ON THE C-INTEGRAL BENEDETTO BONGIORNO

ON THE C-INTEGRAL BENEDETTO BONGIORNO Let F : [, b] R be differentible function nd let f be its derivtive. The problem of recovering F from f is clled problem of primitives. In 1912, the problem of primitives ws solved by A. Denjoy with n integrtion process (clled totliztion) tht includes the Lebesgue integrl nd the Riemnn improper integrl. Two yers lter, second solution ws obtined by O. Perron with method bsed on the notions of mjor function nd minor function. A third solution, bsed on generliztion of Riemnn integrl, is due to J. Kurzweil (1957) nd R. Henstock (1963). It is surprising tht, nevertheless the three integrtion processes re completely different, they produce the sme integrl (i.e. they hve the sme spce of integrble functions nd stisfy the sme properties). In 1986, A.M. Bruckner, R.J. Fleissner nd J. Forn [9] remrked tht the solution provided by Denjoy, Perron, Kurzweil nd Henstock possesses generlity which is not needed for this purpose. In fct the function { x sin 1, 0 < x 1 (1) F (x) = x 2 0, x = 0 is primitive for the Denjoy-Perron-Kurzweil-Henstock integrl (more precisely, F is primitive for the Riemnn improper integrl), but it is neither Lebesgue primitive, neither differentible function, nor sum of Lebesgue primitive nd differentible function (see [9] for detils). The question of providing miniml constructive integrtion process which includes the Lebesgue integrl nd lso integrtes the derivtives of differentible functions ws solved by the following Riemnn-type integrl: Supported by MURST of Itly. 1

2 BENEDETTO BONGIORNO Definition 1. Given function f : [, b] R we sy tht f is C- integrble on [, b] if there exists constnt A such tht for ech ε > 0 there is guge δ on [, b] with (2) f(x i ) I i A < ε, for ech δ-fine McShne-prtition {(I 1, x 1 ),..., (I p, x p )} of [,b] stisfying the condition (3) dist(x i, I i ) < 1/ε. Here by guge on [, b] we men positive function δ defined on [, b], nd by δ-fine McShne-prtition of [, b] we men collection {(I 1, x 1 ),..., (I p, x p )} of pirwise nonoverlpping intervls I i [, b] nd points x i [, b] such tht I i (x i δ(x i ), x i +δ(x i )) nd p I i = b. The number A is clled the C-integrl of f on [, b], nd we set A = f. The mentioned minimlity of the C-integrl s constructive integrtion process which includes the Lebesgue integrl nd lso integrtes the derivtives of differentible functions follows by the following theorem. Theorem 2. ([5, Min Theorem]) A function f : [, b] R is C-integrble on [, b] if nd only if there exists derivtive h such tht f h is Lebesgue integrble. 1. Reltions between the C-integrl nd the Lebesgue integrl, the Henstock-Kurzweil integrl, nd the Riemnn-improper integrl It is well known tht the Lebesgue integrl is equivlent to the McShne integrl ([16], [17]); consequently, by Definition 1 it follows tht ech Lebesgue integrble function is C-integrble with the sme vlue of the integrl. Notice lso tht if f is C-integrble on [, b], then f is Henstock- Kurzweil integrble on [, b] with the sme vlue of the integrl. In

