On the Integration of Abstract Functions Citation Mersman, William Alvin (1934) On the Integration of Abstract Functions. Master's thesis, California Institute of Technology. doi:10.7907/4358-yn76. https://resolver.caltech.edu/CaltechTHESIS:09202024-154237547 Abstract No abstract. Item Type: Thesis (Master's thesis) 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 1934 Record Number: CaltechTHESIS:09202024-154237547 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09202024-154237547 DOI: 10.7907/4358-yn76 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16746 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 24 Sep 2024 16:57 Last Modified: 24 Sep 2024 16:57 Thesis Files PDF - Final Version See Usage Policy. 3MB Repository Staff Only: item control page

ON THE I NTEGRAT IO N OF ABsrr RACT .F'UNCT I ONS Thesis by Wm. A. Iv1ersrnan in partial fulfillment of the requirements for the de gree of Was ter of Science. California Institute of Technolo gy 1934 1. Introductory. The concept of integration has been extended to abstract spac es by three distinct methods. Saks (1) conside rs numerical-valued functions of a n a bstrac t variable and defines a generaliz a tio n of the Lebesgue integra l for such functions. Kerner (2) considers a bstract-valued functions of a 3ingle real variab le a nd defines a generalization of the Riemann integra l for such functiona. Lastly, Bochner (3) considers a bstra ct~valued functions of n re al varidbles a nd defines a generalization of the Lebesgue integral for such functiono. This paper, as s h own in section 4, presents d unification of these three theories by generalizing the Lebesgue integral to the case of a bstr~ct -valued functions of an a bstract var i able . The essentially new contributions are contained in Theorems 3.4 and 3.5, where the integral is shown to exist for a large class of f u nctions a nd bounded continuous functions in certain s p ac~s a re show n to be integr abl e ■ It ~a s found necess ~ry t o p ostula te the exiatence of ~ me~sure function, ~ s none could be discovered for gener~l ve u tor spaces. If such~ function cun be shown to exist, then ~ e shall have a true gene r aliz~tion of the theory of integr~ tionThis p~p er wu s wri tten under the guid d nce of Professor A. D. Michal. (1) • Saks, THEORIE DE L' INTEGRALE, Vhns...;.,w, 1 s,;:>3. (2). Kerner, Prace Matematyczno Fizyczne, 40, P~rt 1, (3). Bochner, Fundumenta. Mathemuticae, 20, 19.:,3. 1~33 ■ 2. Spa ces. In the course of the pap er we shall ha ve occa sion to refer to the follo wing t ypes of s pa ces: B; a comple t e, normed, linea r s pa ce, usually called a Banach s pa ce, whose elements will be den oted by srra ll La tin letters from t he first pa rt of the al pha bet. R; the real numb er system, whose elements will be d enot ed by small Gr e ek letters. X; a spa ce satis f y ing the following postula tes: (a). X is a B. (b). Corresponding to each set, A, of elements of X there ex ists a non-ne gative real number, denoted by \A \ , called the measure of A, with the :properties: oO oO (1). If A ==[. A ., , then \A\~I:. \A"'\ ; eq_uality obtain' ing if and only if the A h are non-overla pping. (2). IA \::::oc if and only if there exists an infinite s eq_uence \ 1l V\t such that A"'-1-,~ An , for all n, and s.ftch that A=f. A"' and lim. \ A.,\:::oe . In such