it tells us more than Theorem A in that it applies to a larger group of integrable functions. In another way it tells less since in the case of Riemann integrable functions, the set of points at which the derivative of the indefinite integral exists and is equal to the function integrated includes the points of continuity of this function.
6.
365
LEBESGUE INTEGRALS OF DERIVATIVES
We note that in the proof of the theorem we could also have used the functions h,(x) = Z, M ( f , En)x (Ell; x) , where u is a subdivision of [a, b] into measurable sets on each of which f(x) is bounded and M(' E ) = 1.u.b. o f f on E. We observe further that if we define p,(x) = J: h,(x)dx and y,(x) = S," g,(x)dx, where
g,(x)
=
C,m(J;
E,,)xfE,; x ) ,
then the functions p,(x) and yi,(x) are absolutely continuous and such that p;(x) 2 f(x) 2 yj:(x) except for a set of zero measure. Moreover, lim,p,(x) = lim,yo(x) = J: f(x)dx for every x. Then g.l.b.opo(b) = limop(b) = lim,y,(b) = 1.u.b. y,(b) = f(x)dx.
Jl
6. Lebesgue Integrals of Derivatives
In considering an extension of fundamental Theorem B of the integral calculus (XI. 1.2) which recovers a continuous function from its derivative function, we note the following : 6.1. THEOREM. If f(x) is continuous on [a, b] and has a finite deri-
vative f '(x) at all points x of [a,b], then f '(x) is a measurable function. We extend f(x) to the left of a and the right of b by the conditions: f(x) = f(a) +f:(x) (x - a ) for x < a, and f(x) = f(b) fl(b) (x - b) for x > b. Then for any x of [a, b] and any sequence h, --f 0, f(x))/h,. Since the function we have f '(x) = lim?,(f(x h J (f(x h) - f(x))/h for fixed h f 0 is continuous on [a, b] and consequently measurable, it follows that f '(x) as the limit of a sequence of measurable functions is also measurable. In the same way we prove:
+
+
+
~
f(x) is measurable on [a, b] and such that f '(x) exists and is finite except for a set E, of zero measure on [a, b], then the function g(x) = f '(x) on CEOand arbitrary on E, is measurable. Consequently if f(x) is of bounded variation on [a, b], the function g(x) = f '(x) where f '(x) is exists and is finite, and arbitrary elsewhere is measurable. We can extend the first theorem to extreme derivates. 6.2. THEOREM. If
f(x) is continuous on [a, b], then the six extreme derivates: Df(x), Df(x), D'f(x), D,. f(x), D-f(x), D-f(x) are all measurable. 6.3. THEOREM. If
366
XI. DERIVATIVES AND INTEGRALS
The difficulty in the proof of this theorem is that the sequence h,, involved in the intervals (x, x h,,,) leading to a particular derivate at a given point varies with x. We proceed to show that because of the fact that f(x) is continuous, it is possible to determine a sequence h, --f 0 independent of x such that for all x of [a, b ] the class of values approached by (f(x h,) -.f(x)) Jhftis the same as that of (’(x h) - f(x))/h as h + 0. It then follows that the extreme derivates are greatest and least limits of a sequence of continuous functions: (f(x h,J - f(x)) /h,Iand consequently measurable. We assume that f(x) is extended beyond a and b by the conditions f(x) = f(a) for x < a, and f(x) = f ( b ) for x > b. Then on the closed rectangles: [a 2 x 5 b, l / ( n 1) 5 h 5 l / n ) , the function (f(x h) -f(x))/h is continuous in x and h, and so uniformly continuous. Then for each n, there exists d,, such that for any rectangle of side at most d,, of [a 5 x 5 b, l / ( n 1 ) 5 h 5 l / n ] , the oscillation of (f(x h) - f(x))/h is less than l / n . Subject the intervals [ l / ( n I ) , l / n ] and [- l/n, - l/(n l ) ] to a subdivision CY by the points h,,, i = 1 ... m,, so that G < derL.Then if l / ( n 1) 5 [h I l / n , there will exist an h,, such that
+
+
+
+
+
+
+
I I
+ +
+
+
I I
1
I (f(x + h) -f(x))/h
-
(f(x
+ h,J
-f(x))/hn,
I
for all x of [a, b ] . Consequently if the h,, are arranged as a single sequence k,, then the values approached by (f(x h) - f(x))/h will be the same as those of (f(x k,) - f(x))/k,, as m + co. Following is a simple but limited extension of fundamental theorem B:
+
+
6.4. T H E O R E M . If the function f(x) on [a, b ] is such that it has a bounded derivative on all of [a, b ] , then f(x) -.f(a) = J,”f’(x)dx
for all x of [a, b]. It is sufficient to prove this theorem for the case x = b. Since f(x) has a finite derivative at each point of [a, b ] it is continuous on [a, b ]. Consequently, f‘(x) is measurable. Since f‘(x) is also bounded, the integral J’:f’(x)dx exists. Extend f(x) beyond a and b by the conditions f(x) = f ( a ) f ’(a) (x - a) for x < a and .f(x) = f(b) f’(b)(x - b ) for x > b. Then (f(x h,) f(x))/h,, with h, f 0 is a sequence of continuous functions, which converges to f ‘(x) for every x as h, --f 0. This sequence is uniformly
+
+
+
6.
