THE FUNDAMENTAL THEOREM OF CALCULUS FOR MULTIDIMENSIONAL BANACH SPACE-VALUED HENSTOCK VECTOR INTEGRALS

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1 Real Analyss Exchange Vol.,, pp Márca Federson, Insttuto de Matemátca e Estatístca, Unversdade de São Paulo, R. do Matão 1010, SP, Brazl, e-mal: marca@me.usp.br THE FUNDAMENTAL THEOREM OF CALCULUS FOR MULTIDIMENSIONAL BANACH SPACE-VALUED HENSTOCK VECTOR INTEGRALS Abstract In the present paper we gve the Fundamental Theorem of Calculus for the varatonal or Henstock vector ntegrals KR α df and KR dα f of R R multdmensonal Banach space-valued functons. Introducton In [1, Proposton 3.2], Bongorno and D Pazza gave a characterzaton of the functons whch are Kurzwel-Henstock vector ntegrals of the form ht = K f dg K for the Kurzwel ntegral consderng the one-dmensonal real-valued case. They state that: a f g and F belong to ACG [a, b] and f : [a, b] R s such that F t = ftg t for m-almost every t [a, b] m for the Lebesgue measure, then f s Kurzwel-Henstock ntegrable wth respect to g we wrte f H g [a, b] and for every t [a, b], F t = K [a,t] f dg. And recprocally, b f g ACG [a, b] and f H g [a, b], then f g ACG [a, b], where f g t = K [a,t] f dg for each t [a, b], and there exsts f g t = ftg t for m-almost every t [a, b]. Key Words: Banach Space-Valued functons, Henstock Integral, Fundamental Theorem of Calculus Mathematcal Revews subject classfcaton: 26A39, 26B12 Receved by the edtors January 1, 1999 I thank Prof. Dr. Marna Pzzott for her careful readng of a great part of the prevew. [a,t] 469

2 470 Márca Federson In order to prove b, Bongorno and D Pazza use the result K fg = K f dg referrng manly to [5, p. 186]. However, n ths reference, or else n [5, p. 186], McLeod affrms that n general K f h = K f d h, where ht = K h. But does not always hold. In [2, p. 37], a counterexample for Banach space-valued functons s gven and we present t n Ex- [a,t] ample 1 below. In the real case t s even possble that [a,t] h [a,t] d h for m-almost every t [a, b]. It suffces to take, for nstance, h : [0, 1] R defned by ht = 1/q f t = p/q and p N such that q 0, q/p and p/q, and ht = 0 otherwse. For condtons n whch holds, the reader may want to consult [2] or [3]. In the present paper we gve the Fundamental Theorem of Calculus for the varatonal or Henstock vector ntegrals K R α df and K dα f of multdmensonal Banach space-valued R functons. 1 Basc Termnology For smplcty of proofs and notaton, we consder only the two-dmensonal case. Let X and Y be Banach spaces and f : R X be a functon defned n a compact nterval R R 2 wth sdes parallel to the coordnate axes. Gven t, s R 2 wth t s.e., t s, = 1, 2, we denote by [t, s] the correspondng closed nterval and we wrte [t, s] = m[t, s], where m denotes the Lebesgue measure. Any fnte set of closed nonoverlappng ntervals J of R such that J = R s called a dvson of R and denoted by J. A tagged dvson of R s a par d = ξ, J, where J s a dvson of R and ξ J for every. Each ξ s called the tag of J. We denote by T D R the set of all tagged dvsons of R. A tagged partal dvson d of R s any subset of a tagged dvson of R and we wrte d T P D R. A gauge of a set E R s a functon δ : E ]0, [ and d = ξ, J T P D R s δ-fne f for each, J B δξξ = { t R ; t ξ < δξ }.