ON THE C-INTEGRAL 3 fct, condition (2) holds for ech collection {(I 1, x 1 ),..., (I p, x p )} of pirwise nonoverlpping intervls I i [, b] nd points x i [, b] such tht I i (x i δ(x i ), x i + δ(x i )) nd x i I i, i = 1, 2,..., p. Now we prove tht both inclusions re proper. Proposition 3. The Lebesgue integrl is properly contined into the C-integrl; i.e. there is C-integrble function f such tht f is not Lebesgue integrble. Proof. Let us consider the function f on [0, 1] given by f(x) = 2x sin 1 x 2 2 x cos 1, with f(0) = 0. x2 A primitive of f is the function F given by F (x) = x 2 sin(1/x 2 ), F (0) = 0. It is esy to check tht F is not bsolutely continuous on [0, 1]; hence f is not Lebesgue integrble on [0, 1]. Now remrk tht F is differentible everywhere in [0, 1] with derivtive F (x) = f(x), x [0, 1]. So f is C-integrble, by the following proposition. Proposition 4. Ech derivtive is C-integrble. Proof. Let F be n everywhere differentible function on [, b], nd let f(x) = F (x) for ech x [, b]. Given 0 < ε < 1/(b ) nd x [, b], by definition of derivtive, there exists δ(x) > 0 such tht (4) F (y) F (x) y x f(x) < ε2 4, for ech y [, b] with y x < δ(x). Given n intervl I = (α, β) we set F (I) = F (β) F (α) nd I = β α. If I is subintervl of (x δ(x), x + δ(x)), then by

4 BENEDETTO BONGIORNO (4) we hve (5) F (I) f(x) I F (β) F (x) f(x)(β x) + F (α) F (x) f(x)(α x) < ε2 4 ε2 2 ε2 β x + α x 4 (dist(x, I) + I ). Therefore, if {(I 1, x 1 ),, (I p, x p )} is δ-fine McShne-prtition of [, b] stisfying the condition p dist(x i, I i ) < 1/ε, then by (5) we get f(x i ) I i (F (b) F ()) f(x i ) I i F (I i ) < ε2 (dist(x i, I i ) + I i ) 2 < ε 2 + ε 2 = ε. Thus f is C-integrble on [, b]. Proposition 5. The C-integrl is properly contined into the Henstock-Kurzweil integrl; i.e. there is Henstock-Kurzweil integrble function f such tht f is not C-integrble. Proof. Let us consider the function (1) nd define f = F in (0, 1] nd f(0) = 0. It is esy to see tht f is Riemnn improper integrble on [0, 1], hence f is Henstock-Kurzweil integrble. To prove tht f is not C-integrble we need the following extension of clssicl Henstock s lemm: Lemm 6. If function f : [, b] R is C-integrble on [, b], then given ε > 0 there exists guge δ on [, b] so tht (6) f(x i) I i < ε, I i f

ON THE C-INTEGRAL 5 for ech δ-fine prtil McShne-prtition {(I 1, x 1 ),..., (I p, x p )} in [, b] with p dist(x i, I i ) < 1/ε. Here by δ-fine prtil McShne-prtition {(I 1, x 1 ),..., (I p, x p )} in [, b] we men collection {(I 1, x 1 ),..., (I p, x p )} of pirwise nonoverlpping intervls I i [, b] nd points x i [, b] such tht I i (x i δ(x i ), x i + δ(x i )) nd p I i < b. The proof of this lemm follows esily by simple dpttion of stndrd technics (see [13, Lemm 9.11]). Assume, by contrdiction, tht f is C-integrble on [0, 1]. Given ε > 0, tke guge δ ccording with Lemm 6, nd let h = (π +2hπ) 1/2, b h = (π/2+2hπ) 1/2. It is esy to see tht h h = h b h =, nd tht the intervls ( h, b h ), h = 1, 2,..., re pirwise disjoint. Moreover, by (1) we hve F ( h ) = 0, F (b h ) = b h, for ech h N. Tke nturl numbers n, p such tht ( n+i, b n+i ) (0, δ(0)), i = 1,, p, nd ε < n+i < 1 ε. Hence p 1 b n+i > p 1 n+i > ε. Now define I 1 = ( n+1, b n+1 ),, I p = ( n+p, b n+p ). Then {(I 1, 0),..., (I p, 0)} is δ-fine prtil McShne-prtition in [0, 1], nd p dist(0, I i) = p i < 1/ε. Moreover f(0) I i = F (b n+i ) F ( n+i ) = b n+i > ε, I i f in contrdiction with (6). Thus f is not C-integrble on [0, 1]. Remrk, moreover, tht the function (1) is Riemnn improper integrble on [0, 1]. Hence next proposition is lso proved. Proposition 7. The C-integrl doesn t contin the Riemnn improper integrl. 2. The vritionl mesure V C F A first descriptive chrcteriztion of the C-primitives ws obtined by A.M. Bruckner et l. [9]. They proved tht function F is C-primitive if nd only if F is the limit in vrition of sequence of bsolutely continuous functions. 1

