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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.11872 |
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| _version_ | 1866913892299440128 |
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| author | Rodríguez, José |
| author_facet | Rodríguez, José |
| contents | 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 $σ$-algebra $Σ$ and for each map $ν:Σ\to X$, the countable additivity of the composition $x^*\circ ν$ for all $x^*\in Y$ 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 $(A_n)_{n}$ in $Σ$ 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 $s(x^*,M):Σ\to \mathbb{R}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11872 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Diestel-Faires type result for multimeasures Rodríguez, José Functional Analysis 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 $σ$-algebra $Σ$ and for each map $ν:Σ\to X$, the countable additivity of the composition $x^*\circ ν$ for all $x^*\in Y$ 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 $(A_n)_{n}$ in $Σ$ 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 $s(x^*,M):Σ\to \mathbb{R}$ 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. |
| title | A Diestel-Faires type result for multimeasures |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2506.11872 |