3 The Fundamental Theorem of Calculus 471 Let J be a closed nterval wth sdes h and k, h k. Gven 0 < c < 1, J s sad to be c-regular f h/k c and d = ξ, J T P D R s c-regular f each J s c-regular. Let I R be the set of all closed ntervals contaned n R. A functon F : I R X s called addtve on ntervals we wrte F AI R, X f we have that F J = F J 1 + F J 2 for any ntervals J, J 1 and J 2, wth J 1 and J 2 nonoverlappng and J = J 1 J 2. A functon F AI R, X satsfes the regular Strong Lusn Condton on R we wrte F r SLI R, X, f for every ɛ > 0, every 0 < c < 1 and every E R wth me = 0 there s a gauge δ of E such that for every c-regular δ-fne d = ξ, J T DP R wth ξ E for each, we have that F J < ɛ. Let 0 < c < 1. We say that F AI R, X s c-dfferentable at ξ R and that fξ s ts c-dervatve we wrte D c F ξ = fξ, f for every ɛ > 0, there s a neghborhood V of ξ such that for each c-regular J I R wth ξ J V, we have F J fξ J < ɛ J. If D c F ξ = fξ for every 0 < c < 1, then F s regularly dfferentable at ξ R wth fξ beng ts regular dervatve we wrte r DF ξ = fξ. We say that F s c-dfferentable at R when F s c-dfferentable at ξ R for every ξ R, and that F s regularly dfferentable at R when F s regularly dfferentable at ξ R for every ξ R. Let LX, Y denote the space of all contnuous functons from X to Y. A functon α A I R, LX, Y s weakly c-dfferentable at R f, for every x X, the functon α x : J I R αj x Y s c-dfferentable at R and we wrte α D c σ I R, LX, Y. If there exsts D c σ α xξ for every c and every x, we say that α s weakly regularly dfferentable at ξ R. We wrte α r D σ I R, LX, Y, f α s weakly regularly dfferentable at ξ R, for each ξ R. A functon f : R X s regularly Henstock ntegrable wth respect to α A I R, LX, Y we wrte f r H α R, X, f there exsts a functon F α AI R, Y such that for every ɛ > 0 and every 0 < c < 1, there s a gauge δ of R such that for every c-regular δ-fne d = ξ, J T D R we have that F α J αj fξ < ɛ. If αt = t, then we wrte smply r HR, X and F nstead of r H α R, X and F α respectvely. In an analogous way, a functon α : R LX, Y s regularly Henstock ntegrable wth respect to f AI R, X we wrte α r H f R, LX, Y, f there exsts a functon A f AI R, Y such that for every ɛ > 0 and every

4 472 Márca Federson 0 < c < 1, there s a gauge δ of R such that for every c-regular and δ-fne d = ξ, J T D R, we have that A f J αξ fj < ɛ. More generally we have: a functon α : R LX, Y s regularly Kurzwel ntegrable wth respect to f AI R, X we wrte α r K f R, LX, Y f there exsts I Y we wrte I = rk α df such that for every ɛ > 0 and [a,t] every 0 < c < 1, there s a gauge δ of R such that for every c-regular and δ-fne d = ξ, J T D R, we have that I αξ fj < ɛ. Analogously, a functon f : R X s regularly Kurzwel ntegrable wth respect to α A I R, LX, Y we wrte f r K α R, X, f there exsts I Y we wrte I = rk dα f such that for every ɛ > 0 and every 0 < c < 1, there [a,t] s a gauge δ of R such that for every c-regular and δ-fne d = ξ, J T D R, we have that I αj fξ < ɛ. Let R = [a, b]. If α r K f R, LX, Y, then we defne αf t = rk [a,t] α df for each t R. And, analogously, gven f r K α R, X, we defne f α t = rk [a,t] dα f for each t R. If αt = t, then we smply wrte r KR, X and ft = rk [a,t] f. We may assocate F AI R, Y wth a functon from R to Y whch we stll denote by F : for R = [a, b] = [a 1, b 1 ] [a 2, b 2 ] we wrte F t = F [a, t] F [a, a 1, t 2 ] F [a, t 1, a 2 ] + F [a, a]. Recprocally, we may assocate a functon f : R Y wth a functon of ntervals of R whch we also denote by f : I R X. In ths case we wrte f [t, s] = fs ft 1, s 2 ft 2, s 1 + ft. Thus, when f r H α [a, b], X, then F α [a, t] = f α t for each t [a, b], and analogously, for α r K f [a, b], LX, Y, we have that Af [a, t] = αf t for each t [a, b], and therefore we can talk about regular Strong Lusn Condton and regular dfferentablty of an ndefnte ntegral f α or α f.