367
LEBESGUE INTEGRALS OF DERIVATIVES
bounded in x for every such sequence h,, since by the mean value theorem of the differential calculus :
(f(x
+ h,) -f(x))lh,,
=f ’(x + exoh,),
0 < OZll < 1.
Consequently, by the Lebesgue convergence theorem VII.8.8.3,
=
I*
lim n J b n [(f(x
[lim (f(x
a
n
+ hJ
+ h,)
-
-f(x))/h.Idx
f(x))/h,]dx
=
j bf’(x)dx. (1
Now
sincef(x) is continuous at x = a, and x = b. This completes the proof of the theorem. Since the boundedness of f ‘ ( x ) plays a decisive role in the proof of this theorem, it is not obvious how to extend it to the case when we know only that f’(x) is finite for all x of [a, b] and J’;f’(x)dx exists. Since for any Lebesgue integrable function f(x) the indefinite integral J’,“ f(x)dx is of bounded variation, we consider the integrability of the derivatives of functions of bounded variation and show: 6.5. THEOREM. If f(x) is of bounded variation on [a, b], and f
’(x)
denotes its derivative where it exists as a finite number, and is arbitrary on the set of zero measure where this derivative does not exist or is infinite, then J: f ’(x)dx exists. Since by XI.6.2 f ’(x) is measurable, it is only necessary to show that there exists a subdivision oY of 0 Iy < co, with uY
I I
368
XI. DERIVATIVES A N D INTEGRALS
6.6. Lemma. If E is any set such that
If
'(x)
I > A for all x of E,
then Ap*(E) 5 v * ( f ; E ) .
Let e > 0. Then there exists an open set G 2 E such that v ( f ; G ) 5 v*(f; E )
+ e.
For any x in E, we can find a sequence of intervals I,, interior to G with x as an end point and lirn,( l ( I J = 0, such that 1 f ( I J l/l(In,) > A. These I,, form a Vitali covering for E and, consequently, for d > 0, we can select a finite number of disjoint intervals I, ... I , from this set such that p * ( E ) - d < Z," l(Ik). Then:
5 v*(f; E)
+ e + Ad.
Since d and e are arbitrary, we conclude that Ap*(E) 5 v*(f; E ) . As a special case of this lemma we have: 6.6.1. If f ( x ) is of bounded variation on [a, b ] , and if E is any Lebesgue measurable set which is also measurable relative to the total variation function v ( f ; x ) , and if ' ( x ) I > A on E, then for x on E Ap(E) 2 v(f; E ) . Let now u y = {y,} be any subdivision of 0 2 y < 03 of finite norm and let E, = E [ y ,< '(x) [ 5 yztl], i = 0, 1, .... Then the sets E, are Lebesgue measurable, and there exists a Bore1 measurable set B , contained in E, such that p ( B J = p ( E , ) . Then by our lemma we have on B , : Y , P f B J 5 vff; 4 ) . Consequently,
If
+
If
C,Y,pU(EJ
=C~Y,P(BJ 5
Cz.v(f; B,) 5 vff;
[a,
'1).
It follows then that I f ' ( x ) I is Lebesgue integrable, and this is also true off '(x). Moreover,
lb(f
'(XI
[ dx 5 v f f ; [a, bl).
We can now complete the extension of fundamental theorem B in:
7.
LEBESGUE DECOMPOSITION OF BV FUNCTIONS
3 69
6.7. T H E O R E M . If f(x) is any function of bounded variation, on [a, b] then a necessary and sufficient condition that ,f(x) -f ( a ) =
f'(x)dx for all x of [a, b] is that f(x) be absolutely continuous. The absolute continuity is obviously necessary. For the sufficiency we note that if f(x) is absolutely continuous on [a, b] it is also of bounded variation. Then the indefinite integral function g(x) = J,"f '(x)dx exists and is absolutely continuous. But by XI.5.4, g'(x) = f'(x) except for a set of zero measure. Consequently, by XI.3.5, g(x) = f(x) c, or g(x) = f(x) -f ( a ) , so that
+
f(x) - f f a )
=
J;m/dx.
7. Lebesgue Decomposition of Functions of Bounded Variation
The above generalizations of the fundamental theorems of the integral calculus to Lebesgue integration throw additional light on the structure of functions of bounded variation. For iff (x) is of bounded variation on [a, b], then f'(x) exists as a finite value except for a set of zero measure, and is Lebesgue integrable on [a, b]. If we set g(x) = J,"f'(x)dx, then g'(x) = f'(x) except for a set of zero measure. If we denote g(x) = J,"f '(x)dx, the absolutely continuous part of f(x), by f,,(x), then the difference f(x) - f,,(x) will be a function which will have a derivative zero except for a set of zero measure, which latter may include the points where f ' (x) is not finite or does not exist. If
the function of the breaks of f(x), then f,,(x) = f(x) - fb(x) f,,(x) will be a continuous function, which has derivative zero except at most a set of zero measure [since fb(x) has zero derivative except at a set of zero measure]. Consequently, we have: 7.1. T H E O R E M . Every function of bounded variation is expressible
as the sum of three functions of different types:
f (x) =.fa, (x) + f b (x) + f,,(x)
9
wheref,,(x) is absolutely continuous, f,(x) is a pure break function, andf,,(x) is continuous but has derivative zero except for a set of zero measure. This decomposition is unique if f,,(a) = f ( a ) .
