Abstract Integration Citation Mersman, William Alvin (1936) Abstract Integration. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/tk6w-3m08. https://resolver.caltech.edu/CaltechTHESIS:11122024-183034102 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Michal, Aristotle D. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1936 Record Number: CaltechTHESIS:11122024-183034102 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11122024-183034102 DOI: 10.7907/tk6w-3m08 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16849 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 13 Nov 2024 00:51 Last Modified: 13 Nov 2024 00:52 Thesis Files PDF - Final Version See Usage Policy. 11MB Repository Staff Only: item control page

ABSTRACT INTEGFATION THESIS by Wm.A.Mersman. In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy, California Institute of Tec)j.nology, Pasadena, California 1936 Introduction. The subject of integ ration was first trea ted abstractly by Fre"chey*. Numero us other authors have contributed to the subject, notably Saks* * ,Bochner*** . and mpst recentl;y Birkhoff t . A comprehensive disc u ssion of the possibilitie s was presented by Kolmogoroff t\-. The purpose of this thesis is to unify so me of the previous theories and to present so me new on es. Chapter I is devoted to the most obvious generalization of the classical Lebesgue theor~. Chapter II extends the definition of Chapter I in such a way that no special treatment is necessary for unbounied funcyions or sets. Chapter III is a discussion of the theory of summation of diverg ent series, an a nplication of which is made in Chapter I V to the question of divergent integ rals. Questions of relative g enerality of these and other int egral s are usually so easily answered that no f ormal proofs were deemed necessary. For exa!uple, the integ ral of Chapter I includes the clas s ical Lebesgue, Stielt j es, and Padon ttt integrals of bounded functions. The integ ral of Ch Cha~ter I I includes those of Saks and Bochner and is included in Birkhoff's. Finall y those in Chapter IV are obviously g eneralizations of the classical Rie mann- and Lebesgue- Stielt j es integrals. Before continuing I wish to express here my appreci a tion to Dr.A. D. Michal for his guidance in the writing of this thesis. Chapter I. Integ ration of Measurable Functions. 1. Spaces. In this paper t h e follo wing t ypes of s paces are considered and t he indicat ed notation use d . B; a Banach s pace, norme d, complete, closed under addition and under multiplication by real numbe r s, el ements d enoted by -l , ~,-\-. "j, T, J , ~, C, , norm by \-- l - R; t h e real numbers -pace, ele ments denot ed by o1. . r:, .e . ~.£.d po s itive integ ers b;y i,j,m,n,p,q,r, M,N, absolute value byl --lV; a metric s pace, the only propert y of distance being that the distance between any two points approaches z ero as t h e distances bet we en each of them and an arbitrary third point app roac h zero. D; a met r ic s pace, ele ments denoted by s,t,u,v,,w, distance by \ -s - 't. \ , with the following postulates: (1). Equality defined, symmetric, r efle x ive, t r ansitive. (2). l-s - t:\-:>,,o . 1--s-H 0 l-l:--:s ~ : \ --s- t;\ ~ o.3:. 'S::t, Th e s pace Dis ~aid to be de nse at t h e points if, for any S; a nmeasurable" space, elements denoted by x,y,z, sets of elements by A,C, E, X, with the prope r ties: There exists at least one set X of elements of Sand a family .¾ of sub-sets of X such t hat: ( 1). A c:: ~ -~ - ( X - A\ E A (2). A "' E X -~ l ~- - A VI,~ E t . (3). There ex ists a f unction, denoted by l. \ , called measure, defined throughout X, Av.. E 'x. -::>. J).0v------ -co to B + .,._ \ 2_.,,,_ GO A ~ \~ such that 2.,. \ AJ . meaning that if the le f t member is i nf inite s o is the ri ght, wh i le if the ri ght member is finite so is the left, and the equality sign ia to obtain if and only if the A .,_ are disjoint. Such sets A will be called measurable sets, the term including not only sets A for wh:lch 3\ A \ =- a finite numb er, = but a lso sets A for which ~ A. ... , \ A . . . \ . A I -.c and consequently \ A\ A ... , so that need not be finite. Throughout it will be explicitly mentioned : wheneve r IA\ is a ssumed to be finite. By means of the convehtions herein ado nted the nature of a function will be made evide nt by the letters used in its expression. Also, throughout, a summation sign with no rang e :i ill indicate a sum of a finite number of terms. 2. The General Princi ple of Convergence. Let u be a variable over a s nace D. Let u' be a fixed ele ment of D, a t which Dis dense. Let f(u) be a function on D to B, no requireme nt's being mad e as to simgle-valuedness. Then the following theorem is true, the proof being exactl y the same as in real number theor;n THEORB]I : A necessary and sufficient condition that f(u) approach a limit as u a r,p ro a ches u' is that (f(u)-f(v)) approach zero as u and v approac h u' independently. Thi s theorem will be referred t o as the g eneral principle of convergence. As a matter of reference we next state Frechet's (1 ) extension of the Borel-Lebesgue theore m: THEOREM: Let 1 be a famil;y of sets I ~.A ~ V A is closed: and com"act, such that , where x 0 A .:, . ;< E: I. ~t . for some I. Then there exists a f inite number of sets such that -x E A .-::) . 'X. ,--.E. +.T \,( T 1-< E: '\: for some k. 3. :Measurable Functions. Fundamental Properties. Definition: The function f(x) is measurable over the set A: =( l ). For any cS , there exists a finite number of non-overlapping , measurable sub-sets of A which almo st co :u e r A; i e . , l. A. ,.,.. o , \ A . . . o \ : \ A - 2 A ... \ =- 0 . ( 2 ) • For any n, d , "X , lj E- A "' 6 ·) ( 3). For any o( , If c -x) - ~ '"!,, \ < d. ( the sub-set of A containi ng all those an d only those points f o r which \\ c x,\.~oZis measurable. For any ~ , the family A "' 6 will be called a finite subdi visio IJ of A. First of a ll we obtain a necessary condition that a function be measurable. THEOREM 3.1. I f f(x) is measurable over A, t h en f(x) is bounded almost everywhere in A. Proof: Choose any 6 . Then, b y the above definition, there are a finite number of sets AhE which a lmost cover A, an d in each o f which \~ cx, -\<..~,\.c:: f. Denote the sum of these sets by A'. Then, f {x) is bounded in A'. For, if we assume t he contrary , ie., for any ~-;, 0, ?i. 'XE A', \~cxl\>C{) 1 then °"3. Y 'X~\'"'-~ l. 1 , - . ; ~ A •. ~l\ (x~J\ is a steadily increasing , diverg ent sequence. Hence, }. ~ -x'.. \ ~, -x ....\ such that for any n, \ ~ c )'"'' - \ ( x·._ __ 1\ ~ f . Therefore, the po i n ts for distinct values of n, mus t lie in dist i nct members of the f amily A iE • Therefore the range of i must be in- finite if A' is t o be covered by \ A ; E~ . Thus we have a co ntrad ictio n . Having obtai ned a necessary condi ti on for measurab ility, we now obtain a sufficient one. THEOREM 3. 