6 BENEDETTO BONGIORNO Note tht, in generl, F is not function of bounded vrition; however, its ssocite vritionl mesure is bsolutely continuous with respect to the Lebesgue mesure. This follows from the observtion tht C-primitive is lso Henstock-Kurzweil primitive nd by [6, Theorem 3]. In fct the lst mentioned theorem gives chrcteriztion of Henstock-Kurzweil primitives. The ide of considering pproprite vritionl mesures to chrcterize the primitives of some integrl hs been lso used in [4, 10, 11] for mny multidimensionl integrls nd in [7] for the Henstock-dydic integrl nd for the Henstock-symmetric integrl. Given function F on [, b], guge δ on [, b], subset E of [, b], nd positive ε, we denote by V ε (F, δ, E) the supremum of ll sums i F (I i) where {(I 1, x 1 ),..., (I p, x p )} runs into the clss of δ-fine prtil McShne-prtitions of [, b] with x i E, i = 1, 2, p, nd p dist(x i, I i ) < 1/ε. It is cler tht (7) V ε1 (F, δ, E) V ε2 (F, δ, E), for ε 1 > ε 2. Then we define V C F (E) = sup ε inf δ V ε (F, δ, E). By the sme rgument used in [20] for proving Theorem 3.7 nd Theorem 3.15, we cn show tht the extended rel-vlued set function V C F is Borel regulr mesure in [, b]. Theorem 8. [2, Theorem 4.1] F is n indefinite C-integrl if nd only if the vritionl mesure V C F is bsolutely continuous with respect to the Lebesgue mesure. This chrcteriztion is used to prove the following theorem. Theorem 9. [2, Theorem 4.2] Every BV function is multiplier for the C-integrl; i.e. if g is BV function on [, b] nd f is C-integrble on [, b], then fg is lso C-integrble on [, b]. Here by BV function we men function g : [, b] R such tht there exists function of bounded vrition g : [, b] R with g = g lmost everywhere in [, b]. It is known tht the product of two derivtives my be not derivtive. A simple consequence of Theorem 9 show tht

ON THE C-INTEGRAL 7 Theorem 10. [2, Theorem 4.3] The product between derivtive nd BV function is derivtive modulo Lebesgue integrble function of rbitrrily smll L 1 -norm; i.e. if f is derivtive nd g is BV function, then there is sequence {φ n } of derivtives such tht fg φ n 1 0. 3. ACG C Functions Definition 11. A function F : [, b] R is sid to be C-bsolutely continuous on set E [, b] (br. AC C (E)) whenever for ech ε > 0 there exist constnt η > 0 nd guge δ on E such tht (8) F (I i ) < ε i for ech δ-fine prtil McShne-prtition {(I 1, x 1 ),, (I p, x p )} in [, b] stisfying the following conditions: (α 1 ) x i E, i = 1, 2,..., p ; (α 2 ) p dist(x i, I i ) < 1/ε; (α 3 ) i I i < η. Definition 12. A function F : [, b] R is sid to be C-generlized bsolutely continuous on [, b] (br. ACG C [, b]) whenever (β) there exist mesurble sets E 1, E 2,... such tht [, b] = n E n nd F is AC C (E n ), n = 1, 2,.... According with R.A. Gordon [12] function F : [, b] R is sid to be AC δ (E) whenever condition (8) is stisfied for ech δ-fine prtil McShne-prtition {(I 1, x 1 ),, (I p, x p )} stisfying conditions (α 1 ), (α 3 ), nd (α 2 ) p dist(x i, I i ) = 0. Moreover F is sid to be ACG δ [, b] whenever (β ) there exist mesurble sets E 1, E 2,... such tht [, b] = n E n nd F is AC δ (E n ), n = 1, 2,.... Therefore ech AC C -function is AC δ, nd ech ACG C -function is ACG δ.