370
XI. DERIVATIVES AND INTEGRALS
For the uniqueness, we have already shown that f , ( x ) is unique, and that f ( x ) - f , ( x ) is continuous. Suppose then f ( x ) is continuous and assume, if possible that f ( x ) = f , ( x ) f i ( x ) = g , ( x ) g,(x), where f , ( x ) and g , ( x ) are absolutely continuous with f,(a) = g,(a) = 0 and f i ( x ) and g,(x) have zero derivative except for a set of zero measure. Then f , ( x ) - g , ( x ) = f i ( x ) - g,(x) has zero derivative except for a set of zero measure, and is absolutely continuous, consequently by XI.3.5, a constant, which is zero. This decomposition of a function of bounded variation is called a Lebesgue decomposition. The function f , , ( x ) is determined by the function f ' ( x ) where it exists finitely, f , ( x ) depends only on the points of discontinuity of f ( x ) , f , , ( x ) would seem to depend on the values of f ( x ) at a set of zero measure. If we think o f f ( x ) as a weight distribution, then f , ( x ) depends on weights concentrated at individual points, fi,( x ) depends on weights distributed over a set of zero measure, and f , , ( x ) is the continuous (derivable) distribution. A simple example of a function of the type f , , ( x ) , which has derivative zero, except at a set of zero measure, is the Cantor function, the continuous monotonic function, constant on the intervals complementary to the Cantor set, which maps the Cantor set of [0, 11 on the unit interval [0, 11. There are however, monotonic continuous functions of this type, where the set of zero measure on which the derivative of f ( x ) is nonexistent or not zero, is everywhere dense on [a, b ] . As an illustration, suppose g ( x ) is the Cantor function on [0, 11. Extend g ( x ) to the interval 0 I x < co by the condition that g(x) = N g ( x - n) for n I x I N 1. Then g ( x ) is continuous and monotonic nondecreasing for x > 0, and has zero derivative on the translations of the Cantor set to [n, IZ I]. Let h ( x ) = Znzog(2"x) /2*" for 0 I x I 1. Then h ( x ) being the sum of a uniformly convergent sequence of positive monotone functions is a monotone nondecreasing continuous function on [0, 11. If E , is the set of points of [0, 11 belonging to the Cantor sets on m/2" < x I (m 1)/2", m = 0, 1, ..., 2" - 1, reduced in the ratio 1 : 2", then g(2"x) has zero derivative on the complement of E,, so that T g ( 2 " x ) / 2 ' " has zero derivative except on 2; E,L,which is of zero measure. Consequently, by the Fubini convergence theorem XI.5.3, h(x) has zero derivative except at most for a set of zero measure E, which may include the points of 22 E,, a set of zero measure. Each of the functions f ( x ) , f , , ( x ) , f , ( x ) , and f , , ( x ) gives rise to an s-additive class of measurable sets, which includes the Bore1 meas-
+
+
+
+ +
+
7.
LEBESGUE DECOMPOSITION OF BV FUNCTIONS
37 1
urable sets, and an s-additive set function. The measurable sets for f , , ( x ) include the Lebesgue measurable sets. Any subset of [a, b ] is measurable relative to f b ( x ) .If E, is the set of all discontinuities of f ( x ) , then for any set E :
+
since f ( x 0) - f ( x - 0) f 0 only if x is in Eb. In the same way one might perhaps guess that forf,,(x) or the corresponding set function f,,(E), there exists a set E, of zero measure such that for certain sets measurable relative to f,,(x), we have f,,(E . E,) = f,,(E). In order to show that this is the case, we make a study of the set function f ( E ) defined by f ( x ) of bounded variation from another angle. For a given function f ( x ) of bounded variation let Q be the class of subsets of [a, b ] which are both Lebesgue measurable and measurable relative to f ( x ) or rather relative to the variation function v(f; x). Then Q is an s-algebra on which both Lebesgue measure and v ( f ; E) are s-additive and which includes the Borel measurable sets. Then we have : 7.2. T H E O R E M . If f ( x ) is any regular function of bounded variation
+
[with ( f ( x 0) - f ( x ) ) (f(x) - f ( x - 0)) 2 0 for all x], and f ( E ) is the corresponding set function, which will be defined on Q, then there exists a Borel measurable set E, of Lebesgue measure zero, such that if f,(E) = f(E E,), then f , ( E ) = f,(E E,) for all E of &. Further, if f,(E) = f(E) - f , ( E ) on Q, then f,(E) is absolutely continuous on &. We prove this theorem first for the case when f ( x ) is monotonic nondecreasing in x, so that f(E) 2 0 for all E of Q. Since f ( E ) is bounded on Q, we let c = 1.u.b. [ f(E) for all Borel measurable subsets E of [a, b ] with p ( E ) = 01. If E , is a set with p(E,) = 0, such that f(E,) 2 c - l/n, then for E, = IZ,,E,,,we have c 2f(E,) 2 c - l / n for all n, so that f ( E , ) = c with p(E,) = 0. If E is any subset of Q, then a similar procedure yields a function f,(E) = 1.u.b. [f ( E ' ) for all E', Borel measurable with ,u(E') = 0 and E' I El, as well as a Borel measurable set EB contained in E such that p ( E , ) = 0, and f , f E ) =f(EL3). We note that the functionf,(E) is s-additive on Q. For if we order sets of Q for which p ( E ) = 0 by inclusion, then for any set E of Q, &(E) = limEof(E,), where the limit is taken relative to the di-
372
XJ. DERIVATIVES AND INTEGRALS
rected set of E, satisfying the conditions p(Eo) = 0, and Eo I E. Because of the additivity of f ( E ) on Q and linearity of limit, we obtain the finite additivity of f l ( E ) . The s-additivity follows from the inequality f l ( E ) I f ( E ) for all E of Q, and the s-additivity of f ( E ) . We can now show that the set Eg, with p(E,) = 0 such that f,(E) = f ( E B ) can be taken to be the set E - E,, where E, is the set E B for the interval [a, b ] . For since there exists a Borel measurable set B 5 E - E, such that f ( B ) = f ( E * E,), with p(B) = p ( E . E,) = 0, it follows that f , ( E ) 2 f ( B ) = , f ( E E,). Then f ( E J =fJ[a, bl) =&(El
+fdW
~ f ( E - E ,+f(CE.E,) ) =f(E,).