2. If V 'l A -s 'S and A is compact and closed, and if f(x) is continuous throughout A, Then f(x) satisfies the first two postulates of measurable functions (We shall see later that the third is a consequence of the first two). -Proof: Tn orde r to prove this we nust ma ke one more postulat e about the measur e f unction, namely that it exists, finite or infinite, for a YJy hyp ersphere, ouen or clos ed , containe d in X when S i s also~ v. Explicit mention will alway s be made when this is to be assume d. Then, since f(x) i s continuo u s, 3 dlE- ,X): Defin e /\ ( )(. . E ) : Then t. E ~ E: \~ -)(.. \<-o(E,Xl _-:). \\{~\-'\(x)\<.f A ( x. , E) • = ·_ \ ~ - x\ --c::::. ct' ( c , --x). A - ::::> • f o r any ( , is a f' ami 1 y theorem of Frechet in section 2). Hence, ~ -x t A ---:::> -x . L ' X: € i"' 't rang e of i finite A ( X; > ~\ - A- { E) for some i. 'XE A - -::>•\ -x - )(.i\ L )< f = n. I, d (t. 'Xi) for some i. A-::>- \ \(x\ -t{)(.;\\.:::: €:/z.. for some i. 1 (see the n Thus we have A =Z A ,- (f) : -X . ~E A ; (() -:>-\fcxJ -1/~11-Lf_ for a ny E . If the were disijoint, our theorem would be proved. Ai Hence it will be sufficient ot const ruct disjoint, measurable sets C1; f such that, for a ny k ' Ci,(~ K ·1 Al{ A= ~ ·, c;,, H'•I Define C , -::. A 1 •. C v1-::. AK-lT ( A-- C.-)-=- AK(A- - .2.r C,,.) •. vf-:;1 , --- . Vl - r =' Now C . t C1.. :.. A, -t A ~ ( A - A,) '::. A, t- A -i.. . ~ ,- Assume that for some r 2 v1 Cu =-- 2 v. A1-i" . ~ C '=- ' I ( ,) IThen 2v. C i,-= t-+ , + L✓, At<:::: An. A- - 2 ,,, C,-. + A-K ) 2~ r t/ -= An- , (A - 1-11 A-w) -t f. A-Vi = I I Therefore, by induction Next c A--J( =- A ' C I C 1.. == C . • A 2.. • ( A- - C.) ::: 0 _ Assume that for some r , Then c ;, 2v. c l{ :;: ., 21-f r+I - i -=t j -::=> - Ci • CJ -: : c t" A- i-t, ( A - c.) --( A -c,_) Therefore, by induction, fo r any i,j, 0) L 'j =-O 1 :t J .::, . C,- Cj -=O . Lastl;)', it is obvious that, for a ny k, Cu s AK. We now show that our definition of measurability ha s many o f the ordinary -properties. THE0 REt'i 3. 3. If f ( x) and Then ex ( x.) are mea s urable over A, f t x) d( x ) is measurable over A. Proof: Let \ ~(xt.::::: q>, \o((-x)\.~ almost everywhere in A. Now and ~ A.if- , 1 -=t j -:>. A ; A:i -=- 0 : l A - 2 A; l ::; 0 . Define Then i,-:thV~:tK ---=> : AiiA..,K-=0. and \ A - 'Z.. 1- A ;~\ -=- \ A - 2 A; • 2 Bj l =\ (A - 2 Ai ) + ( A -2 Bi')\ l A-TA.,\ -t\ A-)_E_i\ ~o . ~ x. ~ E A i j(c l -J • \ t ( x) oZ( x) - Renee t( x) • o!{ x') ~(lj)oU~)\ '=- :: \f-(-,<) ol( ')() -fcx,<i( 1')lt\ t(x)o(l~) <... ~- G/--i-cp + ~- ¾_o( :;_ -fl':J)o/(~'I\ E. is measurable ov e r A. THEO REM 3.4. If f(x) and g(x) are measurable over A, Then Proof : -± 'jC x') is measurable over A. fcx) ~ AI f : \ A - 2 A ; \ -=- 0 : i *J -":J • Ai Aj "' 0 ·- X. ~ E A; f ·-:) • If( "' -h 1< f/2 , '.l) 3~ - :\A-IB-\=o: i-:tJ. _-:,_ B .- B- =o · 'X , uE-Bje -:> -\_g c">') -g(~1\< f/-z, ~t J J -.) Define A ii ( E) =- A i f- • 13 i f 2 Z There f ore \ Therefore \( x, ~ ~ex.) A - A i l -=- o · 1:-Ct Lt V. j .t K ·_"'): Aij A kt< --= 0 • is mea s urable over A. Next we prove an extension of Egoroff's theorem: THEO RE:M 3.5. If ( 1 ) . For a ny n, t 2). t , ( x) ( 3). A)/(~ A \\I\ ( x ) is measurable over A. converges a l most everywhere in A. 1 l A*l - a finite number. Then, fo r a ny cf, ~ A 0 ~ A*: -:\~A 0 \ '7\A:;,-l-S_ such that f"' ( x') converges uniforml y in A6 _ Proof: Let Al .i~~\ Define E "' ' r be a monotonic, decreas ing sequence, lim. o. :: A * · E t Then for a ny r, "-.-::, ·. \\\., cx,--\-,1..c "'\c:::.E'.,.. {. p . q_ >,.~ E 1.1 . t- I: 1.... , . t". Also, for any n,r, • E"' . r is measu rable. For a ny r, \ A "' l :.\t E,. ,Y \ since converges a l most -h., c )() everywhere in A. Therefore, f or a n y r, \/\ ,t-' l ~ ~ i lM. \Ew. , .-\ "'- -'> CO Therefore, for any r, ~ Nl \- , &) ·. II\ ✓/ N .:> . \A*- ~ If , ..- \< d/_ i-- :z. . Define Then converg es uniforml y in THEOREM 3.5'. I f ( 1 ) . For any s in D, f(x,s) is measurable over A. ( 2). Dis dense at t. ( 3). f(x,s) approa c hes a li mit as s - t almost everywhere in A. A""~ A •. ( 4). Then, for any 1 \ A* \ ::: a finite number. d , 1 A& ~ A 'fl. : l \ A 0\ > \ A"' \ - S , such that f(x,s) approaches its limit limiforrnly in A& . THEOREM 3.6. If f(x) i s measurable over A, Then \ f(x) \ is measurable over A. Proof: /!: , '_\ ~ A,E : \ • - I A ; -=-- 0 •. i.. ::t j .:::> - A i Aj -== o . € A ; € ·:-::, : \ \ 1t <x, \ -\ \ l ~ \\ \ ~ \ f ( x) - \-<. 'J' \ .:: f The reiore \f(x) \ is measurable over A. THEOREM 3. 7. If t .. c x.) is a sequence of functio n s , measurable over A, whic h converges to f(x()! uniformly a lmost everyvifh ere .in A, Then f(x) is measurable over A. Proof: For anr n ~ A~t i . ':\ c;. A ~ c ·-J : : \ A - 2 A;\ ~o . ( .,,.i _:, . Ai Ai ::Q_ \ f-. c "") - t"' ( ~, \ < E . ie., f(x) is measurable over A. THEOREM 3.7'. Let f(x,s) be defined t hroughout AD, except when s = t. Let f(x,s) f(x) as s - t, the convergence being uniform almost everywhere over A, For each s, let f(x,s) be measurable over A, and let n be dense at t. Then f(x) is measu rabl e over A. THEO REM 3.8. Let llx) be a sequ ence of functions, mea s- urable over A, which conv erges to f(x) al- most ev e rywhere in A. Let A.,,,"=A'. ~\A""\ finite. Then, i'arr a ny o , ~f\ 6 ~ A ".'. -:!\ A 0\7\A*l- 8" . Ar . such that f(x) is measurable over Proo f : The~rems 3.5,3.7. THEOREM: 3 . 8 '. Let f(x, s) be defi n ed t hro ug ho u t AD exce p wh en s -== t, mea surabl e ov e r A for each s. Let f ( x,s) -- f (x) as s - t, al mo s t ev ery whe re i n A. Let D be den se at t. Let A .i-~ r+. 1 \A* \ f inite. Taan , for a n;y S , ~ AD ~ A * · "3.\ A 3 \ :;, \ A:t' \ - cl, s uch t hat f(x) i s mea s u rab l e over Proof: A 6. Theore ms 3. 5 1 , 3.7'. We now -p ro v e a threore m wh ic h s h ows t hat functions of the t yp es cons i d ere d b y Saks and Boc hner which a re measura.ble a ccording t o ou r d ef i n ition a re a lso me asu r a ble a ccord ing to their defin i tio n s. THEOREM 3. 9 . If f(x) is measurable ov e r A, t h en ~ ~"C~ a sequ enc e of s te p -functio Ls, which -- f( x ) un i f orJply almost everywhe re i n A, and conv ersel y . Proof: Si n ce f(x) is measu r a ble ov e r A, Let ~A i ~ '. ("" be a monotonic decreasing s eque n ce with limi t zero, ie.' ~ \M o ( t)' 'nr,. :,.~ V'A _, ( f) ----::> . <? -- <.. G . Now, f or each i , m, define A, ,,,,_ ::: A ; ( f ""\ and let i ; .,.._E- /t i lM. · For ea ch m define L . . . < x) f-.__ c :x.1 = f (x;_.._) ·_ - 'x'. f. Ai vi-. Such a function t__ c x' i wh i ch has onl y a f inite numbe r of distinct values is called a st ep -function. Now, let 'M For any m, > vv.... a ( E-) \ A - 2- Ai ""'\ :::. C) . ' Therefore \f ( x, - t.,_ cx,\ -: : \\c-,,.) -~C1' j ..._\\ < E..,...:::: E, Therefore t...,.()(.) - fcx) uniformly a lmost everywhe re in A. Converse is obvious. From the definition of measurabilit y it is obvious that if f(x) is mea surable over A and over C, then f(x) is measurable over A -t- c. This can be extended to the sum of any finite number of sets but not to the sum of a d enumerable inf inity mf sets, since f(x) might not be bounded in the sum set. Also, if f(x) is measur able over A and i f C is any measurable sub-set of A, then f (x) is mea surable over c. This p r op erty has alread y been used. We _are now in a position to define an integral for 1 mea surable functions. 