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 $F$.

I will generalize the main result for open and bounded subset of ${\mathbb{R}}^m$