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the case of the descriptive properties. In the case of a function of nvariables, we may project upon an (n - 1) space, an (n - 2) space, etc. The proofs of the preceding theorems are contained in a paper that has been offered to the Transactions of the American Mathematical Society. 1 Ann. Math. Princeton, 18, 1917 (147). 2 For the terminology cf. Denjoy, J. Math., Paris, ser. 7, 1, 1915 (122-125). 3 Cf. for example, Fr&chet, Rend. Circ. Math. Palermo, 22, 1907, p. 1; and Rausdorff,
Grundzisge der Mengenlehre, 1914, p. 211. 4 Hausdorff, 1. c., p. 315. 6 For the case where only measurable sets are admitted cf., for example, de la Valle Poussin, Cours d' Analyse, 2, 1912, p. 114. For the linear case of general (not necessarily measurable) sets cf. Blumberg, Bull. Amer. Math. Soc., 25, 1919 (350). 6 Cf. Denjoy, Bull. Soc. Math. France, 43, 1915 (165). 7 In this connection cf. G&teaux, Ibid., 47, 1919 (47).
GENERALIZED LIMITS IN GENERAL ANALYSIS By CHARIEs N. Moo"E DIPARTMENT oF MATH1mATICs, UNIvERSITY Or CINCINNATI Communicated July 15, 1922
It is well known that to each of the various methods for summing divergent series there corresponds an analogous method for summing divergent integrals. It is readily- seen that similar methods may be used for obtaining types of generalized derivatives of a function at a point where the ordinary derivative fails to exist. Likewise, in any other case in Analysis where we wish to associate a limit with a variable that oscillates, we will naturally be led to make use of methods that have been tried out in the case of divergent series. It would be manifestly poor economy of time and thought to elaborate for each of these special theories such fundamental results as are common to them all, if these results can be obtained in one central theory that includes all the others. According to a principle of generalization formulated by E. H. Moore and stated by him on several occasions,' the existence of such a general theory is implied by the analogies found among the various special theories. It is natural to designate this general theory as the theory of generalized limits in General Analysis. It is the purpose of the present communication to illustrate the nature and usefulness of this theory by outlining the proof of a theorem in it which is-a generalization of one of the important theorems in the theory-of divergent series. This latter theorem is the Knopp- _hnee-Ford theorem2 with regard to the equivalence of the Ceskro and Holder means of order k
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for summing divergent series. Our genreal theorem includes as special cases this theorem, the analogous theorem of Landau, for divergent integrals, and a further new theorem with regard to the equivalence of generalized derivatives of the Cesaro and Holder type. It can also be extended to the case of multiple limits, so that it includes new theorems, analogous to those mentioned, with regard to multiple series, multiple integrals, and partial derivatives. As this latter extension involves more complicated formulas, it will not be dealt with in the present communication. Following the terminology introduced by E. H. Moore,4 we indicate the basis of our general theory as follows:
(e[;,;@i;on@jtog;onSto ;aon eto;S ;fntXon IDto j), where !K = [a] denotes the class of all real numbers a, '=[pI denotes a class of elements p, and e-- [a] denotes a class of sets f of elements p of the [-y],I [ID , and j = [so] are three classes of functions y, range 3; ( t, and p, respectively, on e5 to I (we shall restrict ourselves throughout to the consideration of single-valued functions); s'o is a special function s of the class j; and J is a function on @ to p and on f to a, that is, a functional transformation turning a function of the class @ into a function of the class ! or a function of the class 4 into a function of the class j, denoted by J-y or Jq. In order to make clear- the relationship of our- general theorem to the two special theorems referred to above, we will indicate here what the general basis reduces to in the particular instances III and IV: -
III' [all n =1,.