a cas e l Y.. ➔ oO A is sa id to be the out er limiting set of the seq_uence { A.,1 . V; a metric s pace in which t he triangular ine q_uali ty may not hold, but such tha t the dista nce between any two points a pproaches z ero a s the d i s tance b e t we en ea ch of them and a n arbitrary third point a pproaches ze r·o. The elements of the space X will be denoted by srna.11 Latin letters from the end of the alphabet. Hence, the nature of a function will be completely determined by the letters used in its expression; eg. f{x) denotes a function on X to B, ~ (e) a function on B to R, etc. The letter A will be used to denote a set of elements of the space X. The letters i, j, m, n, will denote positive integers. 3. Integration. Definition and Properties. Definition 3.1. A function g(x} defined over A is said to be elementary if and only if:.A ==I:A" :i t j. => . A:l A-j =- O, where O is ~ the null set. XEAn. J •g(x) =g n. g ~ constant for each n. ( Throughout, the letter [_ with no range indicated will denote finite sums). Definition 3.2. A function f(x) defined over A is said to be measurable if a.nd only if: If A*c: : A is the entir e set of points for which some sequence ff "' (x)} of elementa.ry functions does not converge to f(x), then \.A *\= o. 'fhat is, there exists a sequence [f"'(x)~ which converges to f(x) almost everywhere in A. Hence, every elementary function is measurable. If a function is measurable over each of a finite number of sets, it is measurable over their sum s et. If a function is mea s~ urable over a set A, it is measurable over every sub-set of A. If f(x) is measurable, so is \\ f (x)\\ , because of the triangular property of the norm. Finally, the sum of any f'ini te num- ber of measurable functions is a measurable function, and the product of a measurable function by a real-valued measurable function is measurable. From Egoroff' s theorem ( 1), and its converse ( 2), both of which are valid for functions of the ty pe consider ed here ... since the norm, by means of which convergence is defined, and the measure-function both have the triangular property, and because of postulate b (2) for the spaces X... we have: Theorem 3.1. A function f(x) defined over A is measurable if and only if there exists a sequence of elementary functions which converges nasymptoticallyn ... or tton the avera g en ... to f(x); that is, in every sub-set M, of finite measure, of A, the sequence converges uniformly to f(x) except in a set of arbitrarily small measure. Symbolically: M ~ A. \ M\ finite. ~> O : :) : J !v\ SM. \].~~\>\M- ~. f ,J x) -'> f(x ) uniformly This theorem is essentially the same as tha t of s. Bochner. fhe only :properties of R n that he uses are those of the measure of sets, and these have been preserved. by our postula tes. (2). This property of measurable functions is a fun- damental one, but we have taken the other as our definition bec ~use it is more similar to the definition in the case of functions of a real variable. ( See Saks). (1). Titchmarsh. THE TILlORY OF FUNCTIONS. Page 339. (2). Hobson. FUNCTIONS OF A REAL VARI ABLE. Vol. 2. Page 239. Theorem 3.2. If f(x) is measurable in A and bounded a lmost eve r ywh ere in A , t hen there exists an approximating sequence tf 1-1 (x}1 which is bounded uniformly in A for all n. Proof. Let <p:::!"llf (x)\I almost everywhere in A, and let cp, > cp. Let \g\-\(x)1 be a sequence of elementary functions converging to f (x) a.lmost everywhere in A. Definition f ..Jx) : f"'(x)=g"'(x).