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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
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\Vert {{\mu _n}\left( E \right)... _{n \to \infty }}\left\Vert {{\mu _n}\left( E \right)} \right\Vert < \infty $ for each $ E \in \Sigma $ then $ \left\{ {{\mu _n}:n \in \mathbb{N}} \right\}$ is uniformly strongly additive.

References [Enhancements On Off] (What's this?)

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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

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