Hence equality holds throughout and&(E) = f ( E . E,) for all E of Q. Then ,fl(E) = f ( E * E,) = f ( E * E, * E,) = f,(E - E,) for all E of Q. This completes the demonstration of the first part of the theorem, when f ( x ) is monotone. To prove that f , ( E ) = f ( E ) - f,(E) is absolutely continuous on Q it is sufficient to show that for any E, of Q with p(Eo) = 0, we have f,(E,) = 0, since for an s-algebra of sets this condition implies absolute continuity (see VII.7.3). For given E,, there exists a Borel measurable set B, contained in Eo such that f(E,) =f ( B , ) and obviously ,u(B,) = 0. Then f,(E,) = f (Eo) -A@,)
=
fP,) A@,) 5 0. -
Consequently, f,(E,) = 0. In case f ( x ) is of bounded variation on [a, b ] , we make use of the Jordan decomposition of f ( x ) into the difference of two monotonic nondecreasing functions p ( x ) and n ( x ) . This leads to two positive valued set functions p ( f ; E ) and n ( f ; E ) defined on the sets measurable relative to 'v( E ) . Also by VII.11.6 there exist two disjoint sets E+ and E-, which can be assumed to be Borel measurable, such that p ( f ; E) = f ( E - E ' ) and n ( f ; E ) = -f ( E . E-) for all E measurable relative to v ( f ; E). We can then determine Borel measurable sets E:, a subset of E', and E;, a subset of E-, such that ,u(E;) = 0 and p(E;) = 0, and decompositions of p ( f ; E ) = p , ( f ; E ) p,(f; E ) and n ( f ; El = n,(f; E ) n,(f; E ) , with p ( f ; E - E l ) = pl(f; E ) = p , ( f ; E - E,') and n ( f ; E . E;) = n,(f; E ) = n l ( f ; E E;) for all E of (3, while p,(f; E) and n2(f; E) are absolutely continuous on 6. This leads to functions:
+
+
L(E)
= P,(f;
E)
-
n l ( f ; El and
f p )= P , K
E)
- n,f'
El
7.
LEBESGUE DECOMPOSITION OF BV FUNCTIONS
373
+
and a Borel measurable set E, = E,’ E;, such that f ( E ) = f , ( E) f 2 ( E ) ,f , ( E ) = f l ( E E,) for all E of Q, and f2(E) absolutely continuous on Q. This completes the proof of the theorem. An s-additive set function f ( E ) on an s-algebra for which there exists a set E, of a subclass Q, of Q, such that f ( E * E,) = f ( E ) for all E of Q is said to be singular relative to Q,,.Then the break function f,(E) of a function of bounded variation f ( x ) is singular relative to the sets consisting of a denumerable number of points, the function f , ( E ) above is singular relative to the sets of zero measure of Q. Our theorem could then be stated:
+
7.3. The s-additive set function f ( E ) relative to a regular function of
bounded variation on a finite interval [a, b] admits of a decomposition into the sum of two functionsf,(E) and f2(E) on the class Q of sets which are Lebesgue measurable and measurable relative to f ( E ) , such that f l ( E ) is singular relative to sets of Lebesgue zero measure and f 2 ( E ) is absolutely continuous on Q.This decomposition is unique. The uniqueness of the decomposition is obvious. This decomposition is called the Lebesgue decomposition of the set function f ( E ) . We might note that for a function of bounded variation the set E,’ is a Borel measurable set such that f ( E i ) = 1.u.b.f ( E ) for E ranging over the Borel measurable subsets of [a, b ] for which p ( E ) = 0, while f ( E ; ) = g.1.b. f ( E ) for the same class of sets. To complete the identification of this second decomposition with the one previously obtained, viz., f ( x ) = f , ( x ) +fa,(.) + f , , ( x ) , we show: 7.4. T H E O R E M . If the continuous function of bounded variation
f ( x ) is such that the corresponding s-additive set function is singular with respects to sets of Lebesgue zero measure, then f ( x ) - f ( a ) = f ( [ a , X I ) has derivative zero except for a set of zero measure. We limit ourselves to the case when f ( x ) is monotonic nondecreasing so that f ( E ) 2 0 for all E measurable relative to f. The extension to the case when f ( x ) is of bounded variation can be made without difficulty by using the Jordan decomposition. Suppose if possible that f ‘ ( x ) f 0 on a set of positive measure. Then J: f ’ ( x ) d x > 0. Set f ( x ) = g ( x ) J,S f ‘(x)dx. Then for any Borel measurable set B: f(B) = g(B) JRf ‘(x)dx. In particular, f ( [ a , b ] ) = g([a, bl) Jf f ‘ (x)dx with g( [a, b ] ) < f ( [a, b ] ) . Since f is singular, there exists a set E, with p(E,) = 0, such that f ([a,b] . E, = f ( [a,b ] ) .