8 BENEDETTO BONGIORNO Lemm 13. If F is ACG C [, b], nd E [, b] with E = 0, then for ech ε > 0 there is guge δ on E such tht F (I i ) < ε, for ech δ-fine prtil McShne-prtition {(I 1, x 1 ),, (I p, x p )} in [, b] stisfying the following conditions (α 1 ) x i E, i = 1, 2,..., p ; (α 2 ) p dist(x i, I i ) < 1/ε. Theorem 14. A function F is ACG C [, b] if nd only if there is C-integrble on [, b] function f such tht (9) F (x) F () = x f(t) dt, for ech x [, b]. Proof. Assume tht F is ACG C [, b]. Since F is ACG δ [, b], by [12, Theorem 6] nd by [19, Chpter VII, Theorem 7.2], F is differentible lmost everywhere in [, b]. Let E be the set of points x [, b] such tht F is not differentible t x. Then E = 0. So, by Lemm 13, given 0 < ε 1/(b ) there is guge τ on [, b] such tht F (I i ) < ε 4, for ech τ-fine prtil McShne-prtition {(J 1, x 1 ),, (J p, x p )} in [, b] with p dist(x i, I i ) < 1/ε nd x i E, i = 1, 2,.... If F is differentible t y then, by (5) we cn find positive constnt γ(y) such tht F (I) F (y) I < ε2 (dist(y, I) + I ), 2 for ech intervl I (y γ(y), y + γ(y)). Define { τ(y) if y E, δ(y) = γ(y) if y E, nd f(y) = { 0 if y E, F (y) if y E.

ON THE C-INTEGRAL 9 Given x [, b] let {(I 1, x 1 ),, (I p, x p )} be δ-fine McShneprtition of [, x] such tht p dist(x i, I i ) < 1/ε. Then we hve f(x i ) I i (F (x) F ()) f(x i ) I i F (I i ) < x i E < ε 2 + x i E F (I i ) + x i E F (x i ) I i F (I i ) ε 2 4 (dist(x i, I i ) + I i ) < ε 2 + ε 4 + ε ε(b ) ε. 4 Thus f is C-integrble on [, x] nd (9) holds. Tking x = b we see tht f is C-integrble on [, b]. Now ssume tht f is C-integrble on [, b] nd let F be the C-primitive of f. For ech nturl number n we define E n = {x [, b] : f(x) n}. Then [, b] = n E n. To complete the proof it is enough to show tht F is AC C on E n, for ech n. By Lemm 6, given ε > 0 there is guge δ on [, b] such tht f(x i ) I i F (I i ) < ε 2, for ech δ-fine prtil McShne-prtition {(I 1, x 1 ),, (I p, x p )} in [, b] with p dist(x i, I i ) < 1/ε. Assume tht x i E n, i = 1, 2,..., nd i I i < ε/2n. Then F (I i ) f(x i ) I i F (I i ) + < ε 2 + n i I i < ε. f(x i ) I i

10 BENEDETTO BONGIORNO So F is AC C (E n ), nd the proof is complete. 4. Convergence Theorems. Let f be Lebesgue (resp. Henstock-Kurzweil) integrble on [, b]. We denote by (L) f (resp. (HK) f) the Lebesgue (resp. Henstock-Kurzweil) integrl of f on [, b]. Monotone convergence theorem. Let f 1 f 2 f n be sequence of C-integrble functions on [, b] such tht the limit lim n f n exists finite. Then the function f(x) = lim n f n (x) is C-integrble on [, b] nd we hve f(x) dx = lim f n (x) dx. Proof. Since ech C-integrble function is Henstock-Kurzweil integrble, by [18, Corollry 6.3.5] ech nonnegtive C-integrble function is Lebesgue integrble, nd by [18, Theorem 6.3.3] ech C-integrble function is Lebesgue mesurble. Then we cn pply the Lebesgue monotone convergence theorem to the following sequence Hence (10) (L) 0 f 2 f 1 f 3 f 1 f n f 1 {f(x) f 1 (x)} dx = lim (L) Now for ech n N we hve (L) {f n (x) f 1 (x)} dx = = {f n (x) f 1 (x)} dx f n (x) dx {f n (x) f 1 (x)} dx. f 1 (x) dx.