5 The Fundamental Theorem of Calculus 473 Remark. When X s of fnte dmenson, then r H α [a, b], X = r K α [a, b], X and r H f [a, b], LX, Y = r K f [a, b], LX, Y and such spaces are called spaces of regularly Kurzwel-Henstock ntegrable functons or spaces of Mawhn ntegrable functons. Example 1. Let X = l 2 [a, b] and Y = R. Let f : [a, b] X be defned by ft = e t.e., e t s = 1 f s = t, and e t s = 0 f s t and let α : [a, b] X = LX, R be gven by αt = ẽ t, where ẽ t x = e t, x, for every x X. Then αtft = e t, e t = 1 and therefore K αtft dt = dt = b a, where denotes the Remann ntegral. On the other hand, gven ɛ > 0, there exsts δ > 0, say δ 1 ɛ 2 <, b a 1 2 such that for every d = ξ, t T D wth max {t t 1 } < δ, we have that fξ t t 1 = [ e ξ t t 1 = t t 1 2] 1 2, 1a where we have used Bessel s equalty. But, [ t t 1 2] 1 2 [ ] < δ 1 2 t t = δb a 2 < ɛ. 1b Hence 1a and 1b mply that f = 0, and so αt d ft = 0. Now, f [a, b] s a non-degenerate nterval, then 0 < b a = K αtft dt αt d ft = 0.

6 474 Márca Federson 2 Man Results For the Henstock vector ntegral rk α df we have: [a,t] Theorem 1. Let f r SLI R, X and A r SLI R, Y be both regularly dfferentable on R, and let α : [a, b] LX, Y be such that r DA = α r DF m-almost everywhere on R. Then α r H f R, LX, Y and A = αf. Theorem 2. Let f r SLI R, X be regularly dfferentable on R and α r H g R, LX, Y be bounded. Then αf r SLI R, Y and there exsts r D α f = α r Df m-almost everywhere on R. And, for the Henstock vector ntegral rk dα f we have: [a,t] Theorem 3. Let α r SL I R, LX, Y r D σ I R, LX, Y, let F r SLI R, Y be regularly dfferentable on R and let f : R X be such that r DF t = r D σ α ft t for m-almost every t R. Then f r H α R, X and F = f α. And recprocally: Theorem 4. If α r SL I R, LX, Y r D σ I R, LX, Y and f r H α R, X then f α r SLI R, Y and there exsts r D f α t = r D σ α ft t for m-almost every t R. 3 Proofs Frst we prove the results for the Henstock vector ntegral rk [a,t] α df. Theorem 5. Let f r SLI R, X and α : R LX, Y such that α = 0 m-almost everywhere. Then α r H f R, LX, Y and αf = 0. Proof. Let E = {t R; αt 0} and E n = {t E; n 1 < αt n} for each n N. By hypothess, me = 0. Therefore me n = 0 for every n. Snce f r SLI R, X, then gven n N, ɛ > 0 and 0 < c < 1, there s a gauge δ n of E n such that for every c-regular δ n -fne d n = ξ n, J n T P D R wth ξ n E n for every, we have that fjn < ɛ n 2. n Let δ be a gauge of R such that f ξ E n then δξ = δ n ξ, and f ξ / E n then δξ takes any value n ]0, [. Hence, for every c-regular δ-fne d = ξ, J T D R, αξ fj = αξ fj n fj < ɛ. n ξ E n n ξ E n