+
+
+
374
XI.
DERIVATIVES AND INTEGRALS
But
f ( [ a , b] * E,)
= g([a,
bI . E,)
g ( [ a , bI) < f ( [ a , bI),
which involves a contradiction. In the Lebesgue decomposition by sets, we have f ( E ) = f l ( E ) + f 2 ( E ) where f l ( E ) is singular relative to sets of measure zero, and f,(E) is absolutely continuous, we can decompose f l ( E ) = f b ( E ) + f,,(E), where f , ( E ) is a break function and fl,(E) is continuous, It follows that
f,,([a, XI) =f,,(x) -f(a),
f,(h
XI)
=f,,(x)
=
rf
'fx)dx
where we use the uniqueness of the two types of decomposition. Summarizing, we have that for any function f ( x ) of bounded variation, there are two types of decomposition :
(A)
f(x) =fbcx)
+f,~(~)
where f,(x) is a pure break function, f,,(x) is absolutely continuous and has value J: f '(x)dx, and f , , ( x ) is continuous and has zero derivative except for a set of zero measure; and
(B)
f(E)
+f,~(~)
=fb(E)
where f b ( E ) is a pure break function, that is there exists a set Eb consisting of a denumerable set of points such that for any subset E of [a, b ] :f,(E) = f ( E * E,) = Z,f(P), where P ranges over the points of E * Eb;f,,(E) is absolutely continuous and has as value J E f ' (x)dx for every Lebesgue measurable set; and f,,(E) is singular relative to sets of Lebesgue measure zero, that is there exists a Bore1 measurable set E, with ,u(E,) = 0, such that for all sets E Lebesgue measurable and measurable relative to f, we have f,,( E * E,) = f , , ( E ) ; moreover, f,,(E) is continuous in the sense that if a set consists of a single point P then f,,(E) = f,,(P) = 0.
EXERCISES 1. Show that if f(x) is absolutely continuous on [a, b ] and f '(x) 2 0 except for a set of zero measure, then f ( x ) is monotonic nondecreasing on [a, 61. 2. Show that if f,(x) is a sequence of functions of bounded variation on [a, b ] such that for all [c, d ] of [a,b ] and all n : f,,(d) - f,(c) 2fnPl(d)-
8.
375
THE RADON-NIKODYM THEOREM
fn-l(c), and limnfn(x) = f(x) as a finite valued function, then limnf,,'(x) = f '(x) except at most at a set of zero measure. 3. Is it possible to show that if f(x) is Lebesgue integrable on [a, b ] , then (dldx) f(x)dx = f(x) except for a set of zero measure, without decomposing f(x) into the difference of two positive integrable functions?
J':
4. Suppose that f(x) is such that its upper and lower integrals on [a,b ]
exist as finite numbers. What can be said of the derivatives of the corresponding indefinite integrals f(x)dx and f(x)dx?
7;
s: -
5. Suppose that Of stands for a fixed extremal derivate of f(x) on [a,b], and suppose that Df is bounded on [a,61. Is Df(x)dx = f(x) - f(a)?
s,"
6. If f(x) is absolutely continuous on [a,b ] , and has a zero derivate (zero as a value approached by Af/3x) at all points of a set E whose complement is of zero measure , then f(x) is a constant on [a,b ] . 7. Is it possible to have a continuous function f(x) on [a,b ] such that
f '(x) exists as a finite number at each point and is Lebesgue integrable, and still not have J: f '(x)dx = f(x) - f(a)? 8. To what extent are the theorems relating integrals and derivatives valid for a function Lebesgue (and so absolutely) integrable on (- co, a)?
+
9. Suppose that g(x) is the Cantor function on [0, I ] . Define g(x) on [l, 21 by the condition g(x) = g(2 - x), and then extend g(x) to the interval - GO < x < cc to have period 2, i.e., g(x) = g(x + 2) for all x. Let h(x) = ZnTog(2"x)/2". Show that h(x) has zero derivative except for a set of zero measure. 10. Show that a pure break function has derivative zero except for a set of zero measure. Is this set of zero measure necessarily the set of points of discontinuity of the function? 11. What information does the Lebesgue decomposition of a regular function of bounded variation a(x) give relative to the value of an R-S integral St f(x)da(x), or an L-S integral J E f(x)da(x)? 8. The Radon-Nikodym Theorem
The developments of the preceding section have an equivalent in an abstract setting. The basic result is embodied in the: 8.1. Radon-Nikodym Theorem. If X is a n abstract space, C? an s-algebra of subsets of X , a a finite valued s-additive function on C? with
376
XI.
DERIVATIVES AND INTEGRALS
a ( E ) 2 0 for all E of Q, and P a finite valued s-additive function on Q, then there exists a set E, of @,suchthat a(Eo) = 0, and a point function f on X , Lebesgue integrable with respect to a , such that for any set E of @, we have P(E)
=
P(EE0)
+ j Efda
with
P(E - EE,)
=
SEfda.