4. Integration . Definition and Fundamental Prope r ties. Let f (x) be mea s u rable ove r A and let \A\ be f inite. ~ Ai E •. i -:t j -:::> . A , A j ])efinition: ::: 0 , \ A - 2 A ;,\ ::: 0. J 't"(x\~ A -=- ~ \~ r f-cx if ) G: -o L if the limit exists. I t is clear that i f \Al=oo will not exist for any , the sum (2. THEOREM 4.1. I f f(x) is measurable over A and i f \A\ is finite, then ~ J{ex) 4 . A By the g eneral princi nle of conve r g ence wti ,Jneed to show that 1 ~(E): ~ - -v ~ S(e) ·.-=> ·. \ -r ( ~'I - I (v)\ ..::f _ Let ~l~, =- G/2\ A \ l L<~\ -I l v1 l ~ \ ~ t-cx i r-4 ) -, Ai ~\ - ~ f ex.iv)·\ A.i)\ J I Define \ -r l . \ - r <v) \ If \A\::c.LL\A -- \ ' .I /\ 'i : :. A it-4 - .A. ~-v A iy,- ,AlV ==- \ I f f {t < j ~ ( xj V) 1- \ A ,~ \\ x i ~') - have no points in commo n, If A ; t" . l\.i-v have commou. po i nts, let be aDy such point, so t hat 'X I-.~ ( \ A i j \ -=- 0 Xij A i l'-'-, Aiv Then l T tr-4\- T.<.v)\ ~ l"J;J;.1h){ i,--,-f(')C. ,.)'\ -rhx ;J) -fc)(jv)t·lA ,}\ ' J ~ 7-.- 2-• ~ \\-(x-I I-'-)-·h -xj\ -t\t(t j-,-hx 11)\1 \A ,j·\ _, j - L TH EORF,u 4.2. If -y C-x"\ "1 \ 1 J cL-,c_ ) ) A. Then '\ ~ A \1 t b .)t~C x) ~ ~ :: \ \ <x"l ~ A A. Proof: By theore m 3.4, By theore m 4.1, 'jC x"'l +<x) ~ '-~c-x \ ~ \3 c)(\ cl,,x_ A is meas urable ove r A. ~ \ l t ( X) ~ ~ C x'\ \ c\ )( A Using the notation of these theorems, ::- \ ~ \ { ( )( \ -:\: ~ ( Xl \ 4 ,,.. - ~ ~j \ ' ( X i ~I ~ l\ C"- \ l ') ~ \ A i i O\ \ I +\ 2.~ , , ( )(; ;')~C~('l< ;- ) \ . \Ai io\ - 1 \A \(x)Q.x:i r'j( X) ct"><-~\ A ' < E J if I l ~ ,.) is sufficiently small. But the original left member is independent of (:' THE0 RE~-1 4.3. If ex • Therefore is any real constant and i f "1 ~ tcx, d.x, A Then 1 \ ol.. - t cx, c\ x = ol.. - \ \- ( x\ cl x. _ A A. Proof: It is obvious that a constant is a measurable function. Hence, by theore in 3 .3, ol.. · t cx\ is measurable. t ~ ~ d. • c x\ d. x Hence, by theorem 4 .1, A which is clearly equal to o<. • \ 4 . tcx1 A THEO REM 4.4. If Then A Proof: By theorems 3. 6, 4. 1, "'l \ \ ~ c ><) \ ~ A :.\ 6(E'.\ , "\ <- 8 Cf\ _J .\ ~ h ,._,<k-2.-\-C14 j)\ A; \\< E- -::,.\ ~ \\ c,o\cl)(- Ilt()( ;l\ -\ A;\\ ~ Hence \ \\c -,<1 \ ~ -:;;,, A A l ~ \~()( i)\ ·\ Ai \-E • ) \ I ~-\C >1l ~ \ p. ~\2h "',,~ A1\\ t f'. I Adding, we have -;;, _2_ \ ~ c ~ ; \ Uf\ -, \ - \ ~ ~ <',( i ) \ A; \ \ - :l f ~ - 2 A. \ I\ The left member is inde-pendent of Therefore \ A ~ cx1\ d x - \ e:: \ ~ ( x )q)('\ A >,, O. THEO REM 4 • ~ • If ( 1 ) • ~ \ \ c ')(. ) ct ( x) ~ ) ) o1. ( ~) c\Jx A ,,._ (2). Al mo st everywhere in A, \h ,..)\~~:,cx( x) ~O . Then \ ) tc. -x) o1__ e x) ~ \ ~ <p \ o{ Cx) ~ A A Proof: ~ o (E ) ·. I').:. ~ (E-) ·_ ·y_\ ~ {cx)o<cx,d.,x - L f{x ;)ol(x ; ) f\ \ - T i olc)(i) · \ AJ\ < \~ex)~ \A ;\\< E . I E. A. Therefore \ \ t c -x1 o1. c x, k \.::: € +\~ t-(-x i )dC-,<i)\A ; \\~€ + l" \f()( ;)\\o<'C'l<;)\\A·, ~ L A ~ THEO REM 4. 6 • If ( l ) • S t c x, ex c 'I<.) M 1 \ ~ C x) ~ ~ A (2). Fis the smallest closed linear manifold in B containing all the values of f(x) for x in A . Then 3. \ E ~ ·- ) -tcx) o( C x) ~ ::: I, 1 · \ ol (x) ~ A where f will of cogrse depend on the function ~. Proof: ~-Q ) o/. CX) ~ A \.MA.. J; '\_,,,o • cJ.. ( v .) \A · \ "• I ( , y ( )(; ) ~olCx, <lx A Now o<()(. ; ) 1 \A i \ , \ J. cx,o-ix are real numbe r s. Since Fis a pt.. clos ed linear manifold, the rightmember ab ove i s an element of F , say f. THEOREM: 4.7'. I f (1). f(x,s) is defined thro ughout AD except wh en s =- t. ( 2). F or eac h s, ~ ~ \ c ~. ~'I ~ A ( 3). f ( x, s) .- f ( x) as s ~ t, uniformly almost everywher e i n A. (4). D is den se a t t. Then ~ \ \c x, 4 = /\ ~-\JV\,\.., \ \ c1' . -s\ ~ - .s --t p., THEO REM 4. 8 . If A and C a re non-overlapping s ets, and if ~ ~ ( -,<. ) ~ \ A ' \ \ ( ~, °'>' C Then ~ \ -\ (x\~ =- ~A t At C ('X\ ~ + \ \- (X\ (M( . C Chapter II. Integration of Summable Functions. 1. }!~unctions having the pro-pert y M ( or, belongi ng to the class M). As a generalization of Chapter I we now d efine an integral f or a more extensive class of f unctio n s. In the new definition no restrictions are macle re g arding boundedness, either of functions or sets. Ho d eve r, in the case of s ets the usus.l r' evices seem t o be necessary for the proofs of the fundamental properties of the integ ral. Tn t h e case o f bounded sets, on the other hand, the fundamental prope r ties can be proved at once f or unbounded a s well as bounde d functions, thus giDing a simnlification of the theo ~y as found . 111 T't 1 1 Ctlmar s h( ,.') • Definition: f(x) is of class M over A:= (l ). For any ~ , there exists a denumerable family of disjoint, measurable, boundee sub-sets of A wijich almost covers A,ie., ~ A, v. 00 0 •. \ A. - I.. A ...\ "' o : W\ -:f 1M _--:, • A "' A..,., = 0 . (2). For any n, cf, ""· 'i E A.~ 0 •. \ \<. "' -t',~,\ <- 8 . (3). For any o( , the sub-set of A containing all those and only those points for which \~ c.-x,\-c:::. ol For any d is measurable. , such a family of sets AV\ 6 will be called a s ubd ivision of A. THEO R;3;H 1.1. Any measurable func tion is of cl ass H. If (1 ). f (x ) is of class I.1 ov er A . THEO R1~M 1.2. B is a measurable s ub-s et of A . (2). Th en f (x ) is of class M ove r B . THEO REI-:1 1. 3. I f For any n, f ( x) is of cl!.ass m over A.,. . ( 1). ( 2 ). n -=I' """ _-::,. A"' A. _ = 0. (3). \ A Then f(x) is of class M over A. This de-pen d s simpl]1 on the d enumerability of a denumerable set of o enum erable set s . a, If ( 1 ) • I A - _L., A .,. \ -=- 0 . THEO REli 1. 4. (2). "l \A\,\ A,.\ For a ny n, f(x) is of class I''f ave r A .... ) (3). Then f(x) is of class M over A. Proof: See theore m I.3.2. en Define A Jt' = 2 .. Then \A- A* \-=--O. /:\ V\ ... , 1\ -1 Define C I :. A, C \.\.-::: .. i--q h For a ny n, 2 .. c l< :: 2 '{ I Therefore A.,. Tr ( A "" - C,,) = A ,.,..( 11)/ - Ii- ,( \,} "* AK ro 1: c., - A* WI _--:) . c ~ '. \ A - 2 ., C., \ l-l e ""' =- 0. ::: o. By the pr eceeding theorem, f(x) is of class M over A. THEOP.EM 1. 5 . If (1 ). '3. \ A \ , finite, ( 2 ) • F'o r any <S , 'J A~ ;: A ·. \ A O\ :,, l At - ~. (3). f(x) is of cl a ss M over A J . Then f(x) is of class M over A. ~roof: Let & 111 be any monotonic d ecreasing se qu ~nce with <t:, limit zero. Define Al,<:: AoIll • Then \ A - X A .. l :: 0 . The remainder of t he proof follo ws immediately from the preceeding theorrem. THEO REM 1.6. If (1). A is measurable. ( 2). For any Ait ~ A such that finite, and for a my ~ \ A* \ o :i A O '= A"" : ~\A&\ :> \ A~\- A ~ (3). f(x) is of class M over A~ Then f(x) is of class M over A. Proof: By the nreceeding theorem, f(x) is of ckass M over A~ Let A~ be any sequence of sets of ~inite measure such that Q:, l A- 2 A..,.