A Diestel-Faires type result for multimeasures

Let X be a real Banach space and let Y ⊆ X^* Y ⊆ X ∗ be a linear subspace having the Orlicz-Thomas property, that is, for each σ σ -algebra Σ Σ and for each map ν :Σ → X ν : Σ → X , the countable additivity of the composition x^*∘ ν x ∗ ∘ ν for all x^*∈ Y x ∗ ∈ Y implies the countable additivity of ν 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 σ σ -algebra Σ Σ whose values are convex, σ (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 (Aₙ)ₙ ( A n ) n in Σ Σ the series of sets ∑ ₙ M(Aₙ) ∑ n M ( A n ) is unconditionally convergent and the equality M( ₙ Aₙ)=∑ ₙ M(Aₙ) M ( ⋃ n A n ) =