Dedicated to Prof. J. Kurzweil on the occasion of his 80th birthday

131 (2006) MATHEMATCA BOHEMCA No. 2, 211 223 KURZWEL-HENSTOCK AND KURZWEL-HENSTOCK-PETTS NTEGRABLTY OF STRONGLY MEASURABLE FUNCTONS B. Bongiorno, Palermo, L. Di Piazza, Palermo, K. Musia l, Wroc law (Received November 16, 2005) Dedicated to Prof. J. Kurzweil on the occasion of his 80th birthday Abstract. We study the integrability of Banach valued strongly measurable functions defined on [0, 1]. n case of functions f given by x nχ En, where x n belong to a Banach space and the sets E n are Lebesgue measurable and pairwise disjoint subsets of [0, 1], there are well known characterizations for the Bochner and for the Pettis integrability of f (cf Musial (1991)). n this paper we give some conditions for the Kurzweil-Henstock and the Kurzweil-Henstock-Pettis integrability of such functions. Keywords: Kurzweil-Henstock integral, Kurzweil-Henstock-Pettis integral, Pettis integral MSC 2000 : 26A42, 26A39, 26A45 1. ntroduction n this paper we study the Kurzweil-Henstock and the Kurzweil-Henstock-Pettis integrability of strongly measurable functions. t is well known (cf [7, Lemma 5.1]) that each strongly measurable Banach valued function, defined on a measurable space, can be written as f = g+ x n χ En, where g is a bounded strongly measurable function, x n are vectors of the given Banach space and E n are measurable and pairwise disjoint sets. As each bounded strongly measurable function is Bochner integrable, it is enough to study the integrability only for functions of the form x n χ En. n the case of the Bochner and Pettis integrals, a necessary and sufficient condition for the integrability of a function given by x n χ En is, respectively, the 211

absolute and the unconditional convergence of the series x n E n (see Theorem A). n the case of the Kurzweil-Henstock or of the Kurzweil-Henstock-Pettis integrability, in general the series x n E n is only conditionally convergent. So the conditions for the integrability depend on the order of the terms x n E n. We present one sufficient condition for the Kurzweil-Henstock and the Kurzweil-Henstock-Pettis integrability of such functions. 2. Basic facts Let ]٠, ١[ be the unit interval of the real line equipped with the usual topology and the Lebesgue measure. f a set E ]٠, ١[ is Lebesgue measurable, then E denotes its Lebesgue measure. denotes the family of all closed subintervals of ]٠, ١[. A partition in ]٠, ١[ is a finite collection of pairs P = {( 1, t 1 ),..., ( p, t p )}, where 1,..., p are nonoverlapping subintervals of ]٠, ١[ and t i i, i = ١,..., p. f p i = ]٠, ١[ we say that P is a partition of ]٠, ١[. Given a subset E of ]٠, ١[, we say that the partition P is anchored on E if t i E for each i = ١,..., p. A gauge on E ]٠, ١[ is a positive function on E. For a given gauge δ, we say that a partition {( 1, t 1 ),..., ( p, t p )} is δ-fine if i (t i δ(x i ), t i + δ(x i )), i = ١,..., p. Throughout this paper X is a Banach space with the dual X. The closed unit ball of X is denoted by B(X ). Definition 1. A function f : ]٠, ١[ X is said to be Kurzweil-Henstock integrable, or simply, integrable -بث on ]٠, ١[ if there exists w X with the following property: for every ε > ٠ there exists a gauge δ on ]٠, ١[ such that p f(t i ) i w < ε 1 (بث) =: w ١[. We set ]٠, of for each δ-fine partition {( 1, t 1 ),..., ( p, t p )} 0 f. We denote the set of all integrable -بث functions f : ]٠, ١[ X by, ٠ [)بث ١[, X). The space, ٠ [)بث ١[, X) is endowed with the Alexiewicz norm (cf. [1]) = A f 0<α 1 α (بث) f(t) tل. A family A, ٠ [)بث ١[, X) is said to be Kurzweil-Henstock equiintegrable, or simply, equiintegrable -بث on ]٠, ١[ if in Definition 1, for every ε > ٠ there exists a gauge δ on ]٠, ١[ which works for all the functions in A. 212 0