7 The Fundamental Theorem of Calculus 475 Corollary. If f r SLI R, X, α r H f R, LX, Y and β : [a, b] LX, Y wth β = α m-almost everywhere, then β r H f R, LX, Y and βf = α f. The next example shows us that the hypothess of f n Theorem 5 s really needed. Example 2. Even n the one-dmensonal case when the regularty s not used, we may have that f f does not satsfy the Strong Lusn Condton, then α may not be Henstock ntegrable wth respect to f. Take, for nstance, an nterval [a, b] of the real lne wth a 1, and consder the context of Example 1. Consder arbtrary ξ and [t 1, t ] [a, b] such that ξ [t 1, t ]. Then ft ft 1 2 = et e t 1 2 = t 2 + t 1 2 > a 2 + a 2 > 1, and hence f / SL[a, b], X. We also have that αξ [ ft ft 1 ] = 0 for ξ ]t 1, t [ and αξ [ ft ft 1 ] 0 otherwse, and therefore α does not belong to H f [a, b], LX, Y and nether to Kf [a, b], LX, Y. Proof of Theorem 1. 1 Let E = {t R; there exsts r DAt = αt rdf t}. Hence, gven ɛ > 0, 0 < c < 1 and ξ E, there s a neghborhood V 1 of ξ such that for every closed c-regular nterval J R, wth ξ J V 1, AJ αt cdft J < ɛ J. 2 By hypothess, f r SLI R, X and mr \ E = 0. Therefore, by the Corollary after Theorem 5, we may suppose that αt = 0 for every t R \ E. 3 Snce mr \ E = 0 and A r SLI R, Y, there s a gauge δ of R \ E such that for every c-regular δ -fne d = ξ, J T P D R wth ξ R \ E, AJ < ɛ. 4 Because f s regularly dfferentable on R and hence c-dfferentable on R, there s a neghborhood V 2 of ξ such that for every c-regular J R, wth ξ J V 2, and we have that αξ fj αt cdf t J < ɛ J. 5 Fnally, let δ be a gauge of R such that B δξ ξ V 1 V 2 for each ξ E, and such that f ξ R \ E, then δξ δ ξ and B δξ ξ V 2. Hence for every c-regular δ-fne d = ξ, J T D R, t follows that AJ αξ fj

8 476 Márca Federson AJ αξ fj + ξ E ξ R\E AJ + ξ R\E αξ fj, where the frst summand s smaller than 2ɛ J = 2ɛ R from 1 and 4. The second summand s smaller than ɛ from 3, and the thrd summand s equal to zero because we have from 2 that fξ = 0 for each. Remark. In Theorem 1, we can requre that A r SLI R, Y s regularly dfferentable m-almost everywhere on R. Lemma 6 Saks-Henstock Lemma. Gven f AI R, X, α r H f R, LX, Y, let 0 < c < 1, ɛ > 0, and δ be the gauge of R from the defnton of α r H f R, LX, Y. Then for every c-regular δ-fne d = ξ, J T P D R we have that αξ fj J rk α df < ɛ. Proof. The proof follows the standard steps. Theorem 7. If f r SLI R, X and α r H f R, LX, Y, then αf r SLI R, Y. Proof. Let E R be such that me = 0 and let β = αχ R\E. Then by the Corollary after Theorem 5, β r H f R, LX, Y and βf = α f. Therefore, gven ɛ > 0 and 0 < c < 1, let δ be the gauge of R from the defnton of β r H f R, LX, Y. Then, from the Saks-Henstock Lemma Lemma 6, t follows that for every c-regular δ-fne d = ξ, J T P D R, wth ξ E for each, we have that αf J = αf J βξ j fj < ɛ, snce β f = α f and βξ = 0 for every. Lemma 8. See [4, Theorem 2.2]. If g r HR, X, then there exsts r D g = g m-almost everywhere on R. Theorem 9. Let f r SLI R, X be regularly dfferentable and let α r H f R, LX, Y be bounded. Then the functon t R αt rdft Y s regularly Henstock ntegrable wth rk α rdf = rk α df. R Besdes, there exsts r D α f = α rdf m-almost everywhere on R. R