==-.)lg.,,(x)l\~ q:i,. f ~JX )=0 • ~. \\ g "'( X}\\ ~'P.• rhen, clearly \\f,.Jx)\\<<P,for all x in A a nd for all n. I 1 Next, Definition A*: xt A*~ A: =: i f 'v\(X) \ does not converge to f(x}. Definition A7: xc A; s A*.:=:. 3 an infinite sequence S of n's: n~s .J. f"'(x) ~o . Definition A:: xe A; s A*.:= :. 3 an m: n ">m. °J. f,._(x) =g "'( x ). Then , A*I · A*=O A*=A*+A* . l. • l ").. • • ' • IA*\=\A*\ t \ A*\ \ "l.. • Now, xtA; .:~ :.3an infinite sequence S of n's: nES.::i. \\f(x) - g ,_( x }\\~ \ \\f( x }\\ - \\ g_._(x) \l\ 1cr 1 - <p, almost everywhere • . •• {g"'( x )1 does not converge to f (x) anywhere in A~ except poss.ibly in a sub-set of measure z ero. . .·- .. 3 m: n>m.:::i. l g "'(x}1 does not converge to f (x}. :. \ A~\ = O. :. \ A*l -=- o. :. \fn(X)) converg es to f(x} a lmost everywhere in A. A1 so , s inc e \If ~ ( x )l 1'"'- qi, f or any cp1 > cp , Uf"' ( x ) \\ ~ <p . Definitio n 3.3 A , wh ere If g (x ) is a n el emen t a ry fur1cti on 6.ef ined ov er \Al is fin it e , we define the symb ol r [ g {x) , A) by the relat ion: I t_g (x), .AJ = [. g __ ·\A"\ \-\ wher e g " and An sati sfy Defin ition 3.1. Definition 3.4. If f{x) is measurabl e ov er A and bounde d a lmost everyw here in A, and if for every seq_uence t f ~ (x) \ def ined by ~heorem 3.2 the seq_uence [I Lf ~ (x), A 1\ is converg ent, then f(x) is said to be inte grable over A, and we wr ite Theorem 3.3. If } f(x)d:x exists, it is uniq_ue; i e. , independA ent of the particular choice of the a ppr oxima ting seq_uenc e { f " ( x )} . Proof. Let fg ~(x )1 and th J x )\ be bounded approximating sequences, and let1I [g ~(x ), A1~ and \I[h~(x ), A1\ converge to g and h respective l y. Con.sider the seq_uence lk "'(x)1 , wh ere k,_...(x) '::: g ~(x) and k,_,,,_J x) = h ~(x ). Then clearly tk ,,,{ x)~ is a bounded approximating sequence . Hence, by Definition 3 .4 {r [k ... (x), A1\ converge s. This can be true if and only if g=h. We a Ee now in a position to prove the Existence of the integra l for a large clas s of functions, namely: Theorem 3.4 If f(x) is measurable over A and bounded a l- most everywhere in A , then f f(x)dx exists, if lAI is A finite. Proof. Let t f ,J x)1 be any sequence satisfying the conditions of Theorem 3.2. Let cp?\lf (x)\\almost everywhere in A. Hence, q>~\\fv-.(x)II for all x and n. By Theorem 3.1, given any £":;• 0, there exists a set AEs A, such that \Ac\ ":7\A\ - E. . , and such that in A£ lf vJx l\ approaches f (x) uniformly; ie. , there exists m,n?n., , then\\f"'(x)-f ,Jx)\\.:::.£ an n 0 = n(E.) such that if 0 for all X in A£. Write !"£=A- Ac.. Then, \ FJ<::.£. Now clearly, for any n, I (f . _ (x) , ~ -=- I[f"" ,A~Jt I[f" ,FE1 since A(·F/0 a nd A= A/FE.. Since the f V\(x) are elementary functions, we ha.ve tbat for each n, ,.\ .=~£A¾...._, where f Ari;'/1. . (0 if i-4=j, and fV\(x)=f1,...,.if x is in cA.. '"". S imila.rly, F€ '°E iF.,.~, c.F.,,;\l ..j-=-O if i t j, and t "(x)::: fl\..,. if x is in c:F.,_. Definition I.,..,,: I.,..._:::\\ I [f.,. ( x) , A1 - I Lf.,J x) , A°\\I . Hence :£_..,~\II (f~ (x}, Ac1 -I l_f~(x), A£}\\ +\\ r [f"' ( x J , 1 - r Lt ,J x l , Fc: 1\\. FE By Definition 3 .3 I"W\~\l~:r.,Jf ) L -r f.