A function f : ]٠, ١[ X is said to be scalarly Kurzweil-Henstock integrable, or simply scalarly, integrable -بث if for each x X, the function x f is Kurzweil- Henstock integrable on ]٠, ١[. Definition 2. A scalarly integrable -بث function f : ]٠, ١[ X is said to be Kurzweil-Henstock-Dunford integrable or simply, integrable -ءبث if, for each nonempty interval ]a, b[ ]٠, ١[, there exists a vector w ab X such that for every x X (بث) = ab x, w (1) b a x f(t). tل t follows from [5, Theorem 3] that a function f : ]٠, ١[ X is integrable -ءبث if and only if f is scalarly. integrable -بث The generalization of the Pettis integral obtained by replacing the Lebesgue integrability of the functions by the Kurzweil-Henstock integrability produces the Kurzweil-Henstock-Pettis integral (for the definition of the Pettis integral see [3]). Definition 3. f a function f : ]٠, ١[ X is scalarly integrable -بث and for each subinterval ]a, b[ of ]٠, ١[ and for each x X there exists a vector w [a,b] X such that x w [a,b] = (بث) b a x, f, then f is said to be Kurzweil-Henstock-Pettis integrable, or simply, integrable -ذبث on ]٠, ١[ and we set w [a,b] =: (ذبث) b a f. We recall that a function f : ]٠, ١[ X is said to be strongly measurable if there is a sequence of simple functions f n with n ى f n (t) f(t) = ٠ for almost all t ]٠, ١[. 3. ntegration of strongly measurable function The aim of this section is to give conditions for the Kurzweil-Henstock or the Kurzweil-Henstock-Pettis integrability of strongly measurable functions. We start by recalling the following simple lemma (cf. Lemma 5.1 of [7]). Lemma 1. f f : ]٠, ١[ X is strongly measurable, then there exists a bounded strongly measurable function g : ]٠, ١[ X such that f = g + x n χ En where x n X and the sets E n are Lebesgue measurable and pairwise disjoint. Since each bounded strongly measurable function is Bochner integrable, and then Kurzweil-Henstock (and Kurzweil-Henstock-Pettis) integrable (see e.g. [4]), it is enough to give criteria of integrability only for functions of the form x n χ En, where x n X and the sets E n are measurable and pairwise disjoint. For the Bochner and Pettis integrals we have the following classical result: 213

Theorem A (cf [7]). Let f = x n χ En, where x n X and the sets E n are Lebesgue measurable and pairwise disjoint subsets of ]٠, ١[. Then (i) f is Pettis integrable if and only if the series x n E n is unconditionally convergent; (ii) f is Bochner integrable if and only if the series x n E n is absolutely convergent. n both cases E f = x n E n E for every measurable set E. Here we would like to give similar conditions for the Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrals. Theorem 1. Let f : ]٠, ١[ X be defined by f = x n χ En, where x n X and the sets E n are Lebesgue measurable and pairwise disjoint. Assume that the following condition is satisfied: (A) for every ε > ٠ there exist a gauge δ and k 0 such that given a δ-fine partition {( 1, t 1 ),..., ( p, t p )} of ]٠, ١[ and given s > r > k 0 we have (2) s x k k=r t j E k Then f is Kurzweil-Henstock integrable with = tل f(t) (بث) (3) < ε. x n E n for every interval.. Let ε > ٠ be arbitrary and let δ and k 0 be respectively a gauge and a natural number such that inequality (2) is satisfied. We are going to prove first that the series x n E n is convergent for every. To this purpose we set f m = m x n χ En for every m, and observe that each function f m is Bochner, and hence also Kurzweil-Henstock integrable, with its integral equal to m x k E k. Now let s > r > k 0 and, according to Definition 1, let δ r 1 and δ s be two gauges related to f r 1 and f s respectively, such that 214 r 1 x k E k p f r 1 (t i ) i < ε

