Abstract 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 -algebra Σ and for each map: X ν: Σ → X, the countable additivity of the composition x^* x ∗ ∘ ν for all x^* Y x ∗ ∈ Y implies the countable additivity of 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 -algebra Σ whose values are convex, (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) ₍ (A n) n in 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) =
Juana Rodrı́guez (Fri,) studied this question.