~Jt\ . J\\ r\\ ~ t.Jl.J -~f.... \l~,\ll. J L J J J Let us consider the second term of the right member. By Theorem 3.2 we have that \\f,..{x)\l~ rp. \\z=. f_"'.\,, Fn-I1 - r_ t ..,J.\ ~F..,~111 _,\\~t\r.i l "I--l I::.F""1\\ +\\rJ t n,,J.\~F"'J.\\\I i. j C. ~ if (f\t J ~ 7\£-Fy..~\ ) -= 2 f \ F £ \ • (1). .-. \\~f_,,-\[J -f f i,,j\fi,J\\.c2 cpc: . Now consider the first term of the right member·. :. the right member becomes: it (2) V\ . VI,\ ";,,. V\ 0 l£, l \\ L:(f ,., - f ... jf: ~1\l!,f\\f., i - f,,_)\\f7ll~eI.\f'i7 \=t\ AE\~£\A\ • \ •~ ••~ I•~ Thus from ( 1) and ( 2} we have that Sinc e cp and \.A\ are finite, we ha.ve that the sequence ~ I [f \,\(xl, A1\ converges. But the choice of { f .Jx )~ was arbitrary. Hence every such sequence {I[f,..(x), A1\ converges, and by Theorem 3.3 to the same limit. Hence, ._the expression I[g(x), A1 , where g(x) is an elementary function can be written /g(x)dx. A Before proceeding with the properties of tije integral, we wish to prove a theorem on the measurability of continu@us ., functions. For this purpose we need a theorem proved by Frech~t in an article in the Bull. de la Soc. MatQ . de France (45). The statement of the theorem which we need is the following: If A is compact and closed, and if C\- is a family of sets I such that every element of A is an interior element of some I, then there exists a finite sub-family ~' of ~ same property. A._"''\ 'be '"'- ex"''\ -s\'='~c~ having the V. From this we have Theorem 3.5. If A is compact and closed and if f(x) is con- tinuous over A, then f(x) is measurable over A. Proof. ] 'b= ~( X, t.): .x ,XEA . \\x-x\\ £:~ ~.£):::>: \~f ( X )-f (x )\\..:: [. ., . the function ~( x,E.) determines a. family C\,as in the pr evious theorem. : . ) ~" a finite sub-family with the same property. That is 3 1x ..3tA such that , range of n finite, . xLtl : :) : 3 x " . \\ x-x,..\\«~ . \\f(x)-f(x) \\.:.E. . Definition g(x): Ux-x.._\\..:.~ .=. g(x) =f (x). Hence g(x) I: € n E. is an elementary function, which approaches f{x) everywhere in A as ~ approaches zero. Hence, any denumerable seq_uence from the set l gJ x)1 is an a :pprox- innting seq_uence for f(x). Henceforth we shall consider only bounded, measurable funQtions in a. set A of finite measure, unless the contrary is explicitly stated. Theorem 3.6. If f(x) and g(x) are elementary functions, then / ( f ( X ) ±;g ( X ) ) d.x = / f ( X ) dx ±. / g ( X ) d.x. A A A Proof: A =L, (A\J, where A' A'-=- 0 if mt n, and f (xh:f • = . "' ~ "' S imilarly, A-=-2._ A: , where 1( A::_0 if mt n, and \,-. A -= A' _f'·. Definition A : \.\ k... "" "" \,'\. I'\,'\. As i~ the proof cf Theorem 3.4, A ·, A:CO if mt n, or ""'- "'l if i -:J: j. Hence> S ince x t-A . -=. . f{x) =. f and g{x) =- g""' ,then ""'"' X E-A,_,! = • f{x) ~g(x) = f "'~g_,• .'. f f ( x ) dx :t f g ( x ) dx :::; [_ ( f "' ± g.,J \ A"'j: f (f ( x } ± g ( x ) ) dx • A A ~. ~ A Theorem 3.7. If f"(xl and g{x ) are measura.ble and b ounded over A, so is their sum, and ± fg ( x ) dx. /( f ( x ) t g {x ) } dx -=- / f { x ) dx A A A Proof: f {x) = lim. f "'" (x ) ~~~ and g(x) -= lim. g ,_{x) ~➔~ almost everywhere in A, where f ~ (x) and g "' (x ) are elementary functions as in Theorem 3.2. : . f(x) .-g{x) = limlf(x) - ... ....,c--0 "' ~ "(x}l almost everywhere in A. . . Af i' ( x ) dx -t A/ g ( x ) dx -=- 1"'....,"° im. f f ( x }dx ~ .Q / g ( x ) dx A "' ".,~ " = lim.