Radon–Nikodým theorems in non-separable Banach spaces

Abstract

In this paper, we present two Radon–Nikodým theorems in non-separable Banach spaces using Pettis and variational McShane integrals. The first one works for a dominated additive interval multifunction $\Phi :\mathcal{I}\to ck(X)$, and the second one works for a dominated strong multimeasure $M:\mathcal{L}\to cwk(X)$, where $\mathcal{I}$ is the family of all closed non-degenerate subintervals of the interval $W=[0,1{]}^m\subset {\mathbb{R}}^m$, $\mathcal{L}$ is the family of all Lebesgue measurable subsets of $W$, and $cwk(X)$ $(ck(X))$ is the family of all convex weakly compact (convex compact) non-empty subsets of $X$.