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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Mutual absolute continuity of sets of measures


Author: Bertram Walsh
Journal: Proc. Amer. Math. Soc. 29 (1971), 506-510
MSC: Primary 28.50
MathSciNet review: 0279275
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Abstract: A theorem slightly stronger than the following is proved: If K is a convex set of (signed) measures that are absolutely continuous with respect to some fixed positive sigma-finite measure, then the subset consisting of those measures in K with respect to which all measures in K are absolutely continuous is the complement of a set of first category in any topology finer than the norm topology of measures. This implies, e.g., that any Banach-space-valued measure $ \mu $ is absolutely continuous with respect to $ \left\vert {\langle \mu ( \cdot ),x'\rangle } \right\vert$ for a norm-dense $ {G_\delta }$ of elements $ x'$ of the dual of the Banach space.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0279275-X
PII: S 0002-9939(1971)0279275-X
Keywords: Measures, absolute continuity, vector-valued measures, Choquet simplex, maximal measures
Article copyright: © Copyright 1971 American Mathematical Society