If we set P,(E) = P(EE,,) and P,,(E) = P(E - EE,), then Pa, is absolutely continuous relative to a, and ,B, is singular relative to a, that is, we have a Lebesgue decomposition of P(E). Since P(E) is s-additive and finite valued on the s-algebra &, it is bounded and expressible as the difference of two positive valued functions: P(E) = p ( P ; E ) - n(P; E). Under the circumstances, it will be sufficient to prove the theorem for the case when P(E) 2 0 for all E of @. Let { r } be a denumerable set of real numbers with r 2 0, and dense on the positive real axis. Consider the functions: y,(E) = P(E) - rcc(E) on Q. These will be bounded and s-additive. Then as indicated in IX.4.3, for each r there exist disjoint sets E,' and E; such that X = E,' E; and y,(E) = P(E) - ra(E) 2 0 for all E contained in ET+and yr(E) = P(E) - ra(E) < 0 for all E contained in E;. For any y > 0, let Ey= ZIT<,ET+. Since @ is s-additive, E, will be in (3 for each y. Then the sets E, have the following properties:
+
(1) If y1 2 y z then E,, I EY2.
This is obvious.
(2) I7 E,
= E,,.
Y
For if an element x of X is in E, , then for r < yl, x is in E,' and consequently in all E, with y < yl. On the other hand, if x is in all E, with y < yl,then it is in all E,' with r < y. (3) If E is any subset of E, in Q, then P(E) - y a ( E ) 2 0. For if E is any subset of E,, it is a subset of all ET+for r < y , so that
P(E)
-
ra(E) 2 0
for all r < y .
(4) If E is any subset of the complement CE, of E,, then P(E) ya(E) I 0. Obviously CE, = ZT<,CET= ZTIYE;. Now the set of numbers r was assumed to be denumerable, and so Zr
8.
377
THE RADON-NIKODYM THEOREM
expressed as the sum of a sequence of disjoint sets of the form E; ET;. Since E; - E,; is a subset of E i , we will have
P(E;
- E,;) I r'a(E;
-
-
E,;) < ya(E7; - E,;).
By summing these inequalities and using the s-additivity of a and p, we obtain p(CE,) I ya(CE,). In the same way we show that if E is any subset of CE, in E, then P(E) I ya(E). Since P(E) 2 0 = O . a( E) for all E of & we can set E,=, = X . Then : (5) Z E,
= X.
Y20
On the other hand, suppose Em = L7,ET+= n,E,. Then if E is any subset in & of E,, we have P(E) 2 na(E) for all n. It follows that a ( E ) = a(E,) = 0. Relative to the set X - E,, the class of sets E J X - Em) satisfies the conditions of VI.2.5 for a point function determined by the condition f(x)= y if.x belongs to (E,- 2,,>,E,,) ( X - E z ) , with E [f 2 y ] = E,CE,,. We can extend f(x) to all X by the condition that f (x) = 0 on Em,that is a set of a-measure zero. Then since E, and Em are in &, f satisfies the measurability conditions on &. We show that f is Lebesgue integrable with respect to a and that P(X- Ex) = J X f d " .
Let yk be any subdivision of the positive Y-axis of finite norm. Let E, = E[yk2f ( x ) < Y,+~].Then E, is contained in EVk- EUk+,. As a consquence, yka(Ek) I P f E k ) 5 Yk+lCIfEk)' Then xkYkfffEk) I zkPfEk) 5 P f X ) ' Then Zkyka(E,) is convergent and f(x) is Lebesgue integrable. On the other hand, since the subdivision {y,} is of finite norm, and a ( X ) is finite, it follows that Zky,+,a(Ek) is cmvergent also, and
Sx
It follows that f d a which is the limit of Z,y,a(Ek) as is equal to ZkP(E,) or P(X - E,).
I I 0,
+
0
378
XI. DERIVATIVES A N D INTEGRALS
It is obvious that the same procedure could be applied to the case when X is replaced by any set E of &, and the function f ( x ) is f ( x ) X ( E ; x ) , so that we have
for all E of &. Since J B fda is absolutely continuous relative to a, it follows that P,,(E) = P(E - E,) is absolutely continuous. In particular :
P(E) is absolutely continuous relative to a , then @(Em)= 0, so that for any set E of &, we have
8.2. If
The Radon-Nikodym theorem for the case when X is an n-dimensional Euclidean space was derived by J. Radon in “Absolute additive Mengenfunktionen” : Sitzber. Akad. Wiss. Wien 122, Abt. 2a (1913) 1295-1438. The abstract setting setting is due to 0. Nikodym: Sur une generalisation des integrales de Radon, Fundamenta Math. 15 (1930) 131-179, particularly pp. 169-179. The proof given above is a slight modification of Nikodym’s proof. See also J. M. H. Olmsted: Lebesgue theory on a Boolean algebra, Trans. Am, Math. SOC.51 (1942) 185-191; and C. Caratheodory, “Mass und Integral und ihre Algebraisierung,” Birkhauser, Basel, 1956, p. 190.
Some Reference Books on Integration
Bourbaki, N. “ Integration” (“ ElCments de Mathematiques,” XI11 and XXI, Livre VI). Hermann, Paris, 1952 and 1956. Caratheodory C. “ Vorlesungen iiber reelle Funktionen,” 2nd Ed. Teubner, Leipzig, 1927. Caratheodory, C. “ Mass und Integral und ihre Algebraisierung.” Birkhauser, Basel, 1956. Graves, L. M. “ Theory of Functions of Real Variables,” 2nd Ed. McGrawHill, New York, 1956. Hahn, H., and Rosenthal, A. Press, Albuquerque, 1944. Halmos, P. R. Kestelman, H. York, 1960. Lebesgue, H. Paris, 1928. McShane, E. J. Jersey, 1944.
