INROADS Rel Anlysis Exchnge Vol. 26(1), 2000/2001, pp. 381 390 Constntin Volintiru, Deprtment of Mthemtics, University of Buchrest, Buchrest, Romni. e-mil: cosv@mt.cs.unibuc.ro A PROOF OF THE FUNDAMENTAL THEOREM OF CALCULUS USING HAUSDORFF MEASURES Abstrct This note contins proof of the Fundmentl Theorem of Clculus for the Lebesgue-Bochner integrl using Husdorff mesures (see 2.4). For the rel cse (X = R), this proof uses only the bsics from the Lebesgue integrl theory (see 2.6). 1 Preliminries Throughout this pper, µ denotes the Lebesgue mesure on R, X will be n rbitrry fixed rel Bnch spce, nd the mesurbility is understood reltive to the Borel sets of R nd X respectively. We sy tht given f : [, b] X is µ-mesurble if there exists the µ-null Borel set A [, b] such tht f([, b]\a) cn be included in seprble subspce of X nd f restricted to [, b]\a is mesurble. By n integrl of µ-mesurble f : [, b] X we shll lwys men the the Lebesgue integrl or, to be more ccurte on Bnch spces context, the Lebesgue-Bochner integrl. Recll tht there exists (possibly infinite) integrl f(t) dt for every positive µ-mesurble function f : [, b] R + nd, for µ-mesurble f : [, b] X, one sys tht f is µ-integrble if nd only if f(t) dt < +, nd in this cse the integrl b f dt exists. Obviously, [, b] could be replced with n rbitrry intervl of rels. It hs to be pointed out tht prt from bsic definitions from integrtion theory, we shll use in the sequel lmost nothing but the following esy consequence of the Dominted Convergence Theorem. Key Words: Fundmentl Theorem of Clculus, Husdorff mesures Mthemticl Reviews subject clssifiction: 26A42, 26A46, 28A78 Received by the editors April 2, 1999 381
382 Constntin Volintiru Proposition 1.1. If f : [, b] X nd f n : [, b] X (n 1) is sequence of integrble functions such tht f n (t) f(t).e., then f is integrble nd f n (t) dt provided tht (f n ) n is uniformly integrble. f(t) dt, Note : Recll tht the uniform integrbility of (f n ) n mens f n (t) dt = 0 uniformly for n 1, lim k + f n k nd in this cse the fct tht [, b] is bounded intervl is essentil becuse the proof of the bove proposition needs finite mesure. All the bove re stndrd fcts in this field nd cn be found in lmost every mesure theory books see for exmple [2] nd [7]. A definition of the Husdorff mesure h α would go s follows: if (T, d) is ny metric spce, A T nd δ > 0, let Λ(A, δ) be the set of ll rbitrry collections (C) i of subsets of T, such tht A i C i nd dim (C i ) δ for every i. Now, for every α > 0 define { h δ α(a) def } = inf (dim Ci ) α (C i ) i Λ(A, δ). (1) Then there exists lim δ 0 h δ α(a) = sup δ>0 h δ α(a) nd h α (A) def = lim δ 0 h δ α(a) gives n outer mesure on P(T ) which is countble dditive on the σ-field of ll Borel subsets of T. If T = R n, the Husdorff mesure h n, restricted to the σ-field of the Borel subsets of R n, is identicl to the Lebesgue mesure on R n up to constnt multiple. In prticulr, h 1 (C) = µ(c) for every Borel set C R. With this remrk, the following proposition is the trnsltion of well known inequlity: Proposition 1.2. If C R is Borel set nd F : C X is Lipschitz mp (i.e f(x) f(y) x y whenever x, y C), then h 1 (F (C)) µ(c). Aprt from the bove proposition, the next result is the only one needed in this pper from the Husdorff mesure theory. See for exmple [4] or [3]. Proposition 1.3. If (T, d) is metric spce nd C T is connected, then dim (C) h 1 (C).
