We study the integrability of Banach valued strongly measurable functions defined on [0, 1]. In case of functions f given by ∞∑ n=1 xnχEn , where xn belong to a Banach space and the sets En 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)). In this paper we give some conditions for the Kurzweil-Henstock and the Kurzweil-Henstock-Pettis integrability of such functions.