“
Set Functions.” Univ. of New Mexico
.
“
Measure Theory.” Van Nostrand, New York, 1947.
“
Modern Theories of Integration,” 2nd Ed. Dover, New
“
LeCons sur I’Intkgration, ” 2nd Ed. Gauthier-Villars,
“
Integration.” Princeton Univ. Press, Princeton, New
Munroe, M. E. “ Measure and Integration.” Addison-Wesley, Cambridge, Massachusetts, 1953. Picone, M., and Viole, T. “ Lezioni sulla Teoria Moderna dell’rntegrazione.” Einaudi, Turin, Italy, 1952. Saks, S . “Theory of the Integral” (L. C. Young, trans.). Warsaw, Poland, 1937. Zaanen, A. C. “ An Introduction to the Theory of Integration.” NorthHolland, Amsterdam, 1958. 379
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INDEX
A Abel-Dini theorem, 57, 307 Absolute continuity of interval functions, 313 of set functions, 242 uniform, 247 Absolute integrability, 97, 244, 275 Absolute property, 41 Abstract measure theory, 191 Abstract space integrals, 320 Addition of sets, 141 Additive functions of intervals, 26, 102 of sets, 158, 169 Additive property of classes of sets, 146 Algebra of sets, 147 Almost bounded functions, 229, 288 Almost uniform convergence, 210 a-measurability, 168 a-measure zero or a-null set, 172 a*-convergence, 21 1 Approximation theorems (function) for measurable functions, 21 5 Weierstrass polynomial, 82 Approximation theorems (integral) L-S integrals, 269 R-integrals of interval functions, 34 R-S integrals, 53, 60, 131 Arzkla’s Lemma, 72
B Baire functions, 208 Banach space, 22 Base of a filter, 17 Bernstein polynomials, 82 Bessel’s inequality, 303
Bilinear functional or form, 116, 138 Boolean algebra, 142 Boolean ring, 144 Borel measurable functions, 203 Borel sets, 152 Borel theorem, 16, 47 Bounded convergence theorem (Lebesgue), 249 Bounded linear functionals, 23, 181 Bounded variation continuous part of, 40 extended (BVFI), 275 FrCchet, 116 functions of, 37, 106 Jordan decomposition, 39,108 Lebesgue decomposition, 373 regular, 84, 264 space of, 41 uniformly, 44 BVFI, 185, 275 BV2, 114 Break functions, 40
C Cantor set and function, 357, 370 Cauchy condition of convergence, 6, 28, 21 1 Cauchy left and right integrals, 40, 87 Characteristic functions, 146 Cofmal, 6 Compactness, 46 Complementary property of set classes, 147 Complementation of sets, 142 Complete space, 22, 310 Complete orthonormal system, 305 Compositive property, 3 Content, 154 ff
382
INDEX
Continuity, absolute, 313 equal, 15 semi-, 199 Convergence, almost uniform, 210 a*-convergence, 21 1 relative to a or a*, 211 relative uniform, 27 1, 283 Convergence theorems (integral) of interval functions, 36 Lebesgue bounded or dominated, 249 L-S integrals, 247, 250, 255, 271, 280, 282 Osgood theorem, 71, 312 R-S integrals, 69 ff Cover additive, 197
D Darboux continuous function, 349 Darboux integrals, 27, 60 Darboux theorem, 51 Decomposition, Jordan, 39, 108 Lebesgue, 310 Derivates, 347 Derivatives, 77, 343 ff Diagonal procedure, 45 Difference of sets, 142 symmetric, 143 Differential equivalence, 35, 106 Directed set, 4 of sequential character, 11 Dirichlet formula, 55, 95 Directed limits and R-S integrals, 74 and L-S integrals, 255 ff, 259, 282 Dominated convergence theorem, 249 Double and iterated integrals, 132, 335 Double limits, 13
E Equicontinuity, 15 Existence theorems L-S integrals, 226, 230, 276 R-integrals of interval functions, 28 R-S integrals, 49, 59, 63, 66 Y-GS integrals, 234 Extended bounded variation (BVFI), 275 Extension of classes of sets, 148 Extensionally attainable, 148
F Fatou’s Lemma, 250 FBV, 116 Filters, 17 Finitely additive functions of intervals, 26, 102 of sets, 158, 161 Form, linear, 22, 80, 306 Fourier coefficient, 309 FrCchet bounded variation (FBV), 116 F, set, 173 Fubini convergence theorem, 362 Fubini double integral theorem, 335 Functional, 22 continuous, 23 linear, 22, 80, 220, 306 positive, 220 Functions absolutely continuous, 242, 313 bounded variation, 37 ff, 106, 116 break, 40 measurable, 201 monotone (on directed set), 7 nionotonely monotone, 108 semicontinuous, 197 ff, 314 ff step, 208 Functions of intervals, 25, 101 additive, 26, 102 pseudoadditive, 31, 104 semiadditive, 26 Fundamental theorems of calculus, 77, 339 FV(J) (Frkhet variation), 117
G G, set, 173 General limit, 3 General sum, 12 Gramian, 300 Greatest of limits, 9
H Helly’s theorem, 44, 115 Hereditary property, 209, 324 Hilbert space, 299 Hoelder’s inequality, 294
383
INDEX
