Let X be a real Banach space and let \(Y \subseteq X^*\) be a linear subspace having the Orlicz-Thomas property, that is, for each \(\sigma \) -algebra \(\Sigma \) and for each map \(\nu :\Sigma \rightarrow X\) , the countable additivity of the composition \(x^*\circ \nu \) for all \(x^*\in Y\) 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 \(x^*\in X^*\) the support map \(s(x^*,M):\Sigma \rightarrow \mathbb {R}\) defined by \(s(x^*,M)(A):=\sup \{x^*(x):x\in M(A)\}\) is countably additive. (iii) \(s(x^*,M)\) is countably additive for every \(x^*\in Y\) . As an application, we give a result on the factorization of multimeasures through reflexive Banach spaces.