and s x k E k p f s (t i ) i < ε for each partition {( 1, t 1 ),..., ( p, t p )} of ]٠, ١[ which is both δ r 1 -fine and δ s - fine. Now define δ (t) =,( δ(t } ى δ r 1 (t), δ s (t)} and take any δ -fine partition {( 1, t 1 ),..., ( p, t p )} of ]٠, ١[. Then (4) s s x k E k = r 1 x k E k x k E k k=r s x k E k p f s (t i ) i r 1 + x k E k p s f r 1 (t i ) i + x k k=r t i E k i < ٣ε. This proves that the series x n E n is norm convergent. Now for each i (s1) K i E i U i ; let K i be a closed set and U i an open set such that: (s2) U i \ K i < ٢ i ε/( x i + ١); (s3) if j < i, then U i K j = ; (s4) if i k 0, then U i K j =. k 0 j i t follows from ) ( 3 ) 4 ) that if {( 1, t 1 ),..., ( p, t p )} is a δ-fine partition of ]٠, ١[, then K n i for every n k 0. t i E n The functions f 1, f 2,..., f k0 are Bochner, and hence also Kurzweil-Henstock integrable on ]٠, ١[. Let γ(t) be a gauge on ]٠, ١[ such that m x k E k p f m (t i ) i < ε for each γ-fine partition {( 1, t 1 ),..., ( p, t p )} of ]٠, ١[ and for m = ١, ٢,..., k 0. 215

Define δ 0 (t) =, t ) ىل} ى U c i ), δ(t), γ(t)} if t E i and let {( 1, t 1 ),..., ( p, t p )} be a δ 0 -fine partition of ]٠, ١[. For each m > k 0, by (2) (4) we get m x k E k p k0 f m (t i ) i x k ( E k i ) t i E k m + x k m + x k E k k 0 < x k E k k=k 0+1 t i E k i k 0 x k < ٢ε ٢ k + ٤ε < ٦ε. ( x k + ١) t i E k i k=k 0+1 k 0 + ٤ε x k U k \ K k + ٤ε Thus, the sequence (f m ) is Kurzweil-Henstock equiintegrable. Moreover, since ١[, by [8, Theorem 1] f is Kurzweil-Henstock integrable and ]٠, in f m(t) = f(t) ى m (f m ) converges to f in the Alexiewicz topology. So, in particular, for each we have, tل (t) f n (بث) = tل f(t) (بث) n ى and the assertion follows. 1. Within the proof of the previous theorem it is also showed that condition (ء) implies the Kurzweil-Henstock equiintegrability of the sequence (f m ). t is easy to check that the same proof can be used to prove the reverse implication. So we have also: f the sequence (f m = m x k χ Ek ) is Kurzweil-Henstock equiintegrable, then the function f = x n χ En is Kurzweil-Henstock integrable and = tل f(t) (بث) x n E n for every interval. 2. There exist points x n X and pairwise disjoint Lebesgue measurable sets E n, n = ١, ٢,..., such that the series x n E n is convergent for every, the function f = x n χ En is Kurzweil-Henstock integrable and 216 (بث) 1 0 f(t) tل x n E n.

. Let π be a permutation of such that the series ( ١ ( ١) π(n) ]π(n) + ١[ π(n) ١ ) = π(n) + ١ is convergent but Let (5) f = ( ١) π(n) π(n) ( ١) n n. ( ١) π(n) ]π(n) + ١[χ [1/[π(n)+1],1/π(n)). Remark that the function f can be written also in the form f = ( ١) n (n + ١)χ [1/(n+1),1/n). ( ١) π(n) π(n) Since the function f is Riemann improper integrable on ]٠, ١[, then it is Kurzweil- Henstock integrable, with Note that the series is convergent, but (بث) 1 0 f(t) tل = ( ١) n n. [ ( ١) π(n) ١ ]π(n) + ١[ π(n) + ١, ١ ) π(n) (بث) 1 0 f(t) tل ( ١) π(n). π(n) 3. t follows from Remark 2 that condition (ء) of Theorem 1 is not necessary for the integrability -بث of a function f given by x n χ En, with x n X and E n pairwise disjoint Lebesgue measurable sets. However, there are cases in which the convergence of the series x n E n implies the Kurzweil-Henstock integrability of the function f = x n χ En and the equality (3) holds. 217

