Fundamental Theorem of Calculus for Lebesgue Integration


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1 Fundmentl Theorem of Clculus for Lebesgue Integrtion J. J. Kolih The existing proofs of the Fundmentl theorem of clculus for Lebesgue integrtion typiclly rely either on the Vitli Crthéodory theorem on pproximtion of Lebesgue integrble functions by semicontinuous functions (s in [3, 9, 12]), or on the theorem chrcterizing incresing functions in terms of the four Dini derivtes (s in [6, 10]). Alterntively, the theorem is derived using the Perron or the Kurzweil Henstock integrl nd its reltion to the Lebesgue integrl (see [5, 8]). In this note we give proof of the theorem which uses only stndrd results of the Lebesgue mesure nd integrtion without resorting to ny extrneous mteril. Two of these results, the theorem tht n bsolutely continuous function with derivtive equl to zero lmost everywhere is constnt, nd Lebesgue s theorem on differentition of monotonic functions, hve received n elegnt tretment by elementry mens in this Monthly in the hnds of Michel Botsko [1, 2]. To simplify formultions we employ the following often used terminology. A sttement is true nerly everywhere in S R if it is true in S except for countble subset of S. The ide for the proof of the following key lemm comes from [7]. Lemm 1. Let F :[, b] C be continuous on [, b], let f :[, b] C be Lebesgue integrble on [, b], nd let F (t) =f(t) nerly everywhere in [, b]. Then F (b) F () f(t) dt. (1) Proof. Let F (t) =f(t) for ll t A =[, b] \ D, where D is countble. We my ssume tht the (one sided) derivtives exist t the end points of [, b], otherwise we consider intervls [ n,b n ] with this property, nd stisfying [ n,b n ] [, b]. Let ε>0 be given. Set c i = iε/(b ), i =0, 1, 2,..., nd define E i = {t A : c i 1 f(t) <c i },i N. Since f nd f re Lebesgue integrble on [, b], the sets E i re Lebesgue mesurble, nd A is the disjoint union of the E i. Hence the Lebesgue mesure of A is m(a) = b = i=1 m(e i), nd c i 1 m(e i ) f(t) dt c i m(e i ), i N, E i 1
2 which gives 0 c i m(e i ) f(t) dt ε E i b m(e i). From the countble dditivity of the Lebesgue integrl we conclude tht c i m(e i ) f(t) dt + ε. (2) i=1 For ech i N there exists bounded open set G i R such tht G i E i nd m(g i ) m(e i )+c 1 i ( 1 2 )i ε, i N. (3) Define functions H, M :[, b] R by H() =M() = 0 nd H(t) = c i m(g i [, t]), M(t) = ( 1 2 )j ε, < t b, (4) i=1 u j [,t) where {u j : j N} is n enumertion of D. Both H nd M re incresing. Let x be the supremum of ll t [, b] such tht F (t) F () H(t)+M(t). For proof by contrdiction ssume tht x<b. Suppose first tht x A. Then x E k for some k N, nd from f(x) <c k it follows tht there exists x 1 (x, b) such tht [x, x 1 ] G k nd F (x 1 ) F (x) <c k (x 1 x). Since F is continuous, F (x) F () H(x)+M(x). Then F (x 1 ) F () F (x 1 ) F (x) + F (x) F () c k (x 1 x)+h(x)+m(x), while H(x)+c k (x 1 x) H(x 1 ). Then F (x 1 ) F () H(x 1 )+M(x 1 ), which contrdicts the definition of x. Suppose tht x D. Then x = u m for some m N. Since F is continuous, there exits x 2 (x, b) such tht F (x 2 ) F (x) < ( 1 2 )m ε, nd F (x 2 ) F () F (x 2 ) F (x) + F (x) F () ( 1 2 )m ε + H(x)+M(x); since M(x) +( 1 2 )m ε M(x 2 ), we hve F (x 2 ) F () H(x 2 )+M(x 2 ), which gin contrdicts the definition of x. This proves tht x = b. Hence, by (2), (3) nd (4), F (b) F () H(b)+M(b) f(t) dt +3ε. Since ε ws rbitrry, (1) holds. 2
