Mutual absolute continuity of sets of measures

Walsh, Bertram

doi:10.1090/S0002-9939-1971-0279275-X

Mutual absolute continuity of sets of measures
HTML articles powered by AMS MathViewer

by Bertram Walsh PDF

Proc. Amer. Math. Soc. 29 (1971), 506-510 Request permission

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 $\left | {\langle \mu ( \cdot ),x’\rangle } \right |$ for a norm-dense ${G_\delta }$ of elements $x’$ of the dual of the Banach space.

References

Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
G. G. Gould, Integration over vector-valued measures, Proc. London Math. Soc. (3) 15 (1965), 193–225. MR 174694, DOI 10.1112/plms/s3-15.1.193
Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470
V. I. Rybakov, On the theorem of Bartle, Dunford and Schwartz on vector-valued measures, Mat. Zametki 7 (1970), 247–254 (Russian). MR 260971