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 is absolutely continuous with respect to for a norm-dense of elements of the dual of the Banach space.

**[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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DOI:
http://dx.doi.org/10.1090/S0002-9939-1971-0279275-X

Keywords:
Measures,
absolute continuity,
vector-valued measures,
Choquet simplex,
maximal measures

Article copyright:
© Copyright 1971
American Mathematical Society