Proposition 1. Let (a n ) be a decreasing sequence converging to zero such that a 1 = ١, and let f : ]٠, ١[ X be defined by f = x n χ En, where x n X and E n = ]a n+1, a n ). f the series x n E n is convergent, then condition (ء) of Theorem ١ is satisfied.. Let ε > ٠ be given, and let k 0 be a natural number such that (6) x n E n < ε ٥ n=r, ٥ x r E r < ε ل ف for every r > k 0. Now we define a gauge δ on ]٠, ١[ in the following way: δ(٠) = a k0, δ(x) =, x ) ىل E c n ) if x E0 n (the interior of E n), δ(a n ) = 1 5 ٢ n } ى ε/( x n 1 + x n + ١), a ) ىل n, ٠)} for n = ١, ٢,... Now let {( 1, t 1 ),..., ( p, t p )} be any δ-fine partition of ]٠, ١[. By the definition of δ, the point ٠ has to be the tag of one of the pairs of the partition; we may assume that t 1 = ٠. Let n be the first natural number such that a n+1 1. By the definition of δ(٠) it follows that a n+1 < a k0, hence n k 0. Then (7) t j E n E n and (8) t j E n n. n > ن = ٠ Hence x n n>r t j E n = ٠ for each r n. Besides, for ١ < n < n we have (9) for suitable ε n, ε n ]٠, ١[. 218 t j E n = E n + ε nδ(a n+1 ) ε nδ(a n )

So, for n > r > k 0, by (6), (7), (8) and (9) we obtain x n = x n + x n ( E n + ε nδ(a n+1 ) ε nδ(a n )) n>r t j E n x n E n + ε ٥ + ٢ ε ٥ + ١ ٥ n>n>r n>r t j E n x n E n + ε ٢ n + ١ ٥ n>r n>n>r n>n>r ε ٢ n < ε. x n δ(a n 1 ) + n>n>r x n δ(a n ) This completes the proof. Theorem 1 and Proposition 1 yield Theorem 2. Let f : ]٠, ١[ X be defined by f = x n χ En, where x n X, E n = ]a n+1, a n ) and (a n ) is a decreasing sequence converging to zero such that a 1 = ١. f the series x n E n is convergent, then f is Kurzweil-Henstock integrable and = tل f(t) (بث) x n E n for every interval. -بث. Suppose that f is defined like in Theorem 1 and it is integrable. Does there exist a permutation π of such that for each? = tل f(t) (بث) x π(n) E π(n) By applying Theorem 1 the following two results follow: Theorem 3. Let f : ]٠, ١[ X be defined by f = x n χ En, where x n X and the sets E n are Lebesgue measurable and pairwise disjoint. We assume also that the following conditions are satisfied: (a) x n E n is weakly convergent; (b) for every ε > ٠ and every x X there exist a gauge δ and k 0 such that for each δ-fine partition {( 1, t 1 ),..., ( p, t p )} of ]٠, ١[ and each n > m > k 0 we have n x (x k ) < ε. k=m t j E k 219

Then f is Kurzweil-Henstock-Pettis integrable and for every and x X we have = tل f(t) x, (بث) (10) x (x n ) E n. Theorem 4. Let f : ]٠, ١[ X be defined by f = x n χ En, where x n X and the sets E n are Lebesgue measurable and pairwise disjoint. We assume also that the following conditions are satisfied: (a) x n E n is weakly Cauchy; (b) for every ε > ٠ and every x X there exist a gauge δ and a natural number k 0 such that n x (x k ) < ε k=m t j E k for each δ-fine partition {( 1, t 1 ),..., ( p, t p )} of ]٠, ١[ and each n > m > k 0. Then f is Kurzweil-Henstock-Dunford integrable and for every and x X we have = tل f(t) x, (بث) (11) x (x n ) E n. Theorem 5. Let f : ]٠, ١[ X be defined by f = x n χ En, where x n X, E n = ]a n+1, a n ) and (a n ) is a decreasing sequence converging to zero such that a 1 = ١. f the series x n E n is weakly convergent, then f is Kurzweil-Henstock- Pettis integrable and = tل f(t) x (بث) x (x n ) E n for every interval and every x X. f the series x n E n is weak -convergent, then f is Kurzweil-Henstock-Dunford integrable and = tل f(t) x (بث) for every interval and every x X. 220 x (x n ) E n

