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Proceedings of the American Mathematical Society

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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.

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

  • [1] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
  • [2] G. G. Gould, Integration over vector-valued measures, Proc. London Math. Soc. (3) 15 (1965), 193–225. MR 0174694
  • [3] Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470
  • [4] V. I. Rybakov, On the theorem of Bartle, Dunford and Schwartz on vector-valued measures, Mat. Zametki 7 (1970), 247–254 (Russian). MR 0260971

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Keywords: Measures, absolute continuity, vector-valued measures, Choquet simplex, maximal measures
Article copyright: © Copyright 1971 American Mathematical Society