Mutual absolute continuity of sets of measures

Author:
Bertram Walsh

Journal:
Proc. Amer. Math. Soc. **29** (1971), 506-510

MSC:
Primary 28.50

DOI:
https://doi.org/10.1090/S0002-9939-1971-0279275-X

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]**N. Dunford and J. T. Schwartz,*Linear operators*. I:*General theory*, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR**22**#8302. MR**0117523 (22:8302)****[2]**G. G. Gould,*Integration over vector-valued measures*, Proc. London Math. Soc. (3)**15**(1965), 193-225. MR**30**#4894. MR**0174694 (30:4894)****[3]**R. R. Phelps,*Lectures on Choquet's theorem*, Van Nostrand, Princeton, N. J., 1966. MR**33**#1690. MR**0193470 (33:1690)****[4]**V. I. Rybakov,*K teoreme Bartla-Danforda-Švarca o vektornyh merah*, Mat. Zametki**7**(1970), 247-254. MR**0260971 (41:5591)**

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Additional Information

DOI:
https://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