# Institute of Mathematics

## Variational Henstock integrability of Banach space valued functions

#### Luisa Di Piazza, Valeria Marraffa, Kazimierz Musiał

Abstract:  We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by $\sum\nolimits_{n=1}^{\infty} x_n\chi_{E_n}$, 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 $\sum\nolimits_{n=1}^{\infty}x_n|E_n|$ 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.
Keywords:  Kurzweil-Henstock integral; variational Henstock integral; Pettis integral
Classification MSC:  26A39
DOI:  10.21136/MB.2016.19