2025/06/13 by José Rodríguez, Rodríguez, José
Computer Science · Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2506.11872
openalex publication_date 2025/06/13 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28
Let X be a real Banach space and let Y ⊆ X^* be a linear subspace having the Orlicz-Thomas property, that is, for each σ-algebra Σ and for each map ν:Σ→ X, the countable additivity of the composition x^*∘ ν for all 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, 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 in Σ the series of sets ∑n M(An) is unconditionally convergent and the equality M(\bigcupn An)=∑n M(An) holds. (ii) M is a multimeasure, that is, for every x^*∈ X^* the support map s(x^*,M):Σ→ ℝ defined by s(x^*,M)(A):=sup \x^*(x):x∈ M(A)\ is countably additive. (iii) s(x^*,M) is countably additive for every x^*∈ Y. As an application, we give a result on the factorization of multimeasures through reflexive Banach spaces.