3 Theorem 1. (Fundmentl theorem of clculus.) Let F :[, b] C be continuous on [, b], let f :[, b] C be Lebesgue integrble on [, b], nd let F (t) =f(t) nerly everywhere in [, b]. Then F is bsolutely continuous on [, b], nd f(t) dt = F (b) F (). (5) Proof. By Lemm 1, F (v) F (u) v u f(t) dt for ny subintervl [u, v] of[, b]. Since the Lebesgue integrl is bsolutely continuous, so is F, nd (5) holds by Lebesgue s theorem on integrtion of derivtives of bsolutely continuous functions. Let <b. We sy tht complex vlued function f is Newton integrble on (, b) if there exists complex vlued function F ( generlized primitive of f) continuous on (, b) such the F (t) =f(t) nerly everywhere in (, b), nd such tht the one sided limits F (+), F (b ) exist. The function f is bsolutely Newton integrble on (, b) ifbothf nd f re Newton integrble on (, b). The complex number F (b ) F (+) is the Newton integrl of f on (, b), written (N ) f = F (b ) F (+). The definition of the Newton integrl is independent of the choice of generlized primitive: This is gurnteed by the following well known result (see, for instnce, (8.5.1) in [4]), from which it follows tht two generlized primitives to f differ by constnt. Lemm 2. Let < b, let F : (, b) R be continuous, nd let F (t) 0 nerly everywhere on (, b). Then F is incresing on (, b). An elementry proof of this lemm in the spirit of Thomson [11] nd Botsko [1] cn be bsed on properties of full covers of [, b]. Theorem 2. Let <b.iff :(, b) C is both Newton nd Lebesgue integrble on (, b), then f(t) dt =(N ) f. (6) Proof. Choose sequence [ n,b n ] of subintervls of (, b) such tht [ n,b n ] (, b), nd set f n = fχ [n,bn], where χ [n,bn] is the chrcteristic function of [ n,b n ]. Then f n f pointwise on (, b), nd f n f for ll n N. By Lebesgue s dominted convergence theorem nd by Theorem 1, f(t) dt = lim n n n f(t) dt = lim n (F (b n) F ( n )) = F (b ) F (+). 3
4 We show tht n bsolutely Newton integrble complex vlued function is lso Lebesgue integrble, nd the two integrls re consistent. Theorem 3. Let <b nd let f :(, b) C be bsolutely Newton integrble on (, b). Then f is Lebesgue integrble, nd f(t) dt =(N ) f. (7) Proof. Assume first tht f is nonnegtive. By Lemm 2, generlized primitive F to f is n incresing function on (, b) nd so, by Lebesgue s theorem on differentition of monotonic functions, f is Lebesgue integrble on ny compct subintervl of (, b). Choose sequence [ n,b n ] of subintervls of (, b) such tht [ n,b n ] (, b). By Theorem 1, 0 n n f(t) dt = F (b n ) F ( n ) F (b ) F (+). Writing f n = fχ [n,b n], we hve f n f, nd the monotonic convergence theorem ensures tht f is Lebesgue integrble on (, b). For the generl cse we observe tht the complex vlued function f is Lebesgue mesurble s it is the limit of continuous functions F n (t) =n(f (t + 1 n ) F (t)) convergent nerly (nd therefore lmost) everywhere in (, b). By the first prt of the proof, f is Lebesgue integrble on (, b). Then so is f, nd Theorem 2 pplies to complete the proof. References [1] M. W. Botsko, The use of full covers in rel nlysis, Amer. Mth. Monthly 96 (1989), [2], An elementry proof of Lebesgue s differentition theorem, Amer. Mth. Monthly 110 (2003), [3] D. L. Cohn, Mesure Theory, Birkhäuser, Boston, [4] J. Dieudonné, Foundtions of Modern Anlysis, 2nd corrected nd enlrged edition, Acdemic Press, New York, [5] R. A. Gordon, The Integrls of Lebesgue, Denjoy, Perron nd Henstock, GSM 4, Amer. Mth. Soc., Providence, [6] E. Hewitt nd K. Stromberg, Rel nd Abstrct Anlysis, Springer, Berlin, [7] J. J. Kolih, Lebesgue through Newton integrl, Austrl. Mth. Soc. Gz. 30 (2003),
5 [8] P. Y. Lee nd R. Výborný, Integrl: An Esy Approch fter Kurzweil nd Henstock, Austrlin Mthemticl Society Lecture Series 14, Cmbridge University Press, Cmbridge, [9] W. Rudin, Rel nd Complex Anlysis, 3rd ed., McGrwHill, New York, [10] C. Swrtz, Mesure, Integrtion nd Function Spces, World Scientific, Singpore, [11] B. S. Thomson, On full covering properties, Rel Anlysis Exchnge 6 ( ), [12] P. L. Wlker, On Lebesgue integrble derivtives, Amer. Mth. Monthly 84 (1977), The University of Melbourne, VIC 3010, Austrli 5
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