ON THE C-INTEGRAL 11 Therefore, since lim n f n exists finite, we cn use (10) to infer tht f f 1 is Lebesgue integrble; hence f = (f f 1 ) + f 1 is C-integrble. Moreover f(x) dx = = lim = lim {f(x) f 1 (x)} dx + f n (x) dx f n (x) dx. f 1 (x) dx f 1 (x) dx + f 1 (x) dx Dominted convergence theorem. Let f 1, f 2,, f n be sequence of mesurble functions such tht (i) f n (x) f(x) lmost everywhere in [, b]; (ii) g(x) f n (x) h(x), lmost everywhere in [, b], with g nd h C-integrble on [, b]; then f is C-integrble on [, b] nd f(x) dx = lim f n (x) dx. Proof. By (ii) we hve 0 f n g h g, lmost everywhere in [, b]. Moreover h g is Lebesgue integrble, since non negtive nd C-integrble. Then, by Lebesgue dominted convergence theorem, f g is Lebesgue integrble on [, b] with (L) {f(x) g(x)} dx = lim (L) {f n (x) g(x)} dx.

12 BENEDETTO BONGIORNO Therefore, by f = (f g) + g we infer tht f is C-integrble on [, b] with f(x) dx = = (L) = lim = lim {f(x) g(x)} dx + {f(x) g(x)} dx + f n (x) dx f n (x) dx. g(x) dx g(x) dx + g(x) dx g(x) dx Definition 15. Let E be subset of [, b] nd let {F n } be sequence of rel vlued functions defined on [, b]. It is sid tht {F n } is uniformly AC C (E) if for ech ε > 0 there exist constnt η > 0 nd guge δ such tht sup n F n (I i ) < ε, i for ech δ-fine prtil McShne-prtition {(I 1, x 1 ),, (I p, x p )} in [, b], stisfying conditions (α 1 ), (α 2 ), (α 3 ). Definition 16. A sequence {F n } is sid to be uniformly ACG C [, b] if there exist mesurble sets E 1, E 2,... such tht [, b] = k E k nd {F n } is uniformly AC C (E k ), k = 1, 2,.... Anlogously we define the notions of sequence uniformly AC δ (E) nd uniformly ACG δ [, b]. It is esy to see tht if sequence {F n } is uniformly ACG C [, b] then {F n } is uniformly ACG δ [, b]. Controlled convergence theorem. Let {f n } be sequence of C-integrble on [, b] functions such tht (i) f n (x) f(x) lmost everywhere in [, b]; (ii) the sequence F n (x) = x f n(t) dt is uniformly ACG C [, b]. Then f is C-integrble on [, b] nd

(11) ON THE C-INTEGRAL 13 f(x) dx = lim f n (x) dx. Proof. Since ech C-integrble function is Henstock-Kurzweil integrble nd since ech uniformly ACG C sequence is uniformly ACG δ, then by [3, Theorem 4.1] f is Henstock-Kurzweil integrble on [, b] nd (12) (HK) x f(t) dt = lim (HK) x f n (t) dt, for ech x [, b]. Now, by (ii), there exist sequence of mesurble sets {E h } such tht [, b] = h E h, for ech ε > 0 nd ech h there exist constnt η h > 0 nd guge δ h such tht (13) sup n F n (I i ) < ε, i for ech δ h -fine prtil McShne-prtition {(I 1, x 1 ),, (I p, x p )} in [, b], stisfying conditions (α 1 ), (α 2 ), (α 3 ), with E h for E. Set F (x) = (HK) x f(t) dt. Since F n (x) = x f n (t) dt = (HK) x f n (t) dt, by (12) we hve F (x) = lim n F n (x), for ech x [, b]. Consequently, by (13), for ech h N nd for ech δ h -fine prtil McShne-prtition {(I 1, x 1 ),, (I p, x p )} in [, b] stisfying conditions (α 1 ), (α 2 ), (α 3 ), with E h for E, we hve F (I i ) = lim F n (I i ) ε. Thus F is ACG C [, b] nd by Theorem 14, F (x) = x f(t) dt for ech x [, b]. In conclusion (11) follows esily by (12).