9 The Fundamental Theorem of Calculus 477 Proof. 1 Gven ɛ > 0, let δ 1 be the gauge of R from the defnton of α r H f R, LX, Y. 2 Snce f s regularly dfferentable, then gven 0 < c < 1 and ξ R, there s a neghborhood V ξ of ξ such that for each closed c-regular nterval J R, wth ξ J V ξ, we have that fj c Dfξ J < ɛ J. 3 Let δ be a gauge of R such that for each ξ R, δξ δ 1 ξ and B δξ ξ V ξ. Hence, for every c-regular δ-fne d = ξ, J T D R, we have that αf J αξ cdfξ J α f J αξ fj + αξ fj c Dfξ J, where the frst summand s smaller than ɛ by 1, the second summand s smaller than ɛ α R for the supremum norm by 2 and the boundedness of α, and the frst part of the theorem holds. The second part comes mmedately from Lemma 8. Proof of Theorem 2. Ths comes mmedately from Theorems 5, 7 and 9. Now we treat the Henstock vector ntegral rk dα f. In general, the proofs R for rk R dα f are analogous to those for rk α df. However, the reader may R want to have a look at Theorem 13 whch, unlke Theorem 9, does not need the boundedness hypothess of the Henstock vector ntegrable functon. Ths fact allows Theorem 4 to be precsely the recprocal of Theorem 3 note that Theorems 1 and 2 are not the recprocal one of another. Theorem 10. Let α r SL I R, LX, Y and f : R X wth f = 0 m- almost everywhere. Then f r H α R, X and f α = 0. Proof. Let E = {t R; ft 0} and E n = {t E; n 1 < ft n} for each n N. By hypothess, me = 0. Therefore me n = 0 for every n. Snce α r SL I R, LX, Y, then for each n, gven ɛ > 0 and 0 < c < 1, there s a gauge δ n of E n such that for every c-regular δ-fne d n = ξ n, J n T P D R wth ξ n E n for each, we have that αj n < ɛn2 n. Let δ be a gauge of R such that f ξ E n then δξ = δ n ξ, and f ξ / E n then δξ takes any value n ]0, [. Hence for every c-regular δ-fne d = ξ, J T D R, αj fξ = αj fξ n n ξ E n n ξ E n αj < ɛ.

10 478 Márca Federson Example 3. Even n the one-dmensonal case, f α does not satsfy the Strong Lusn Condton, then f may not be Henstock ntegrable wth respect to α. In the context of Example1, we have that αt αt 1 = ẽt ẽ t 1 = { = sup ẽ t x ẽ x t 1 } ẽ t e t ẽ e t 1 t = 1, x 1 for arbtrary ξ and [t 1, t ] such that ξ [t 1, t ] [a, b]. Hence α / SL [a, b], LX, Y. We also have that [ αt αt 1 ] fξ = 0 for ξ ]t 1, t [ and [ αt αt 1 ] fξ 0 otherwse, and so, f / H α [a, b], X. Corollary. Gven α r SL I R, LX, Y, f r H α R, X and a functon g : R X such that g = f m-almost everywhere, then g r H α R, X and g α = f α. Proof of Theorem 3. 1 Let E = { t R; there s r DF t = r D σ α ft t }. Hence, gven 0 < c < 1, ɛ > 0 and ξ E, there exsts a neghborhood V 1 of ξ such that for every c-regular J I R wth ξ J V 1, F J D c σ α fξ ξ J < ɛ J. 2 By hypothess, α r SL I R, LX, Y and mr \ E = 0, therefore we may suppose that ft = 0 for every t R \ E by the Corollary after Theorem Snce mr \ E = 0 and F r SLI R, Y, there s a gauge δ of R \ E such that for every c-regular δ-fne d = ξ, J T P D R wth ξ R \ E for each, we have that F J < ɛ. 4 Because α r D σ I R, LX, Y, there s a neghborhood V 2 of ξ such that for every c-regular J I R wth ξ J V 2, αjfξ D c σ α fξ ξ J < ɛ J. 5 Fnally, let δ be a gauge of R such that B δξ ξ V 1 V 2 for ξ E, and such that f ξ R \ E then δξ δ ξ and B δξ ξ V 2. Hence, for every c-regular δ-fne d = ξ, J T D R, t follows that F J αj fξ F J αj fξ + ξ E ξ R\E F J + ξ R\E αj fξ,