\ =D . Then, by theorem II.1.4, f(x) is of cl a ss M over A. THEOREM 1.7. If A is a compact clos ed set in a measurable space V and if f(x) is continuous over A, Then f{x) is of class M over A. Proof: ~heorems I.3.2, a nd II.1.1. THEOREI/I 1.8. If f(x), d.Cx) , are of class Jr and bound ed almost everywhere over A Then ol(x) .t (x) is of class H OV F: r A . THEO nErn 1.9. , f f(x),g(x) are om class 1.1 over A, Then fcx'\ ± ~C x) is of class M over A. THEO RE:.'I 1.10. If f(x) is of class M over A. Then \ f(x) \ is of class M ove r A. THEOREl.-1 1.11. If (1). (2). :b1 or any n, t..,. C x) is of class Hover A. t,. c"')---tcx) uniformly almost ever;ywhere in A• • Then f(x) is of class H ove r A. THEO REM 1.11'. If (1). Dis dense at t. (2). For any sin D, f(x,s) is of class L1 over A. (3) f(x,s)---f(x) ass -+t, uniformly almost everywhere in A. Then f(x) is of class M over A. We also have the extensions of Egoroff's theorem analogous to theo r ems I. 3.5,3.5'. THEO R:,M 1.1:c::. If (1 ). For any n, t~ c x) is of class 3'1 over A. (2). {~ex) converges al most everywhere in A. (3). A"'-=cA · ~\A ~ \ Then for any ~ such that f"' Cx) finite. , ~ Ag ~ A'I< •. ;l\ AO\:>\ A~ l- a". converges uniformly in AJ _ THEOREM 1.12'. If (1). D is dense at t. (2). For any sin D, f(x,s) is of class M over A. (3 )• f (x , s ) - f ( x ) as s - t , al mo st everywhere in A. ( 4). A" ~ A •. 1 \A,... I finite. Then, for any & , 1. A8 ~ A* 'l \A ,d ~tA"\- d . such that f(x,s)- f(x) uniformly in A6 . THEO REH 1.13. If (1). For any n, ( 2). f" C f,,_Cx) x) -+ hx.) is of class M over A. almost eve r;sT\vhe re in A. Then f(x) is of class M over A. Proof : Thi3 ©"re ms I I .1 . 12 , 1 . 11 , 1 . 6 • THEOREH 1.13 '. I f (1 ). J) is den se at t. (2). For any sin D, f( x, s) is of cla s s M over A. (3). f(x,s)---f(x) as s ---- t. almost everJrwhere in .A . Then f(x) is of class M over A. Proof: Theorems I I.l.12',1.11',l.6. Thus we see that functio ns of class M have all the ordinary pro perties of me asurable functions in real number theory. Tn partic ular , the class H is closed urid er limiting n rocesses a pplie d t o f unctions or sets. DEFIN I TI ON : f (x) is an el ement a ry fu n ctio n over A: = : 1A"' , \ A,..\ finite ,"'"" \. 2. ,- - ·) A"'~ A, v..--;:IM.-J - A"'A...,-==-0. a'.) \ A - L A" \ 0 . -x '= A" ;::, . f cX ) = -\ ~ , -== ""'' \~ a constant for aach n. THEOREM 1.14. f(x) is of class M over A if and onl y i f 3 ~.. Cx) a sequence of element a r y functi ons which conve rg es to f(x) uni forml y al most everywhe r e i n A. This shows that any fuhction which is measu rable a cco rd ing ~o Saks or Bo chner is of class M. THEOREM 1.15. In the definiti on of functions o f the class M, postul a te 3 is a consequence of the first 2. -P r oof: ('3) { • Let T,_,.Cx) be an approxi mating sequenc e of ele- ment ary functions, a s in the pr eceeding t heorem. Then, clearly postulate 3 is satis f ied by eac h t_ <. x ) Let o<. be a ny -positive re a l numbe r . For any rn, ~ -\-V\ \ ~ Cx\ - ( x) ""' Cw- = Define ex,\< '/2w. fV\ IN\ A · E \ \ , \/\ Then, in G ,. __ , \ ~ c x )\ < '/:l..""' 1 Cx)\ -c:: o< - o<: _ Conversely, if \, c 'X)\ <- o< - f Let Then \ , V \ - ( -x'I < < \ ~ c .,.,\ .\ ~ - -:: d - G + -;.,... ..:::: o< - ½ .,.... 00 Therefore A ·Ef\\( ,n\<-o<} -= A·2 G..._ except for a set of I measure zero. Hence postulate 3 is satisfied. 2. Surmnable Functions. Definition: f(x) is summable over A: =- (l). f(x) is of class M over A. ( 2). For any a , any subdivision A "' 6 , any Q) the series T.. \tcx \\\/\ \ converges. w -:= l "" \I'\ A:) Hence, the series .2 Fc-xH)\ A..\ converg e s. IJ::: I l'.' e begin by finding sufficient conditions that a function be sumrnab:Fe. '.PHEO REM 2.1. If (1). f(x) is of class M over A . (2). f(x) is bound ed over A. ( 3 )• \ A\ is finite. Then f(x) is summable over A. Proof: Let lb Then 2 (\l ~ l \ c ..,,_,\ \ f C" ,J\ \ A ,J ~ cp •\ A~ But, any bounded series of positive terms is convergent. THEOREM 2.2. If (1 ). f(x) is of class M ove r A. J ( 2). For any t some subdivision A v..6 co any "f...,,. t A._ .\ 3. L \ h 'I( ",\ \ A ..\ n~I ( 3). For any n, ~ A1-1 ""' is a subdivision ' of A.,,.. ,~"'"' E A VI.Y\A co Then ~ 2- \ \(x.,, •.:-\\ 1\ .,.=\ , the manner of summing I being immaterial since all the te rms are positive. (b Proof: Le mma. 1 b (\)""\A "\ where ~"' is the l ea st upper bound I of f(x) in A"' . :B' rom the definiti on of the class JvI of func~ ~"' tions it is clear that , any n, ~ - Also, by the definiti on of a lea s t u-pper bound, for an;y n, d,E' , ~ 'X,._ E: A"' ~ =- Choose such that ~~I').. , D J: !,'.. €. ~"' ~ 2.\ ~ c x "')\ . Then (b Therefore (\)\,1. - l, cx .,)\ -:: Q:) cp,,.\A~\ ~ 'J...2-\\C-x.~l\\A,J and, by hypothesis, the right member o f the ine quality exists. d:) <b J\ lA ('.I:) <'b 2 ~~lA~\ A"'=\ 2. .,.2 .,._I l f ( X., •• M ... \ .,,, 'PIA ]_\ ,,,..:: , '1 =I and, by the lemma, the extreme right membe r ex ists. Now, .:. ~ THEOREM 2.3. If (1). \ A\ is finite. (2). f(x) is of class M over A. (3). For any [J , some s ubdivision A. . r a, any 'X..._ f /\ "' , 1 l \ t ( x ")\ \. A ... \ h: I Then f(x) is summable over A. Proof: By the de f inition of summability it must be showm that if, for a n y cS , B"'~ a::, then ~ I. \f c'J .. )\ \~ ,J Let Then is any sub division o f A, ~"" ~,1' . ., . ¢> «> L \ (c~ .....'I\·\~-\ = L 0) a, L \ tc~ ...'I\\A" ... \ (I) <X) <b .: : z.., J_ ., l t(~...,'1-t()(.,.,.\\\A.,..,\ +-.Z: ~ \t'(X.,,..1\\A\,1\,1,,\ a, <- lO ~ -\A\ + Z:,, Z., \ \C x.,. ...,\\ A""'\ a By the preceed j ng theorem, th e rig ht member is, for any finite. THEOREM 2.4. If (1 ). \ A \ is fini t e. (2). f(x),g(x) are Then Proof: Let \ e x ) ~ ~ (x) summable over A. is summable ove r A. A.. ,'D ."' be the res p ective a/ -:i__ subdivisions of A f or f ,g. Define Then A~.,,.. is a & subdivisio n of A for Therefore, by hypothesis 2, ~ ~C')(,l ~<jCx'I Cb ,2_ \ ~cxV\..,.') ~ S(xV\,,_ 1\lA.,,.½'\ 11 , ,,., I 'l1herefore, by theore m II.2 .3, \e x ) ~ ~ Cx\ is summable ove r A. THEO"'Ell 2. 5. If \A\ is fin i te, then any measura ble function is summable. THEO PE:T 2 .6. I f f ( x) is summable over A, so is \ f(xH and conversely. TH EOREM 2.7. I f f(x) is summable over A, a nd i f Bis a measura ble s ub-set o f A, then f (x ) is s mmnable over B. THEORK·.~ 2.8. If f(x) is s ummable over A an d ov er B, Then f ( x) is S'l!limmabl e over A +- B, AB. 11.'HEOREM 2.9. If f(x) is summable ove r li, and if ex is any real numb e r, then c,1.f Cx) is s ummable over A. THEO PEM 2.10. If ( 1 ) . ~( x ') , summable. "=;!- 0 (2 )• ex Cx) --;::.,,,. 0 . of class M. ( 3). ex c )<._) 0>C x"') . ~ ( 4). \A \ is finite. Then o{ C x ) c subdivisions of A for ""' . r-, respect- Proof: Let A ..,. , ~ "' be ].·vel y . The11 is summable. A...._::.- 0" -"'. 1\:l "" 1.·s a S suba1.·v1.·s1.