I Ideals, 172 Integrals Cauchy left and right, 87 Lebesgue-Stieltjes (L-S), 221 ff Mean Stieltjes, 86 Riemann Stieltjes (R-S), 47 ff, 121 ff Young-Lebesgue, 231 ff Young-Stieltjes, 88 Integration by parts, 53, 86, 93, 127, 342 Interval functions, 25, 101 Interval sets, 72, 188 Iterated integrals, 79, 122 and double integrals, 132, 335 Iterated limits, 13 Iterated limits theorem, 14
1 Jordan decomposition, 39, 108
L Lattice, 41 Least of limits, 9 Lebesgue decomposition of a function of bounded variation, 370 a set function, 373 Lebesgue measure, general, 163 special, 179 Lebesgue integral definition, 224 de la VallCe Poussin modification, 230 as measure of a plane set, 333 Young definition, 231 Lebesgue integration postulates, 219 L-S abstract integrals, 320 1' space, 257 L' space, 290 IT, 307 ff Lp space, 291 ff L2 space, 296 ff Left and right Stieltjes integrals, 87 Limit, directed, 3 double, 13 greatest and least, 9 iterated, 13 properties of, 5 of sequence of sets, 145 Linear functionals or forms, 22 continuous, 23, 81, 98, 306
Linear independence, 300 Linear spaces, 20 Lower content, 156 Lower integrals, 28, 59, 234 Lower measure, 166 Lush's theorem, 216
M Mean Stieltjes integral, 49, 86 Mean value theorem for R-S integrals, 68 of differential calculus, 348, 352 second of integral calculus, 68 Measurable functions, 203 ff Measurable sets a-measurable, 168 Bore1 measurable, 162 Examples, 171 Lebesgue measurable, 176 in two dimensions, 188 Measure theory, abstract, 191 Measure zero, 64, 172, 323 Metric density, 360 Metric space, 310 Modified Stieltjes integral, 49,96 Monotone functions on a directed set, 7, 16 in two variables, 108 Moore Smith limit, 4 Multiplication of sets, 139 Multiplicative property of a class of functions, 41, 245 of a class of sets, 147
N Negative part of a function, 206, 245 Negative variation, 39 Net (subdivisions), 102 Nonmeasurable sets, 180, 181 Norm (N) integral, 27 Norm of linear space, 21 of subdivision, 27 Normalized function, 300 Null set (a-null set), 172 w(SfAg; I), 30,50 w(Sf; I), 30
384
INDEX
w(f; I), 26, 50, 101 w(f; x), 63, 202
0 Orthogonal functions, 300 Orthonormal system, 300 complete, 305 Oscillation functions of a function at a point, 63, 202 of a function on an interval, 26, 50, 101, 106 of a Riemann sum on an interval, 30, 50 Osgood’s theorem, 71, 98, 312
P Partition (subdivision), 27 Positive linear functional, 220 Positive part of a function, 206, 245 Positive variation, 39 Product measures, 327 ff Pseudoadditive, 31, 104
R Radon-Nikodym theorem, 375 Rectangular sets, 328 Refinements of subdivisions, 27 Regular function of bounded variation, 84, 264 Relative uniform convergence, 271, 283 Riemann-Stieltjes (R-S) integrals Cauchy left and right, 49, 87 in one variable, 47 ff in two variables, 121 ff mean, 49, 86 modified, 49, 96 on infinite intervals, 97 Young-Stieltjes, 88 Riesz-Fischer theorem, 304 Riesz representation theorem, 82, 99 Ring of sets, 147
S s-additive class of sets, 146 s-additive function of sets, 169 s-algebra of sets, 147 s-multiplicative class of sets, 147
s-ring of sets, 147 Schmidt orthogonalisation process, 301 Schwarz inequality, 294 Semiadditive interval functions, 26, 102 Semicontinuous functions, 199, 317 Separability, 296 Sequential character of directed sets, 11 Set functions, additive, 169 s-additive, 169 singular, 373 cr-finite set function, 325 cr-integrals, 27 ff Simple step or break function, 40 Singular part of set function, 373 Spaces, 20 BV, 37, 41 ; BV2, 114; FBV, 117 1’, 251 L’, 290 ff, 307 ff; L2,296 ff; LV, 291 ff M, 229, 288 Staircase (interval step) function, 218 Step function, 208 Subdivision (partition) into a-measurable sets, 231 into subintervals, 27, 102 Substitution theorems interval function integrals, 35, 105 L-S integrals, 271 R-S integrals, 53, 91, 126 Subtractive property, 147 Sums, general, 12 Symmetric difference of sets, 143, 147
T t-additive, 147 t-multiplicative, 147 Total variation, 37, 107, 117
U Ultimately bounded, 6 Uniform absolute continuity, 247 Uniform bounded variation, 44 Uniform relative convergence, 271, 283 Upper content, 156 Upper integral, 28, 59, 234 Upper measure, 163 Upper measure (u*) convergence, 211
385
INDEX
v Value approached, 7 Variation, negative, 39, 108 positive, 39, 108 total, 37, 107, 117 Vector space, 20 Vitali covering, 354 Vitali nonmeasurable set, 180 Vitali theorem, 355 Vitali variation, 106 v-measure zero. 64
C D € F C H 1 J
8 9 O 1 2 3 4
w Weierstrass approximation theorem, 82 Weierstrass-Bolzano theorem, 44
Y Young (L-S) Y-integral, 231, 235 Young (R-S) integral, 88
Z Zero measure, 64, 169
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