We recall that a function G: ]٠, ١[ is a primitive -بث if and only if for each N ]٠, ١[ with N = ٠ and for each ε > ٠ there exists a gauge δ in N such that p G( i ) < ε for each δ-fine partition {( 1, t 1 ),..., ( p, t p )} anchored on N. Here G(]a, b[) = G(b) G(a). n this case the derivative of G exists almost everywhere in ]٠, ١[ and G is the there). primitive -بث of G (see e.g. [2] and the bibliography Theorem 6. Let f : ]٠, ١[ X be defined by f = x n χ En, where x n X and the sets E n are pairwise disjoint intervals. Then f is Kurzweil-Henstock-Pettis integrable and = tل f(t) (ذبث) (12) x n E n for every if and only if the following conditions are satisfied: (a) x n E n is weakly convergent for each ; (b) for every ε > ٠, every x X and every N ]٠, ١[ with N = ٠, there exists a gauge δ in N such that (13) x (x n ) E n p i < ε for each δ-fine partition {( 1, t 1 ),..., ( p, t p )} anchored on N.. f f is integrable -بث and condition (12) is satisfied, then condition (a) follows directly from (١٢). Now we prove condition (b). By hypothesis, for every x X the scalar function x f is. integrable -بث Set (بث) = α(t) t 0 x, f(s) sل = x (x n ) E n ]٠, t[. Since α(t) is the primitive -بث of x f, hence for each N ]٠, ١[ with N = ٠ and for each ε > ٠ there exists a gauge δ on N such that p α( i ) < ε for each δ-fine partition {( 1, t 1 ),..., ( p, t p )} anchored on N. 221

Therefore x (x n ) E n p i p x (x n ) E n i = p α( i ) < ε. Assume now that conditions (a) and (b) are satisfied, and fix x X. By condition (b), for every ε > ٠ and every N ]٠, ١[ with N = ٠, there exists a gauge δ on N such that inequality (13) holds for each δ-fine partition {( 1, t 1 ),..., ( p, t p )} anchored on N. Set once again α(t) = x (x n ) E n ]٠, t[. Then, by (a) and (13), we infer p α( i ) = α( i ) + α( i ) + = x (x n ) E n i + x (x n ) E n i < ٢ε, + where + and denote the set of all indices i = ١,..., p such that α( i ) is positive or negative, respectively. Therefore α(t) is a. primitive -بث As the sets E n are intervals, it follows easily that α (t) = x (x n )χ En almost everywhere in ]٠, ١[. Then α(t) is the primitive -بث of the function x f = x (x n )χ En. Consequently (بث) 1 0 x, f(t) tل = x (x n ) E n = x, x n E n. Hence the function f is integrable -ذبث and (12) holds.. s Theorem 6 still true if the sets E n are arbitrary pairwise disjoint Lebesgue measurable sets? References [1] A. Alexiewicz: Linear functionals on Denjoy integrable functions. Coll. Math. 1 (1948), 289 293. Zbl 0037.32302 [2] B. Bongiorno: The Henstock-Kurzweil integral. Handbook of Measure Theory, E. Pap ed., Elsevier Amsterdam, 2002, pp. 587 615. Zbl 0998.28001 [3] J. Diestel, J. J. Uhl: Vector measures. Math. Surveys, vol. 15, AMS, Providence, R.., 1977. Zbl 0369.46039 [4] D. H. J. Fremlin: The Henstock and McShane integrals of vector-valued functions. llinois J. Math. 38 (1994), 471 479. Zbl 0797.28006 [5] J. L. Gamez, J. Mendoza: On Denjoy-Dunford and Denjoy-Pettis integrals. Studia Math. 130 (1998), 115 133. Zbl 0971.28009 222

[6] P. Y. Lee: Lanzhou Lectures on Henstock ntegration. World Scientific, Singapore, 1989. Zbl 0699.26004 [7] K. Musial: Topics in the theory of Pettis integration. Rend. stit. Mat. Univ. Trieste 23 (1991), 177 262. Zbl 0798.46042 [8] C. Swartz: Norm convergence and uniform integrability for the Kurzweil-Henstock integral. Real Anal. Exchange 24 (1998/99), 423 426. Zbl 0943.26022 Authors addresses: B. Bongiorno, L. Di Piazza, Department of Mathematics and Applications, University of Palermo, via Archirafi 34, 90123 Palermo, taly, e-mail: bbongi@math. unipa.it, dipiazza@math.unipa.it; K. Musia l, Wroc law University, nstitute of Mathematics, Plac Grunwaldzki 2/4, 50-384 Wroc law, Poland, e-mail: musial@math.uni.wroc.pl. 223