·on for cJ. and Q:) for ~ ~ 2 \ ~ ( x "-,\ \ A ""'\ "'•"" I and, by hypothesis (1), the right member exists. Therefore, by theorem II.2.3, THEO REM 2 .11. If ( 1 ) . ~( x ) is summable. \ A \ is finite. (2). For a n y n, \ .... ( x.) is summable over A. ( 3). t._,c x, - tc x) uniformly al most every- where in A. Then f(x) is summa ble over A. Proof: Let A""' . ,_ be a o/3 subdivision of A for t ,,_ Cx ) Let m be such that \ \ (-,<) - \.....,.c x,\ < S/3 almost everyv,rhere in A. Then A"' , ,. is a d subdivision of A for f(x). a, ~ z. \ \ C)( .3- ·L.c )<..)\\ A~ 3, Q'.) V', -\ I: \ \""' ( ')\ .. )\ \ A" \ n:: I and, by hypothesis 2, the ri g ht t rn embe r exists. THEO REM 2.12. If (1). \ A is finite. ( 2). For any n, t,,cx) is surnmable over A. (3). For every n, \\ ~ <.-x)\<-cJ,(x") where ri.Cx) is surnrnabl e ove r A. t"' Cx\ - ( 4). tCx\ as "'- ___,._ co almost eve ryvvhere in A. Then f(x) is summable over A. Proof: By theo r em ~I.l.13, f(x) is of class M ove r A. Hence, by theorem I I.2.1O, f(x) is summable over A. 3. I nteg ration of Su mmable Func ti ons. ll) Definition: ~( <2) =L h x.,) \ A J where A,.,,, is any € subdivision of A for f(x), -x.,.. E A ,., , . Definition: S-te.,.,,~ ·- ~ F(f} if the limit exists uni qu el y . A e -,o ~ e beg in with existence theorems. I f (1 ). f(x) is s ummable over A . THEOR '-]!T 3.1. (2). \A\ is finite. Then "1 ~ ~ c x'l ~ - A Proof: I t is s u fficient to sho w that FCE'') satisfies th e Cauchy con d ition. lb © \ F ( c.=) - f ( o) ::: I~ f ( x" I') \ AV\ E' \ - :z... h )( .,.. /) \ F'\ ._ £\\ \ As in the n roof of ~heor em I.4.1, eo Define A.. ..,,,= /\Vlf · A =~ since A.., .,,.. Then Then \ A a r e disjoint. Let = F (.E) - F ( :::: 2 ", . . . \A.,,,.,..\ ,x.,....._ E AV\,.,,,.. ~ 0 \ 4.: \ 2.. (~C x..,f ) -~c~ ... ..,\\\ A., ,,,.\\-\-\ Z. {\c"X.,.._') - ~C-x ... 5\\\A .. ,J\ 11 , "'1 :: )_ \ \,.. , "' ao Q:) Cx.,,,_',-~c)(."'.,..."'l Al.\ ,,.,,.\-\- 2. \t (x .,, .,,.,.'>-hx ... 0 \\\A\,\ v,,. A and the ri ght me mber can be made arb i tra rily s mall b y making f:. ~ su:ffici entl:':,; smaibl. The above rnani , ulations of se ri e s are permissibl~ since e~ch is absolutely converg ent. THEOREM 3. 2 . If (l ). f(x) is measurable ove r A. \A\is finite. (2). Then the two definitio n s of its integ r a l are equiva lent. Proof: Theorems I.3.1,4.1,II.2.l,3.l, and the general ~rinci ~le of convergence. Thus if a function of cltss M ha ppens to be measurable, there will be no a mbi guity in the expression: \ t c')(, ~A THEOREM 3.3. If \A \ is finite, then a nee essary and suff;:,. (~ icient condition that ~ J \ c x, 6.1)( is that '1 \ \\ c)()\ ON.. ; then \ \ , c·,<\~ \ ~ \ \\ cx)\ <il><. . A f\ Proof: By theorems II.2.6, 3.1, the equiva lence is evident. The remainder of th e nroof is exactly the same as that of t h eorem I.4.4. THEO REM 5,4. If ~ \ \c x, 4 , ~q( x)ck ,IA\ finite, A A Then "3. \1 f cx, ± ~Cx'\\ 4 = r f cx) ~ i \ 3<- x, M'- . /\ /'I A Proof: Theorems II.2.4,3.1, remainder of proof as in theore m I. 4 .2. THE OREM 3. 5 . I f :\ \ ~ e x) ()._I)(. I\ Then ~ Proof: Theorem II.2.9. and is any real number, o(. ) o(_ t ( x )Q.!x =o<. ) t-C x ) cl/,>< A A THEOREM 3 • 6. If ( l ) • ~ ~ ct< X'\ f c x, ~ 1 \ 0\ CX) ~ A f-. (2). \~Cx)\""cp)o< c x) 3-0 almost everywhere in A. Then \ \ cJ.. Cx ) ~Cx) ~ \ ~ q> I\ \ ol. Cx ) ~ r\ Proof: Same as that of theor em I. 4 .5. THEO REM 3.7. If (1). ;\ \o1. cx ) f c x ) ~ , \o<'.C x)M_ A A ( 2). F is the s mallest closed linear mani- fold in B containing all the values of f(x) for x in A. ::.t € F Then such that \oZ(x) t(x)th = f- \°'-C x) OIX . ~ A where f will of course depend on the function ~. Proof: ~ame as that o f theore m I.4.6. THEOREM 3.8. If (1). \ A\ is finite. ( 2). ~C -x) ~ o '=- oUx) ~ ~C x ) al most everywhe r e in A. ~e x ) (3). Then 1 \ oUx) ~ is summable over:r·A. ~Cx ) , \ ~ ~ 'l \ olC )'.") ~ . A A Proof: By theorem II.2.10, ol c x"\ is summable. By theorem II.3.1, 3 \ olC x 'l ~ ) A ~e x., ~ . 3. \ /\ Inequalit y is obvious. THEOREM 3. 9. If ~ \ \ex,~ - Then Att3 \ ~( x'l ~ A + \ ~(. x) 4 . ~ Proof: Theorem rr.2.s. THEOREM 3.10. If 3. \ -\("J<J ck , B:;1 is a measu r a ble sub-set of A, A Then Proof: Theorem II.2.7. l S ~ex) c\.;x. B THE OREM 3.11. If (1 ). \ A \ is finite. (2). For any n, ~ \ \,Jx)l,x _ A (3). -\..._C.x) - \cx, uniformly a lmost everywh ere in A. Then ~ ) ~ c x\ 04"' 1~ \ \-"' C?<-) 4. A Proof: ~ ----. co P.. By theorem II.2.11, f( X) is summable over A. By theorem II.3.1, 3 s \ C x ') ~ - A Remainder of proof is the same as that of theore m I.4.7. THEORETJ 3.12. If f(x) is of class M over A, where IA\ is finite, then a n ecessary and suff i cient condition that ~ S h x) ~ is that the series A converge, where for amy m, Proof: For any E , let A v..f be any f subdivi s ion of A. By theorems II.2 . 2, 2.3, 3.1, we mus t prove that the condition stated in the theorem is a necessar;y and sufficient co condition for the convergence of t h e seri es 2 ... ... \ \ C)<. .,..,~ )\ \A ~E A... \ COROLLARY 1. If \ A\ is finite, and if l \ fcx,k. I\ where 00 Proof: Since 2 - V\,\_" I A..,_ \ converges, But \\i\ · \ ~ M \ -::: co \\I\ . L _ \ A -.\ (V\ COROLLARY 2. The necessity does not depend on the finiteness of \A\. co Therefore 2.- \ A .__\ Therefore \~ 1 must converg e. is finite. CO ROLLARY 3. Si mil a rl;y, if for any posit iv e number cl , ~o1:: A- £ '\ al~ \~ c. ")<.)\1 1 \ ~ C x) ~ - and if /'I. Then \ ~ o{ \ i s fin it e • THEOREM 3.13. Define ~"' Cx) ::. \" C. ")(.) •. ~ •. \ \ ( ,n\ < lt\ ~ 0 •. ::: •. \ \' ( 'I{")\ ❖ \I\, ~ c -x'\ 4 -: ~ vvv-.. . \ ~ ~ ( ""-> ~ . If \A\ is finite and if ?i. \ . I\ Then \ ~ c x) 4 A A \S.-"> = Proof: Using the notation of the preceeding theore m and its corollaries, \ \ °\ ( X '\ -~ - \ A A For any «> --\- .Z: <lO ~ , let A. ._ ~ be a I- \~c-x .,_ 0 ..._1\\A-..d.._\ \.. ~ ~ \. ( ><) ~ \ ~ \ ) °t ( X) ~ \ S Ek subdivision of A. " -¾-- 2.,_ 2--. ( ...... .,,.~\A "'-r,..__\ \,, (0 co \• -\- L- \A.,..\ + 1- Iv. -\ A-\ By the preceeding theorem, the third term can be made arbitraril y small by a suit a ble choice of h. Similarly for the second term. After h has bee n so chosen, the first term can be ma de arbitrarily small by a suitable c h oice o f r , from the definition of the integral . is finite, COROLLARY • If\ A\ 3. \ ~cx1 ~. A Then 1_ \ \ -\-c-,,.,\ 4 A S\~ . . c.~,\ ~. =- ~ . .,.,_co A We now prove a quasi-converse of the above theorem: THEOREM 3.14. If, for any n, ~ \~ .,__C."K'I\~ ?:. "' 1 ] ~ \ \ ~ ~ (. . .)\~ , \ A \ \,-..,\ e , "----+ro A using the notation of the previous theorems, Then ~ \ ~<: x'\6-1)(. = A ~ \ ~--c "'' ~ . "' - I\ oo Proof: By theorems II.3.1,3.13, it is sufficient to show that f ( x) is summable over A. Hence, by theorem II. 3.12, it is sufficient to show that the series 2- ~ \ A~ \ converges. Now, ~ \A. o ( C\ : \\I '>,.M -:,~1/\. o -J.\ \\~N ( 1<., \ ~ - \ \ \ k.(-x.'\\~\< ( l'I Ps \ Si C ><)\ <l. X \ < f . AMt ··· +AN. Let A..,. d be any d subdivision of A, N "'N M ""'lV\ N (l'.) M ' A.IA. .,,,,._-=- A..,:F;A""", "X"'.,..E A.,. Wl Then 2~ ~\ A~\ f l ~ 1\-\\ A.,. v..\ ~ 2W\ L \fc,<. .. .,..'\ ·\ A" ""'\ N ::: \ T~ J.: l t (Xi.