VARIATIONAL HENSTOCK INTEGRABILITY OF BANACH SPACE VALUED FUNCTIONS

We study the integrability of Banach space valued strongly measurable functions defined on [0, 1]. In the case of functions f given by Sigma(infinity)(n=1) x(n) chi(En), where x(n) are points of a Banach space and the sets E-n are Lebesgue measurable and pairwise disjoint subsets of [0, 1], there are well known characterizations for Bochner and Pettis integrability of f. The function f is Bochner integrable if and only if the series Sigma(infinity)(n=1) x(n)vertical bar E-n vertical bar is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of f. In this paper we give some conditions for variational Henstock integrability of a certain class of such functions.