{ /f (x)dx ± /g (x)dx ). \,\. -')c,0 A "' A "' By the preceeding theorem, / f {x)dx ~ /g{x)dx = lim. /(f {x) ± gi X) ) dX -: :. flfu)i9lY-,}¼ - A A \.-\ ➔"" A " A By mtherna.tical induction the preceding theorem can be extended to the case of the sum of any finite number of functions. Theorem 3.8. If f(x) is measurable and bounded, so is \\ f (x)\\, and. \\ [ f (x} dx \\ ~ [\\t ( x}\\ dx. A A Proof: Let {f,Jx)1 be an ap:proximting seq_uence. A=<[.. A ~ V\~ f ,J X )== f •, A~"""- A ~o if mtl t for any n. "'-.If.. •=-. X t:-A .___• i,,.,__ .•. l/ /t(x)a.x\(~ \\l im. ft"'(x)dx\\::::lim. \\ [f..(x)dx\\. A "'-1 ""' A """"".a A 7:: ~~~,:II E f._J 11__ j\ ~ 1..~;~;~Hfhl\A.. J ::; lim /\\fJ x)\\ dx = j \\f ( x )\\ dx. \l\ -:,o0 A A Theorem 3.9. J°'- f (x)dx = o< ff (x)dx. A A Proof: a../f(x)dx :c:...(lim. 1( f (x)d.x= lim. d.. f f .Jx)dx A ho --.>""' A "' "'..._, A. -::. lim.ot[ f ·\A }:::. limL.ifi. • \A,,,.J \,\. -=,IX) M \,\....::,o() -=- 1 i m. ~ /o( "- --:>"° A "'-"""- "'-~ ~ f ,J x ) dx = [ o( f ( x ) dx. A Theorem 3.10. If f is a particular element of B, then /tdx-:::: f· IAI. A Corollary. If ex is any rea.l number, then Proof obvious. Theorem 3.11. If \\ f(x)\\~ol.. , then \\/f(x)d.x \\:::<>l\A\ • A Proof: By Theorem 3.8., l\ff( x}d.x \l-E:, A ]\\f(x)\\ dx A :. \l/f(x)dx \\ ~ lim. [Hf ~(x) \\ dx -=- lim.Lll f \\ \A \ . A "'" ...... \iil.....,. oo o,c) \;'\,'I, 1tlo'\A 'A.k'\ Pt How, by Theorem 3.2, \lf ... J\~. ;. )l / f (x) d.x ll ~ lim. Lo(\ A "J=ol A\ .• A "'-_,oo w.. Theorem 3.12. If 1<x} is an integrable function on A to R, ~k~.a and if f (x) ~o( , then /cp(x)dx >✓--o<\A \. A Proof: /cp(x)dx ::: lim. A ......... "° / q,(x)dx :: lim.2-fi.,J A" ,,J A "' 1s. ....;, o0 "" ?: lim. L DZl A"'J ==ol\A\ • V\ -'>oO ""'- 4. Important Special Instances.(l). a. Saks. Saks defines a completely additive family of pointsets in a vector-space , and the measure of such sets by means of a. group of postula tes identical with our set (b). With him an elementary function is one whose set of values is denumerable at most, so that a function which is elementary according to our definition is a lso elementary according to his. He defines measurability as does Titchmarsh, but proves the theorem tha t every non-negative measurable function is the limit of a monotonic, non-decreasing se~uence (1). For references, see section 1. of measurable, non-negative finite functions each of which has only a finite number of values. This is obviously a special instance of our cla ss of measurable functi ons. He then :proceeds to o. ef ine an integra l exactly as we have done. Thus Theorem 4.1. If Bis R, then the integral of Definition 3.4 is an integral as defined by Saks, and conversely. b. Bochner. Bochner considers abstract valued functions of n real variables. Mea.sure bas already been defined for such spaces (cf. Hobson), and. satisfies our :postulates. Bochner defines an elementary function, a measurable function and an integral :precisely as we do. Theorem 4.2. If X is R~ , an n-dimensio:nal real space, the integral of Definition 3.4 is an integra l as defined. by Bochner, and conversely. c. Kerner. If X is R, we have i mrne dia tely from 'I1heorem 3 . 5 The orem 4.3. If f(x) is continuous throughout a closed. int erval, its nRiema1mn int egra l i s eq_ual to its int egra l. 11 Lebesguen