[
(1, 2,... n)nI];
@-t-0-[~all y,7,sson e5to ]; 90(on) = n (n); (Jy) (or) - 7((Ti) + Y(7l) .+ .... +(T() (n). V' == [all a>0]; e- [or (all x 0<x
0)]; @
[all functions that are finite and integrable
(Lebesgue) on every finite interval (0, a)]; -[all q = Jf Sylj x > 0) ];W [all sop (f~11 I x >0) ] spo(aa) = a (a); (Jy) (a.) = 7',y (7ya); (Jn) (Oa) = f (ta). We next proceed to make certain postulates with regard to the nature of the, elements in our basis, readily seen to be verified in the specific instances cited. Thus we require the class @ to have the linear property (L) as defined by E. H. Moore,4 and the property (P) defined by
(P)
Yl * 772 **17@,
the arguments of the two 7y's being the same. It will then follow that * * ,, is of the class @. We require the class to have the =P 71.2 properties (L) and (P), and the further property of being a subpcass of the
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w5which property we shall denote by SG. We require the class j to have the properties (L) and (P), and the further property of being a sub-class of the; class &, which property we shall designate as SH. Hence is J also on j to j. We require the operation J to have the linear property (L) as defined by E. H. Moore, and the properties M1 and M2, defined as follows: (M1) a,<'yl
a2[(J-Y2) (O')-(J-Y2) (o')].
For the explanation of the other postulates we need certain auxiliary definitions. By the notation a' < a'-, a">', we shall mean that '" contains all the elements of a-' and at least one element not found in a'. For a given 0 on e to !f, a given a, and a given a' such that there exists f< a" 'we shall write iim uO(a) =a ..~~~~~~~~~~~~.. in the case that, corresponding to an arbitrary positive number e, there exists a ce < a' such that for every a- having the property cie C Or
(U)5 Either cr'. . a U [a<'] or $[ a'r>ao- Z-au [aa'] a 5-an[a> a,i. (n)s ,
In the typical instances in view in the formation of this general theory, of the two alternatives in (U) one holds and the other does not hold; however it is not assiumed that this disjunction between the two alternatives shall be presupposed. In order to avoid notational reference to these alternatives, it is convenient to introduce a property-of sets -a of e; if the first altemative holds,, every - of eS has the property-, in notation 6; if the first; alternative doeg not hold, the sets a are the sets a> a-; further for brevity a-property *,(the negation-of-) is used; thus every a is a a or a T.
We define a-i Uta!< l, (af), * ¢l -~~n [or>r' sn [a>n (a+ ' (af-(f-)( ff) or (P(af),= 0 f
(fn
2,3- *)
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and with r.egard to the functions so we postulate (F) all functions sp are of the class 0. We also define the notation
(DO) (a) = a(a)
(1)
with regard to every 0, a on e to W, to mean that
lim o')-e'o(o a|a6
=a(of) (a).
We further postulate as to the class e *
(R)
a(or,)-
="
W(a),
for the class ,o a. .D- a Iimn -q(af)=
-(C)
'(a)
and for the class W a D
(AN)
ja
-'
For the operation J we postulate (D(J,q)) (T) = 27(0T) (a) -S (Ij) vP 0 (J(Dv)) (a) = jp(cr) (o). We wish finally to postulate certain properties of the special functioun "Po. We need first to introduce a further definition. We shall meai bty the notation =(a lim() (ID)
that for every positive e there exists a or (dependent on e) such that for every c'>ae, .t (o)>e, or in symbols
iao,cr >a,D* =q(a) > e. This definition also is based on E. H. Moore's definition of a limit referred to above. We also introduce for the sake of brevity the following notaitions e: D
(2) pO.(or)
=
spO(a) jpo(orl) -p(02).f -Po(n-1) (n>1), AoI(C) -0(a) 1 ( > (3) .,"(of1)
We now postulate for (po that
(I)
s)o is a real and positive function of 0,
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292
lim,
(II)
(III)
1
(Pox (ai)
p[VO(a,)
-
PRoc. N. A. S.
io(o)I = 1 (a'n),
U"v) (of) as function of ao is of the class 0 (n),
the symbol J"'j indicating that the operation J has been repeated n times (IV) po(o,,) as a function of a is of the class 5 (n), (V) lim (O,,(a) = oo (n)
(VI) (Djpog) (a) = nfpo,,_-1(a)>0 (a,n>1), (Dpoj) (a) n(a,n> 1) (DA") (al) = _ o(fpo(C) (VII)
= 1
(a),
We have then as the foundation of our theory:
2-(8[ '; @; (on eto W, LP; 'ton S5 to !ff LPSGC; W~on G to ? LPSH A;A. I II11 IV V VI VII. ion ( to 4. on D to 5. LMIM2IDIJ) We can now readily prove the two following properties of the operations D and J as applied to the class a:
(4) (5)
(D[((P2]) (a) = V2(a) (D(pi) (a) + sop(a) (D(P2) (r) (a), (J[(2(D(pi)]) (a) = (pl(a)p2(of)- (J V.,(DV2) 1) (a) (a).
