Variational henstock integrability of banach space valued functions

Abstract

We study the integrability of Banach space valued strongly measurable functions defined on [0, 1]. In the case of functions f given by (formula presented) where xn are points of a Banach space and the sets En 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 (formula presented) 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.