A Proof of the Fundmentl Theorem of Clculus 383 We conclude this section with some nottions. Throughout this pper we fix, for n rbitrry intervl [, b] with < b, sequence of divisions n = { = t (n) 0 < t (n) 1 < < t (n) k n 1 < t(n) k n = b } for n 1 (2) such tht n n+1 in the sense tht every t n i some t (n+1) j in n+1, nd from n cn be found s δ n def = k n 1 mx ( i=0 t(n) i+1 t(n) i ) 0. (3) For every g : [, b] X nd n 1, let n g = k n 1 i=0 g(t (n) i+1 ) g(t(n) n ) t (n) i+1 t(n) i 1 (n) [ t i,t (n) i+1 ), where, for ny C [, b], 1 C (t) = 1 if t C nd 1 C (t) = 0 if t C. Remrk 1.4. For the given F : [, b] X nd f : [, b] X, if the derivtive F (t) exists.e. nd F (t) = f(t).e., then n F (t) f(t).e. Indeed, if F (t) = f(t) nd t n n, i.e. lmost everywhere, then there exists n uniquely defined sequence of open intervls ( n, b n ) n 1 such tht n, b n re consecutive points in n nd t ( n.b n ) for every n. Since b n n 0 by 3, n F (t) = F (b n) F ( n ) b n n f (t). The finl im of this proof for FTC is to show tht under some conditions, ( n F ) n is uniformly integrble. 2 Min result We fix f : [, b] X, continuous function F : [, b] X, nd we ssume tht there exists Borel set B [, b] such tht: the derivtive F (t) exists nd F (t) = f(t) for every t [, b]\b.
384 Constntin Volintiru In the end, µ(b) = 0 will be n dditionl hypothesis, but this is of no consequence for the moment. Since f(t) = F (t) for t [, b]\b, f restricted to [, b]\b is mesurble, therefore f restricted to [, b]\b is mesurble, ny set of the form C = { t [, b]\b f(t) ρ } is Borel set, the (possibly infinite) integrl f(t) dt exists, nd so on. C Lemm 2.1. For ny fixed ρ > 0 there exists sequence of mutully disjoint Borel sets (C k ) k 1, such tht [, b]\b = k 1 C k (4) nd for every k 1, F (x) F (y) (x y) f(y) ρ x y whenever x, y C k. (5) Proof. For every fixed rtionl t Q denote by A t, the set of ll those y [, b]\b such tht y + t [, b] nd F (y + t) F (y) t f(y) ρ t. Since f restricted to [, b]\b is mesurble nd F is continuous, the mp y F (y + t) F (y) t f(y) is mesurble function on the set ([, b]\b) { y y + t [, b] }. It follows tht ech A t by 3, is Borel set nd, with the sequence (δ n ) n given B n def = t Q, t δ n A t is Borel set, s n intersection of countble fmily of Borel sets. Since F is continuous nd x = y + (x y), y B n if nd only if F (x) F (y) (x y) f(y) ρ x y (6) for every x [, b] such tht x y δ n,
A Proof of the Fundmentl Theorem of Clculus 385 nd from F (y) = f(y) for y [, b]\b, it follows [, b]\b = n 1 B n. Let E 1 = B 1, E 2 = B 2 \E 1,..., E n = B n \(E 1 E 2 E n 1 ),.... The Borel sets E n (n 1) re mutully disjoint, E n B n (n 1) nd B n = [, b]\b (7) n 1 E n = n 1 Finlly, for every n 1, let n be the division given by 2, nd define [ ) = E n, t (n) if 0 i < k n 1, E (i) n t (n) i i+1 nd E (kn 1) n [ ] = E n t (n) k, n 1 t(n) k n. The sequence (C k ) k 1 is the result of n rbitrry enumertion of the sets E n (i). Indeed, if C k = E n (i) for some n 1 nd 0 i k n 1, then x y δ n for every x, y C k, nd 5 is n esy consequence of 6. C k = E (i) n E n from 7 B n, Remrk 