A Diestel-Faires type result for multimeasures

Abstract

Let $X$ be a real Banach space and let $Y\subseteq X^{\ast }$ be a linear subspace having the Orlicz-Thomas property, that is, for each $\sigma$-algebra $\Sigma$ and for each map $\nu :\Sigma \to X$, the countable additivity of the composition $x^{\ast }\circ \nu$ for all $x^{\ast }\in Y$ implies the countable additivity of $\nu$. We show that the Orlicz-Thomas property allows to test countable additivity of set-valued maps. Namely, if $M$ is a map defined on a $\sigma$-algebra $\Sigma$ whose values are convex, $\sigma (X,Y)$-compact, bounded non-empty subsets of $X$, then the following statements are equivalent: (i) $M$ is a strong multimeasure, that is, for every disjoint sequence $(A_n{)}_n$ in $\Sigma$ the series of sets ${\sum }_{n}M(A_n)$ is unconditionally convergent and the equality $M({\bigcup }_n{A}_n)={\sum }_{n}M(A_n)$ holds. (ii) $M$ is a multimeasure, that is, for every $x^{\ast }\in X^{\ast }$ the support map $s(x^{\ast },M):\Sigma \to \mathbb{R}$ defined by $s(x^{\ast },M)(A):=\sup \{x^{\ast }(x):x\in M(A)\}$ is countably additive. (iii) $s(x^{\ast },M)$ is countably additive for every $x^{\ast }\in Y$. As an application, we give a result on the factorization of multimeasures through reflexive Banach spaces.