A Diestel-Faires type result for multimeasures

Abstract Let X be a real Banach space and let $$Y \subseteq X^*$$ Y ⊆ X ∗ be a linear subspace having the Orlicz-Thomas property, that is, for each $$\sigma $$ σ -algebra $$\Sigma $$ Σ and for each map $$\nu :\Sigma \rightarrow X$$ ν : Σ → X , the countable additivity of the composition $$x^*\circ \nu $$ x ∗ ∘ ν for all $$x^*\in Y$$ x ∗ ∈ Y implies the countable additivity of $$\nu $$ ν . 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 $$\sigma $$ σ -algebra $$\Sigma $$ Σ whose values are convex, $$\sigma (X,Y)$$ σ ( 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 $$(A_n)_{n}$$ ( A n ) n in $$\Sigma $$ Σ the series of sets $$\sum _n M(A_n)$$ ∑ n M ( A n ) is unconditionally convergent and the equality $$M(\bigcup _n A_n)=\sum _n M(A_n)$$ M ( ⋃ n A n ) =