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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Saeki’s improvement of the Vitali-Hahn-Saks-Nikodým theorem holds precisely for Banach spaces having cotype


Author: Paul Abraham
Journal: Proc. Amer. Math. Soc. 116 (1992), 171-173
MSC: Primary 46B20
DOI: https://doi.org/10.1090/S0002-9939-1992-1095219-9
MathSciNet review: 1095219
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Abstract: We prove that a Banach space $X$ has nontrivial cotype if and only if given any $\sigma$-field $\Sigma$ and any sequence ${\mu _n}:\Sigma \to X$ of strongly additive vector measures such that for some $\gamma \geq 1,\lim {\sup _{n \to \infty }}\left \| {{\mu _n}\left ( E \right )} \right \| \leq \gamma \lim {\inf _{n \to \infty }}\left \| {{\mu _n}\left ( E \right )} \right \| < \infty$ for each $E \in \Sigma$ then $\left \{ {{\mu _n}:n \in \mathbb {N}} \right \}$ is uniformly strongly additive.


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Keywords: Vitali-Hahn-Saks-Nikodym Theorem, Vitali-Hahn-Saks Theorem, Banach spaces having cotype
Article copyright: © Copyright 1992 American Mathematical Society