~"'>\\ A~~\ - J\11 + \ \ ,. ( ">( ) \ dX \ A Mt · ··-'rAN \ AM-'r···'tAi-.1 • The last term can be made small by choosing :M sufficiently larg e. For an;y choice of M,N, the first term is small with S which does not appear in the left member of the inequal.ity. THEOREM 3.15. I f (1). (2 )• \A \is finite. ~ \ \ C-x) d._ X . />I <O ( 3). A=- J_... B..,. where the Bvv, are dis joint, mea m1rabl e. Then ) ,ex) 4 ,Z =- A Proof: Let A"'"s Let Ap be "Y C><-) \ 11 4. E"' o subdivision of A. any be defined as in theore m 3.12. Define A..,. ..,...~ :: A"'! ~""'" A?· Then \ \tCx) A ~ -~J.~ \ \ :: \ - .f..,, \Cx') k \ E"' i ' f ~ ~ - ~ \A M "' " t--\. < j 2 .. D ~ -o ~.... ~ \ ..,\ \ ( ~) k I \(x)~ " + p t'\ -t lM M l 2- 2p (p -\ 1)\ IV\ 00 \1\M a:, C)-"' A Cl:> 2", p ( pt 1) \ A~ ~" \ I p\ I Now the series 2p Ct-t11 \ AP\ conve r ges and is equal to f IP <0 ) I L lp (ptl ) lB~ At'\ Therefore the s econd term of the right member of the above inequality approaches zero with 1/M. For any M, the first term approaches zero with 6 , which does not appear in the left member. COROLLARY. If (1 ). \ A\ is finite. ( 2 )• 3 \ \ ( x) ~ . A (3). A~ i. B"" where the B""' are disjoint, Ill:. J measurable. Then THEO REM 3.16. If (1). \ Al is finite. ( 2}. :! \ \ c X) dr,(__ A. ( 3 ) • B.,. ~ A ~ Then. 3. \ l3 ~ \ ·. Q ~ \ ~ "'\ ~o . Proof: It is sufficient to prove the secon« conclusion. Consider the cohstructiom. §'.. A :: A- 2 "8 I '1 \.\ co ~ A 3 =- 8 1 -B-i.{B, + 2 .,. B.J 3 3 = VI -¼ I .a.:\-c. Then, evidently for any n, there is an integ er \{"'(-=.2+ ~ such that B"'" ~ 2 ,c:c, \/\ ----+ co . I{~ Also A ✓ 2 11 A"" •. II\ -=t V¼ . --.:, A .,._A"""==O . ~ \ \ ( x ) \ ~ ==-C) _ Hence, by the~rem II. 3 .15,, 2 ,- A y v("' Therefore THEO REM 3 . 17. Analog of Lebesgue's conve r gence theorem. If (1). \ A\ is finite. (2). For any n, ~ \ \"'-c~1 ~ /\ (3). v; .. c-,q\ "":::o<Cx)almost ~ \ ol Cx ) 4 (4 )• A everywhere in A. . (5). ·\-(-x) -=- Sl.:...... \_,(x)almost everywhe r e in A. Then 3. \ ~C x.1 d.t)( -;. /\ ~~ \ -\-- ( ><-\ 4. v.. - CV A Proof: Same as t hat of Titchmarsh, with the use of theore m II.2.10. 4. Relations of this t o other recently defined integ rals. THEOREM 4.1. If f(x) is summable over A, then a necessary and sufficient condition that f(x) be integrable over A i s that if x..., .~"' E A. ....._E (l) \Z.. ~\Cx._') -\ cj ~')\ A"\\< E. \ Proof: This is equivalent to the Cauc hy condition, sin~e \ F ( c ) - F ( t1\ -5 2.., \ c ~ \ ')<. .;; , \ A " €' \ - 2.,.,""' I t C')(. "' - 1 \ A v. f A "'Y) \ \ I / a:, co 1 l ~ \ 2 ...... ~c~ .. ,,..') \A l,\ 6 A ..,.r:> \ - ].,,, ~ ex... ,') \Aw. , \\ < if G, '1 suf ficientl y small. We might, wi t h Birkhoff (4 ) , take simply this condition as our definition of integ rability. Taht is , f(x) is integrable if and only if the series ]'° \fc)(. J \A..\ converg es and this Cauchy condition is s atisfied. However, this definition does not seem to yield Egor,ff's theorem. For, consider the simple real function Let A be the int erval For any €, 1 \ C~ ~ ;<. -1/-... lD, '1. 1 . L, , the interval ~E =- \.. ~ , 11 can be subdivided into a finite number of sub-interval s in each of which the oscillation of f(x) is le s s than <=/.,. , so that : D ... == S< . tA. .~ . \ \C -,c.) - \(_~~\ Hence, defi n e so that \A..,.\ :: - 3E 00 Then 2 \ f C~ ..,\ \ AJ :- 2 \. \ Ct< --,\ \ A J A ., ~' and the sum over B contains only a finite numb er of terms, {D whil e, Q) 3 f 2 H: I --- - =---2 l ::- 2 11 :::1 - 3 € so that f(x) is summable over the g iven subdivision. a:, Furthermore, 2 .. O"" \ A-..\ == .[. D\.\. \A,,.\ -\- L- ·• A ~ (£> fi 2 "L""-\ 1... so that this Camrhy condition is s atisfied. But, ~"'.,_2_"/f which, for any f , is not uni f ormly bounded in n. That is, the function is not of class M with this choice of subdivisions, and t he proof of Egoroff's theorem rests directly on the properties of functions of th e class M. However, in such a simple exanrnle the di f ficulty is removable. There do exist subdivisions satisfying the postulat e s of the class M. The question, then, of the sufficiency of the condition of theorem 4.1, for the existence of such subdivisions still rema i ns an o~en one. It is evident from the definitions that this integ ral includes those of Saks and Bochner, and Radon. Also, Birkhoff's integ ral is more g eneral than this only in so far as "unconditional" convergence is more g eneral than absolute. If these are equivalent, then so are the integrals. Chapter III. Theory of Diverg ent Series. Before proceeding t o the ne xt and last integral concept, we n eed some elements of the theory of divergent series. ' (. 5 ) 1. The Cesaro Theory. <b Consider th e series I"' QI< c) Define ( vt.-1) o. :: (v.. . ""'\ ::c 0 \I\. .::. > .... Ar = "' "/ 0 - """'0 2- \._'< -\- ""'- -' ,""' \ A "' IA - = 0 g . ._ ( (r- t lA. , VI) o(:-:: Q:> 00 If A -= 2 Q"' , t ) -o. 2 are summable (C,r), then b"' (s) . the f ollowing properti es are well known.: l )• G. ,,_-';, 0 2). (A - A ( rJ C 0 .., ") -'7 o Q:) 3). z"' (a . . - a., ( C, r 1 - ~ QO -t, ") ( c. r- >. () 4). A change in one t erm of A will affect its sum in the obvious manner. 5). The additio n of a constant other than zero to each term of A wi ll de s troy its summability. Q:> , then I Q .,.. 6). If CL "3- o ~ 7 )• is convergent. ~ ~~ 2., ( o. ... , \, ..\ -=- A +C) .J . A ~ l c. " ) ,) 8). Q.,. ~ \:, "' Cl) 9). c:t~~ b~--:, - 1 b.. -ah converges to (B-A). l1 c() We also need the following two more i mportant theorems: co a). If A 7- I.,. Cl"' = Then p/ ::. 2 ~ is summable ( C, r), 0 - z C<."' is summable (C,r-1) and VI 4 \ ()VI Cl) is summable ( C, r-1) to A, where ., =o Q'.) J_...,, al V\ Q.,,,.._ ( C, ' -,) . '""' -\- I V\ co b) • If A= l . . (X"' Q'.) is summable ( C, r-1), 0 is summable ( C, r) to A, where Then J..._ Q"' () Cl---- -=- ("' . _ ,\ ( o!.,._ -cc.. . ,) Vl We shall adopt the following notation: so that A .... ~ A ., ._ ::: - 2 ..... C, Q_ )",-, + I o/.. 0 - .,)_ "'--\- \ The purpos e of this section is to prove the all n. co (b (2). A ~..20.,. 1 B: 2 b.... {) are summable (C,r), l) is surnmabl e ( C, r) and A ::: D ~ B . Proof: The theorem is obviously true for r =o. We proceed by induct ion on r, which is a po si ti ve integral variable. Let o!"' , f'"' be defined as in theorem (a). We have the following results immediately: Lemma 1. o< . . . ~ (s "' all n. Proof: Theorems 8 and (a). co Lemma 2. 2 ( ~ - .,._-:.o ,._) converges to ( B-A) • Proof. Theorems 9 and (a), lemma l. Hence ~ (~ --~--' ::0 . .. --, (I) Lemma 3. cx +D . . - A... :: ~ ,, - ( B. . - D ..., 0 Proof: Since cl. c. -A.,..-=- o<'...._ -\ , , it follows from lemma 1. all n. ~ o - \3.,.-:. r-, .... ..,., Hence it is permissibl e to speak of the interval. -rv.~\._ct -\D.,. - A.. , ~b-C~ .... - b~,'\ • Clearly. I ---. <-\o1 0 0 ,~:) all n. Lemma 4. The sequence of intervals I ~ defines a r eal numb er. Proof: We must show that all n. 2). limit(leng th of I ... ) = o. C1 early. D.._ - A .,,. "? o '= B "' - D... since Q ..._ s ~"' ::: \::i .,_, - Thus to prove (1), we need only show that This is true since D """ ' - As for (2), the (length of I " ) =-t\, .. -~ 0 -~ "-*A \A : : . ,_,__ R ... \ -ol.. "'- ... \ and the result follows from lemma 2. Let d o be the number defined by the sequence of inter- vals I ... . Then d.. 0 ~ ie., <S0 &,, =~ 0 -,(ol. 0 -\'b..._-A..._ ~o., ::.