Full characterizations of the variational McShane Integral on $m$-dimensional compact intervals

Abstract

In this paper we consider the additive interval functions defined on the family $\mathcal{I}_{m}$ of all non-degenerate closed subintervals of the cubic interval $C_{m} = [0,1]^{m}$ in the $m$-dimensional Euclidean space $\mathbb{R}^{m}$ and taking values in a Banach space $X$. We give necessary and sufficient conditions for an additive interval function $F : \mathcal{I}_{m} \to X$ to be the primitive of a variational McShane (or strong McShane) integrable function $f : C_{m} \to X$ in terms of the convex cubic average range of $F$.