2.2. Within the hypothesis of the preceding lemm, since 5 is symmetric for ny fixed C k nd x, y C k, we hve F (y) F (x) f(x)(y x) ρ y x, therefore x y f(x) f(y) = f(y)(x y) f(x)(y x) F (x) F (y) f(y)(x y) + F (y) F (x) f(x)(y x) 2 ρ x y, hence f(x) f(y) 2 ρ for every x, y C k so, if we define k = inf { f(t) t C k }, then 0 k f(x) k + 2 ρ for every x C k. (8) Lemm 2.3. If (C k ) k 1 is the sequence given by lemm 2.1 for ρ > 0, then for every k 1, h 1 (F (C k )) 3 ρ µ(c k ) + f(t) dt. C k
386 Constntin Volintiru Proof. Let ( k ) k 1 be given by 2.2, nd fix some C k. If x, y C k, F (x) F (y) from 5 ( ρ + f(y) ) x y by 8 (3 ρ + k ) x y, hence h 1 (F (C k )) by 1.2 (3 ρ + k ) µ(c k ). (9) Since k f(t) for every t C k, with 1 Ck (t) = 1 for t C k nd 1 Ck (t) = 0 if t C k, we hve (3 ρ + k )1 Ck 3 ρ 1 Ck + f 1 Ck, therefore 3 ρ (3 ρ + k ) µ(c k ) = 1 Ck (t) dt + (3 ρ + k )1 Ck (t) dt f(t) 1 Ck (t) dt = 3 ρ µ(c k ) + f(t) dt C k. Finlly, h 1 (F (C k )) by 9 (3 ρ + k ) µ(c k ) 3 ρ µ(c k ) + f(t) dt. C k Proposition 2.4. (Fundmentl Theorem of Clculus) Suppose tht for the given f : [, b] X, there exists F : [, b] X, which is continuous, the derivtive F (t) exists nd F (t) = f(t) outside µ-null Borel set B [, b] such tht h 1 (F (B)) = 0. (10) Then f is µ-mesurble nd if we ssume the integrbility of f, F (b) F () = f(t) dt. (11) Proof. Let F : [, b] X nd B [, b] s bove. Recll tht the integrbility of the (µ-mesurble) f : [, b] X is equivlent to f(t) dt < +. (12)
A Proof of the Fundmentl Theorem of Clculus 387 Let (C k ) k 1 be the sequence given by lemm 2.1 for n rbitrry ρ > 0. Since µ(b) = 0 nd the sets C k re mutully disjoint, b = k 1 µ(c k) nd h 1 (F ([, b])) = h 1 F (B) F (C k ) k 1 h 1 (F (C k )) k 1 for every ρ > 0, therefore by 2.3 3 ρ (b ) + k 1 h 1 (F ([, b])) [ 3 ρ µ(c k ) + f(t) dt < + h 1(F (B))=0 by 10 ] f(t) dt C k f(t) dt. (13) By the continuity of F, the set F ([x, y]) is connected in X, hence F (b) F () dim (F ([, b])) by 1.3 h 1 (F ([, b])), which, tking into ccount 13, gives: F (b) F () f(t) dt. (14) Since the sme rgument s bove cn be pplied for every intervl [x, y] [, b] insted of [, b], if we define ϕ : [, b] R +, by from 14 we hve ϕ(x) = x f(t) dt for x b, (15) F (y) F (x) ϕ(y) ϕ(x) for x < y b. With the nottions from the lst prt of section 1, it follows tht n F (t) n ϕ(t) for every t [, b]. (16) As consequence of the definition 15, ( n ϕ) n is uniformly integrble, so ( n F ) n is uniformly integrble by 16 nd n F (t) f(t).e. by 1.4. It
388 Constntin Volintiru remins to observe tht f(t) dt [ kn 1 = lim (F (t (n) n i+1 ) F (t(n) i=0 i ) by 1.1 = lim n F (t) dt = n ] = lim n [ F (b) F () ] = F (b) F (). The choice of bounded intervl [, b] is due only to the fct tht this includes the hrd prt of the bove proofs. Since n rbitrry intervl of rels is the union of sequence of bounded intervls, the following proposition is the result of 2.4 vi some stndrd rguments: Proposition 2.5. Let I R be n rbitrry intervl, with u = sup I nd v = inf I. Suppose tht for the given f : I X, there exists F : I X, which is continuous, the derivtive F (t) exists nd F (t) = f(t) outside µ-null Borel set B I such tht h 1 (F (B)) = 0. Then f is µ-mesurble nd if f is integrble, then F (v 0), F (u + 0) X exist nd F (v 0) F (u + 0) = v u f(t) dt. (17) Remrk 2.6. The previous propositions include the sitution X = R, in which cse the condition h 1 (F (B)) = 0 is equivlent with µ(f (B)) = 0, where µ stnds for the Lebesgue outer mesure on R. In this cse n explicit use of the Husdorff mesure theory is not necessry in the bove proof the inequlity from 1.2 becomes µ((f (C)) µ(c) nd, insted of 1.3, it suffices to observe tht ny connected subset C R is lwys n intervl, so dim (C) = µ(c). With this observtion, the entire rel cse proof for FTC presented in this pper becomes fully ccessible in n introductory course of mesure theory. Remrk 2.7. In the cse of positive f : [, b] R + the function F from 2.4 is monotone nd the integrbility of f is the result of the inequlity f(t) dt F (b) F () < +. Consequently, in the proposition 2.5, if f : I R +, the integrbility condition is not necessry nd 17 holds, with F (v 0) F (u + 0) = + being possibility. It should be pointed out tht this prticulr cse cn be viewed s n independent result coming from nother ides. If follows from lem 2.1, pplied to F for fixed ρ > 0, tht F restricted to every C k is Lipschitz mp, therefore µ(f (C)) = 0 for every µ-null set C [, b], s consequence of the condition
A Proof of the Fundmentl Theorem of Clculus 389 µ(f (B)) = 0, i.e. F verifies the Luzin condition (N), hence F is bsolutely continuous by Bnch-Zrecki Theorem ( see for exmple [1] ). Finlly, 11 holds s direct consequence of the Lebesgue-Rdon-Nikodym Theorem. 3 Comments The uthor of this note hppens to be fond of Husdorff mesures, but hs no serious informtions on the FTC reserch re, nd this proof hs been found lmost by ccident. Although for the rel cse (X = R), 2.4 cnnot be explicitly found in Sks s book (see [6]), it is there, somewht between the lines, ultimtely s consequence of the Theorem 7.7, pge 285. Perhps becuse of the fct tht the rguments given in [6] re full of techniclities, pprently 2.4 becme lmost bsent in the subsequent generl presenttions of the Lebesgue integrl. The cse B =, i.e. F (t) exists everywhere in [, b], is much better known see [5] for exmple. The uthor wishes to thnk to professor Brin S. Thompson for the observtion ( mde in the rel-nlysis miling list ) tht 2.4 cn be somehow found in Sks s book, nd to the referee for mny helpful remrks mde on the previous version of this pper. References [1] A. M. Bruckner, B. S. Thompson, J. Bruckner, Rel Anlysis, Prentice Hll, 1996. [2] J. Diestel, J.J.Uhl Jr, Vector Mesures, Mth. Surveys 15, Amer. Mth. Soc. 1977. [3] G. A. Edgr, Mesure, Topology, nd Frctl Geometry, Springer-Verlg, 1990. [4] G. Federer, Geometric Mesure Theory, Springer-Verlg, 1968. [5] W. Rudin, Rel nd Complex Anlysis, third edition. McGrw-Hill, 1987. [6] S. Sks, Theory of the Integrl, Hfner Publishing Co., 1937. [7] E. Schechter, Hndbook of Anlysis nd its Foundtions, Acdemic Press, 1996.
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