(~t>~\.E"-\.'.>.._1) lies in I ._ , all n. all n. But, sine e ol. 0 \- it follows that A . ,._ -=- 6., ~ t) ..._ - c,/.. 0 - r 0 - (. B . ._ - D " "'> A - -:: 6 0 - \i) "'- ~ ~" - \.S. '-'- and hence the result~ We are now able to complete the proof of the theorem stated at the beginning of the parag raph. "Proceeding by inductio n , assume the truth of the thell:> ~ i> <> orem for ( r-1). By theorem (a), ].ex .. , 2 f-- are summable ( C, r-1). By th e induc tion hypothesis and le mma 5, l> 00 Hence, by theorem (b), 2.. Q ., _ is summable (C , r). i) Then A ~ D \::>. by theorem s. ~ Since we are discu s sing the subject of diverge nt s eries , it seems convenient to introduce here the following new methofl of s ummation. 2. New Summation Method f or Diveg-g ent Ser i es. The method to be g iven here is a modi f icatio n of that ('1) due to Euler-Kno pp • For the weighted means of the -partial sums we use the binomial coeff icients, but instead of beginning with th e first we begin with the cent ral one,ie., with the g reatest. Thus the initial terms al ways receive the g reates t weight, a s in the Cesaro ! Holder method. «> Therefore let us consi d er the serie s r>< Q '< . ,. with the partial sums. A "' = IK a 0 0 I-<: 0 and v If there exists ~.:,__ -s:~ "5"then --S \-is called the ri sum 11 of the se r ies. Our first problem is to find t he relation of this .... method of summation to t ha t of Cesaro. To do this we recall the following well-kno wn equalit y co (X) ( \ - -,('{ I ... C (1 ) A: 'K ~ ~ 2..,.. A I.A -x " D botij series converging absolutely i n the open interval ( .; ;.:i,.r tJ. Hence, equating coef fi c ients we obtain the partial sums of ' the original series as linear functions of the Cesaro means. " = 7 ( -,\ K'"'° 11(\(r-\-¼-'-< ,v-""\o£< ---- ~ r . A .1-V. \I\. \ ' ----\( = 0 ... lv.. A .... -=- 0 Thus we can express our means as linear functions of Cesaro's. ~<.... =- i"' WI 2 () 1,{ l - I'/,;_ ( :t-w-.H, ""-.- "'\ (__ " ll<\ (. ,- ... II\ - I,(, 1-\ o<'. :-1.< I')._,.,_,,__ 0 ..... r and or, collecting coefficients and rearranging ..... -,- and J ... ,- 2 j d j 1 J..., [ •• 1 \ + 0 J Hence we may write 0 J ¼-1- • . 2 .. (_ -l)"'- j (,_"""- H,Yh -"'-\( r,\1\- ~)\_Y- ~i,~} by defining / )._,.._....,,_: ~,_,r+\. J Now we make use of the following Toeplit's theorem: Consider the transformati on: 0 In order that whenever the latter exists, it is necessary and sufficient that 2 )• .SL.~- <l..._~ =-0 all j • ..,, _ ( t ) t{ ...., 3 ). Lj \O. ..... ~\ -< K all m, K inde p endent of m. 0 Thus we shall not need the values of the coefficients .... -~ theorem depend on the .s behavior of the a.:_~ as m - Cl . for m ~ r, since all three of the conditions of the co . Now let us consider the first condition, which is that We have Interchanging the order of summing , we have - ,- w, I .. (y . ., . . ""' J· (?... ""' -\ \, ,-- l J· Q - ~ ::. o ., ... O \'l,\ - \-1) - J.~j. c- 1) c,- -+ j. j { ,-, -J ) k-J ) 11 O "' .., _.... r+ I .2.,, ( -1).,, (:2..wi t-1,m-n ) - , ( t-;1d We can write this as / 2 .,_ ,,,,. 0 " l i ( - l) J ( r +..i . i) ( \"' . \I\ - ~) where () }_"' • and J By means of the recursion relations ( ! , \,( ) :c (j -\, \.( -1) we find that --r L r,"'\ -=- -T \_ Y-, ¾. -l1 t-TL ,- - ,, \.-1. -~1 +lS- i- ~) .,. 1 ( 1--1, 1.-\1 \,\ by actual calculatio n . Thus, by induction, T~r. ..."\ -=- (- t') for every n,r. Hence, - ,- . ~ 1 for every r,m. There f ore 2 >·Q 0 "'\\ Toeplitz's first condition is satisfied. Now let us consider the second condition, namely any r, j. J t-r- We have • : :. 2., (-1)"'-J (1..m -tl, Wi-1-1)(t-, "1 -j)(1- +-j , j)/22.,-. J ; 11 ,r ?:t . J t-1- . s . ·) \/1.· ) \t . _ ,\ 15"' c -n"' - \i-- .. -i ) tC """ - "'"'•)--(h< · P H l "-'t "'- tu · ·c-+~ 1--'<Hi -:: le \- ~ > . 1 l 1.. ----- ... 1) . 2 (}'\,I _J . Wt t J -1- 1 -+ " · \ L ~ • • J where there are n-j fac t ors in the firs t bracket and j + r-m in the second, or r in all. That is, ea ch term in the sum is a pol yno mi a l in m of de g ree r. The leading coeff icient in '"- · \ e~ch term is <.- 1\ (, .... -~\ ; hence, for the entire s um the le a d .it\· ing coe ff icient is r ... ( -,, ..., _l, ~ 6-. ~-i \ -:. (\- I\ • =0 ... Thus the en tire sum is a p ol~nomial in m of de g ree r-1 at ~o ~g most. Denote it by P. (\- + i , i\ \ · l • ;'.-) .,, (J.""-1..j -1 ) ·1(-;i.. ..... -'2 ~ -\ 0 ' " h .. ~ -ti\ 1- V where the brac k et in the numera tor has j "° l factors and that in the denominator j t- r t- l. \ ur_~\ Therefore ~ ( \"' ti, i) · 6 C Yi,.-) Hence Now for the third conditinn, which is: K independent of m. j substitute h for n - j , yielding I- -== 2 1, (. - l) \.. ( Y, ~) ( r "° i , ~ ~ l 1.. V'-'- ., ' , VI,\ - ~ . ~ \ IJ... 'l."'°' ((t\ , we have \ ~.l -:,/ f ,· ti 7 () , Now, in the s t a tement of Abel's lemma f \, ::. (10 ) and (i_w.. ,.,, . ""'--i -\.\ Lr -\ i , ~) /1'},."" n~ 7. ( _ \I\, c ,- . \,, ._ s """ ~ K r u"' h (.) .!5. l. "' :\ - 2ti ( - 1) (r,, h) -:.(-/} (Y-/, if) . C \ <\ '> and hence , agai n b y footnote ~ l , ~I Id "' ~ v. ( f v. - 0\.. f\,i = 1, 0 or, sine e fKt" "'S \- -=..O ~-, I Q : ~ :: l ~ ( -,)~ ( r-1. w) Cr+ J ,j){ (2 w.. H, --~-u\ - (_ ,._ VV\H , V\,1-j -l,(-t)\ 2 ').."" (> Now, in the s n ecial case Cl.').._ ~ = ~ l ~ t, ""l. , ~ ) {l'l..""'- " \ , 'M. -~} -1-.l'l-""'- .... ' · "" -~ -\) ( i """""' ~' ,_ 0. ,_ while ... s 2 i \ Q :_ , \ :: ~ - [l Cl . ""' ~ " ::. l - ,c l "" -\- 2 l- x+, a. , = l '\,- X, ,._ )_ , Q . ""'J los~ Therefore 2.-\ \ Q~-~ d -\ 6~ ,._ Q ~j .,_ K \ + 4 ( 'L """ -\\} \_ "l...,.,..__ ~, 'J_ L. -"2_ 2 - (,) 0 ::, ~ + 1 , \_ J :;, \{, 2 J· Q~ _ "" VII\, -\- - ~ - 2-. ~ .... )(_ "3, 4 -\- ( '.l."'1. ~l, l¾ -j - :i.1 \ / J...i""' ~ + 6J n "'\ / 2 ,,., .,..,..c....,,.,.. - ~ , \ l i~ ( - h-- "' ;i..-~ ~- ~ ~ ~o Q ....,.-~ ~ ~ H ' C5t ( ,\ -::::o ..... ~ r :: :t. , -4:(_'l...""'- ",;'-\ \_ -\- 2 'l..'""- " ' \v,,.,..\_ (""'- H :\ \ . ( j \,\) l ~t-1..\ l'-" - "l..i 1:. h ~ - 1') 2 - l"'- - ~")\ l ""'- "' ~ " ~ "')\_ 0 K L l- " ~ ( ~ \- \H j -\- l..") , Y'- - ,_ ~ ..._ f. i - ) J \~ .., >--'\ l"""' "'°'' ~ (__ 1... ""'- ... 1.) \ _ \\ J..."L."- "' '\M.\ , LW1. +\I, Thus, we need only show that ~ ( 'l-""'- -u\ \_ is a bounded function of m. and, using Stirling's theorem -:I..""' "\"'{.,_ ~ ,-:i..--- -\. -:,.> - we have ""'- "<\ -[.;; e _ ..,.,_ e \ e m Hence we have proved the THEOREM 1. Any series which is summable (C,2) is summable by this method to the same sum. Next we investigate the relation of this method of summation to that of Euler-Knopp. Using Rey-Pastor's ( n,) result I) we find that where Cl""' . . - r We wish to see if the matrix Is it true that 2- o"",... - \\ a .,.,_\\ is a Toeplitz matrix. l as VIA - co ? r:. o We have l) 0 ~ -:: l.. ..,._"" ( "2.-. .... \, ~ IM..