14 BENEDETTO BONGIORNO References [1] B. Bongiorno, Un nuovo integrle per il problem delle primitive, Le Mtemtiche, 51 (1996), No. 2, 299 313. [2] B. Bongiorno, On the miniml solution of the problem of primitives, J. Mth. Anl. Appl. 251 (2000), 479 486. [3] B. Bongiorno, nd L. Di Pizz, Convergence theorems for generlized Riemnn Stieltjes integrls, Rel Anlysis Exchnge, 17 (1991-92), 339 362. [4] B. Bongiorno, L. Di Pizz, nd D. Preiss, Infinite vrition nd derivtives in R n, J. Mth. Anl. Appl., 224 (1998), No. 1, 22 33. [5] B. Bongiorno, L. Di Pizz, nd D. Preiss, A constructive miniml integrl which includes Lebesgue integrble functions nd derivtives, J. London Mth. Soc. (2) 62 (2000), no. 1, 117 126. [6] B. Bongiorno, L. Di Pizz, nd V. Skvortsov, A new full descriptive chrcteriztion of Denjoy-Perron integrl, Rel Anl. Exchnge 21 (1995/96), No. 2, 656 663. [7] B. Bongiorno, L. Di Pizz, nd V. Skvortsov, On vritionl mesures relted to some bses, J. Mth. Anl. Appl. 250 (2000), no. 2, 533 547. [8] B. Bongiorno nd V. Skvortsov, Multipliers for some generlized Riemnn integrls in the rel line, Rel Anl. Exchnge, 20 (1994-95), 212-218. [9] A.M. Bruckner, R.J. Fleissner, nd J. Forn, The miniml integrl which includes Lebesgue integrble functions nd derivtives, Coll. Mth., 50 (2) (1986), 289 293. [10] Z. Buczolich nd W. Pfeffer, On bsolute continuity, J. Mth. Anl. Appl. 222 (1998), No. 1, 64 78. [11] L. Di Pizz, Vritionl mesures in the theory of the integrtion in R m, Czechoslovch Mth. J. 51(126) (2001), no. 1, 95 110. [12] R.A. Gordon, A descriptive chrcteriztion of the generlized Riemnn integrl, Rel Anl. Exchnge, 15 (1989-90), 397 400. [13] R.A. Gordon, The integrls of Lebesgue, Denjoy, Perron, nd Henstock, Studies in Mthemtics 4, Amer. Mth. Soc., Providence, 1994. [14] R. Henstock, The generl theory of integrtion, Clrendon Press, Oxford, 1991. [15] J. Kurzweil, Generlized ordinry differentil equtions nd continuous dependence on prmeter, Czechoslovc. Mth. J., 7 (1957), 418-446. [16] E.J. McShne, A unified theory of integrtion, Amer. Mth. Monthly, 80 (1973), 349 359. [17] E.J. McShne, Unified integrtion, Acdemic Press, New York, 1983. [18] W.F. Pfeffer, The Riemnn Approch to Integrtion, Cmbridge University Press, Cmbridge, 1993. [19] S. Sks, Theory of the integrl, Dover, New York, 1964.

ON THE C-INTEGRAL 15 [20] B.S. Thomson, Derivtives of Intervl Functions, Mem. Amer. Mth. Soc., 452, Providence, 1991. Deprtment of Mthemtics, University of Plerme,, Vi Archirfi 34, 90123 Plermo (Itly) E-mil ddress: bb@mth.unip.it