11 The Fundamental Theorem of Calculus 479 where the frst summand s smaller than 2ɛ J = 2ɛ R from 1 and 4, the second summand s smaller than ɛ from 3, and the thrd summand s equal to zero because fξ = 0 from 2. Lemma 11 Saks-Henstock Lemma. Gven α A I R, LX, Y, f r H α R, X, let 0 < c < 1, ɛ > 0, and δ be a gauge of R from the defnton of f r H α R, X. Then for every c-regular δ-fne d = ξ, J T P D R, we have that αj fξ J rk dα f < ɛ. Proof. The proof follows the standard steps. Theorem 12. If α r SL I R, LX, Y and f r H α R, X, then f α r SLI R, Y. Proof. Let E R be such that me = 0 and let g = fχ R\E. Then, by the Corollary after Theorem 10, g r H α R, X and for each t R, we have that g α = f α. Gven 0 < c < 1 and ɛ > 0, let δ be the gauge of R from the defnton of g r H α R, X. Then, from the Saks-Henstock Lemma see Lemma 11, t follows that for every c-regular δ-fne d = ξ, J T P D R wth ξ E for each, we have that f α J αj gξ < ɛ, f α J = snce g α = f α and gξ = 0 for every. Theorem 13. If α r D σ I R, LX, Y and f r H α R, X, then the functon g : R Y defned by gt = r D σ α ft t s regularly Henstock ntegrable wth rk R g = rk dα f. Besdes, there exsts r R D f α m-almost everywhere on R and, n ths case, r D f α t = r D σ α ft t. Proof. 1 Gven ɛ > 0 and 0 < c < 1, let δ 1 be the gauge of R from the defnton of f r H α R, X. 2 Snce α r D σ I R, LX, Y, then for each ξ R there s a neghborhood V ξ of ξ such that for each c-regular J I R wth ξ J V ξ, αj x D c σ α xξ J < ɛ J.

12 480 Márca Federson 3 Let δ be the gauge of R such that for each ξ R, δξ δ 1 ξ and B δξ ξ V ξ. Hence, for every c-regular δ-fne d = ξ, J T D R, f α J c D σ α fξ ξ J f α J αj fξ + αj fξ c D σ α fξ ξ J, where the frst summand s smaller than ɛ from 1, and the second summand s smaller than ɛ R from 2, and the frst part of the theorem follows. The second part comes mmedately from Lemma 8. Proof of Theorem 4. The proof follows drectly from Theorems 10, 12 and 13. References [1] B. Bongorno and L. D Pazza, Convergence theorems for generalzed Remann Steltjes ntegrals, Real Analyss Exchange /92, [2] M. Federson, Fórmulas de substtução para as ntegras de gauge, MS Thess, Insttuto de Matemátca e Estatístca, Unversdade de São Paulo [3] M. Federson, Substtuton formulas for the gauge ntegrals, Semnáro Braslero de Análse , [4] C. S. Höng, The fundamental theorem of calculus for regular multdmensonal Banach space valued Kurzwel-Henstock ntegrals, Semnáro Braslero de Análse , [5] R. M. McLeod, The generalzed Remann ntegral, Math. Ass. Amerca, Carus Math. Monographs ,