-\I\\ /-;_,,_.._ \ l "J.. - \\ 0 Thus the first condition is satisfied. We prove that the third conditi on is also satisfied by showing that Cl,._1- ~ 0 , for y- 2 ,Z.,. (_- II all mt r. YI - 1- (\\ , \") b . """'-·H , ""'--\1\.) i- - ~ -,- - 2 \,( 1,{ ( -\) (_~V¾ -H , \.(J \wt -~ . ~, 0 Now, in the statement of Abel's lemma we have ~ \.""'--¼,"! / =:. ~l(H (.q\ "7 0 . ~ t -f I( :: I,( ( V¼-'-( , rl - ('\Iv\.- ~-\ .\- )-;, ( M - K -\Jr-':) t\ and Therefore '3(~1 ~ Now let 11. - 1- 2 v.. ( -l) I< h . W\, +\- S , K ~ (}ll,t -l.(-s , 1--5) 0 The same method shows that Therefore, by induction ~ ( ':> ') 3 c c) :::- ~ ( s -\ I.} . =-- s ( y-") _ But, D Also =- Therefore 0 0 To prove condition 2, we have O ""-r -- 2 ,- (_ "l..W\-V--) \ (\""-\- :=: J...-,_w- 2 ,- ·\·2-:.···l""l..'--H\·1 l-i.""'--~r-H) --- (."l.""'--v- \ \ WI\ ( \'\,\ - \-)\ where the brac ket in the numerator contains r factors. Therefore '2.. "'"" . \IV,.. \ where the brackets in the numerator have respectively m-r and r factors. Thus where the bracket c pntains r factors. Therefore and hence 0. ..... ,- -- o as VV\ - QJ . Thus vrn have proved the THEOREM 2. Any series summable (E,l) is summabl e by this method to the same sum. Chapter IV. Divergent Integrals. The possible methmds of g eneralizing the integral concept have been well discussed by Kolmogoroff t \- . He suggests that we may (1). Take infinite instead of finite subdivisions. (2). Consider abstract functions. ( 3). Generalize the mea sure -H -\- • He overlooks , however, the possibility of generalizing the sum, which is presented by (1). We propose to make this gen- ' eralizatio n , with particular reference to the Cesaro summation method. Since, however, t h e sum of a divergent series does not re main invari ant under commutation, we must alway s be able to order the subdivision sets according to a definite rule. For this reason we find it convenient to consi der onl y real functions of a real variable in this chapter. We present g eneralizations of both the Riemann and Lebesgue integrals. l. Riemann Integ ration. Let A be a semi-open interval La. ,\, (b may be infinite}. Let ~(~1 be a real function of a real variabl e defined throughout A except pos s ibly at b, and have the following properties. , A oan be covered by a a enumer- ( 1 }. For any € > 0 able infinity of iHosed intervals Av..€ , any two having at most one point in common, such that o< ~ ~ if c::/.f A \,\ , interval. r f A IA 4<\ and ~(oZ°\ is bounded in each . . Definition. ~ld) is summable in A· ==- • For any € , the series co is summable by a regular method (always the same), where cJ....,,. A V\. E G G -o {\ if the right member exists. By a "regular" summation method is meant one with the following properties: a). ~ Z (a_ ,. ~') z_ Q * Z- 6 b). If the ordinary sum exists, bo~h exist and are equal. c). Changing a finite number of terms affects the sum in the obvious manner. ' As wa s shown in III.l, the Cesaro methods of integral order are thus regular. Now define Q:) Define Q2 (f') T ., - cp~(O \ A"'~\ q:, ~ Cf') -- 2 .,. q>_,.f \A'-'-~t- :::. 1 - Then ~ Cr) 7,, ~ <.. r') - - ~ ~ (<,:) for any £ , from property d. Thus we obtain the first existence THEO REI'.'T 1. A necessary and sufficient cond i tion for the existence of S q>Co()<lol is the existence of A the common limit Now, for any ( where in the , any subdivision A. ,._ f , define I.. the terms are associated in any manner such that all the A . ,._ E in any group have a total leng th '.:" f'. Similarly for ~• Cf'\ Then we can prove the more powerful existende THEO REM 2. Let A . . . c:: be any f' subdivision, and XS "'- E any resubdivision of 1--\..,_f (in particular, B may be A itself). Let cE" A ( ~ 1 Let the terms being associated as above. Similarly, define Then, a necessary and sufficient condition for the exisjrence of ) ~c,1.',<\ol. is that it be possible {\ to enclose ci) (6 ~ (15.,. - A ' ~ , hl!13 ' ~ leng th - 0 with E in an interval whose • Proof: Let A... € .I\ "-~ be two subdivisions, 1=>"'- the product subdivision, so that B satisfies the hypothesis rala~ive to both A's. Let ~ I refer to the I_ with B in it, associated so that the nth tenn contains precisely those B's which add up to /\ "'- e • Similarly, in A"'-~ • Let g_> to 111 g? 1\ those in the nth group add up be the simple unassociated series of the B's. Then, firstly, all the series considered are summable, -::C-1 ~ 1 i'h\' ~ ) by d , and w ,K"(I\ because (? is summable. Now, to show integrability, we need only show that ~u,, ~ _(£=\ _ , ~( o\ ~ ~- ( o\ - lie in an interval of arbitrarily small length. Well, \ ci" cn -\- \ ~ - cli \I I ~ ,, -\ \ ~ \1 1 - F \ ili Co)\ and the result follows from the hypotheses. It is clear that \cp(_d.) <\d.. is additive with respe c t (\ to er and with respect to A. We might have allowed the values of the function to be abstract, but then we should not have had these existence theore ms. 2. Lebesgue Int~gration. Let A be a measurable set of real numbers, ~Col) a real function defined and measurable in A. Then, as in classical theory, let -· Definition. ..:: ~ ~(o!'I The s e ri es - 1.-.::: ~ -'-.::::.. ~ o::: D .:::_ '-\~ ~~--..__-<::. is summable in A: =:- : ~ ( E'.) - ] 11 f CV cex~\ \ A J -\- q C £¥. J t I\ - \,\ I t is summa.bl e for any E , with ol... ('. A "'-· Definition Then the analogs of the two preceeding theorems are valid. Here also we might have generalized to the case of an abstract independent variable, since both the ordering of the sets and the a bove ex istence theorems depend only on the realness of the values of the function. Finally, it is evident that these integ~als are true generalizatio ns of the classical Riemann and Lebesgue integ r als. 3. Examples. The function f-c -i< ) ::. (-2')"' , o ~ V\ ~ x ..:..V'--H, o\\ V\, is integrable in either of these senses, using the EulerKnopp summation method, the value of the integral being s i. m simply 1/3. although this function is not integ r able according to any other definition with which we are f amiliar. Bibliog raphy. * Bull . Soc. J.Iath.France,43, (1915), 248-265. ** Thoorie de l'Integrale, Warsa,w, 1933. * ** Fund. Math. 20,(1933),262-276. t Trans.Am. Math. Soc. 38, (1935),357-378. t t Math. Annal em. 103, ( 1930), 654-6 9 6. 111 Vienna Sitz. Ber. 122 2 ,(1913),1295-1438 . (1 ). Bull.Soc. Math.Framce,45,(1917),1-8. ( 2). Kolmog o roff, Ergeb. Math. 2, ( 1933), p. 2 3 3. ( 3) • W. M. Why burn, Bull .Am. Math. Soc. 37, ( 1931), ~• 561. (4 )• t above. (5). Hard y and Littl ewood, Math.Zeit.19,(1924),p.67. (6). Kn opp, Math. Zeit. 15,(1 922), 226-253. (I). s.chapman,Proc.Lon. Math. Soc. 9,(1910-11),369-409. ( 8 ) • Toe Tll it z, Prac. Mat. Fi z. , ( 1911 ) , 113-119. (9). W'b.ittaker and Watson, Modern Analysis, 16-17. I (10). El Lucas,Theorie des Nombres,p.21, e qu a tion E. (11). E.Lucas, ibid., p.239, section 134, formula 4. (12). Titchmarsh, Theory of Functions, section l.87. ;· (13). Teoria se los Algoritmos Lineales de Convergencia :. y I / Sumacion, in the Buenos Pub.Fac·.cien.Fis.y Nat., Univ. Aires, Ser.B,no.12, (1932),51-222.