We next formulate. a further limit definition that is also based on the definition of E. H. Moore to which reference has already been made. We shall say that tj(a) approaches a limit a as to a if, corresponding to every positive e, -we can find a a, such that for a > ', we have I '(O) - a I <e, orin -symbols linmv(X) = a:-= :-e: D : ,a,e >a6.d.il (or) - al <e. (6) We are now ready to define the two generalized limits with which we are concerned. Given any function t(a), we set n!) ) (a) (7) (CG,,) (a) = [n!/49o,(a) (o), (= (PO() Po(i) vo(y J- )(
](JU
(8)
(9)
Wn^) (a-)-= [l/POo(O-)1Uq)(f), (H,l) (() = (Mn) (or),
where C and H are used, as is customary, to connote Ceskro and Holder. If for -a fixed n lim (C,,) (a) exists, we define this limit as the generalized limit of type (Cn) for q(a). If lim (H,,) (or) exists, we define this limit as the generalized limit. of type (Hn) for (o).
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-We can now prove the equivalence of these two generalized limits by a method similar to that devised by Schur6 for proving the special theorem in the case of series. The proof breaks up into three lemmas and a theorem. LnmmA 1. If we represent by E the identical functional operation, EO = 0(0), we have the identity (10)
'
(("- M+n E) (C,q))(o-)
LEMMA 2. I lim sp(or)
=
=
(M(Cm -1I)) (r).
a* I o(a) I
= na,
where cpm(a) = 'Po,n-i()SI(-) (n22), spo(a) = (a), (Pon (a) being defined by (2). LEMMA 3. If we set
w(a)= l (a)+
n
Vow1 (Jfp) (a),
Slimaw (o) = a.**[lim o(u) = a. a
a
Theorem. sI lim
(C.,q) (a)
=
a
(I lim *
(H.1) (or)
=
a.
1 E. H. Moore, Introduction to a Form of General Analysis, The New Haven Mathematical Colloquium, Yale Univ. Press, 1910, p. 3; "On a Form of General Analysis with Applications to Linear Differential and Integral Equations," Atti IV Cong. Inter. Mat. (Roma, 1908), 2, p. 98; "On the Foundations of the Theory of Linear Integral Equa-
tions," Bull. Amer. Math. Soc., Ser. 2, 18, p. 339; "Definition of Limit in General Integral Analysis," these PROCEEDINGS, 1, 1915, p. 628. 2 K. Knopp, "Grenzwerte von Reihen bei der Annaherung an die Konvergenzgrenze," Inauguraldissertation (Berlin, 1907); W. Schnee, "Die Identitat des Cesaroschen und Holderschen Grenzwertes," Math. Annalen, 67,1909, pp. 110-125; W. B. Ford, "On the Relation between the Sum-Formulas of H6lder and Cesaro," Amer. J. Math., 32, 1910, pp. 315-326. 8 E. Landau, "Die Identitiit des CesAroschen und Holderschen Grenzwertes fur Integrale," Leipzig Berichte, 65, 1913, pp. 131-138. 4 Cf. the first three references in the first foot-note; also E. H. Moore, "On the Fundamental Functional Operation of a General Theory of Linear Integral Equations," Proc. Fifth Inter. Congr Math. (Cambridge, 1912), 1, 1913, -pp. 230-255. 6 For the definition of these symbols see page 150 of the monograph-referred to in the first footnote. 6 I. Schur, "Ujber die Aquivalenz der Ces&roschen und H6lderschen Mittelwerte," Math. Annaken, 74, 1913, pp. 447-458.
