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2000/2001 Existence of Measures with Dominated Translates
Inder K. Rana
Real Anal. Exchange 26(1): 449-452 (2000/2001).


Let $G$ be a topological group of second category, ${\cal B}_G$ be its Borel $\sigma$-algebra and ${\cal B}$ a $\sigma$-algebra of subsets of $G$ such that $(G,{\cal B})$ is a measurable group. For a probability measure $P$ on $(G,{\cal B}),$ let $P_g(E):=P(gE)$ for $g\in G, E\in{\cal B}.$ The aim of this note is to show that there exists an inner-regular probability measure $P$ and a $\sigma$-finite measure $\mu$ on $(G,{\cal B})$ such that $P_g\ll \mu\; \forall\; g\in G,$ iff $G$ is locally-compact and in that case $P_g\ll \lambda_G\ll \mu\;\;\forall\; g\in G$ on the $\sigma$-algebra ${\cal B} \cap {\cal B}_G,$ where $\lambda_G$ denotes a Haar measure of $G.$


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Inder K. Rana. "Existence of Measures with Dominated Translates." Real Anal. Exchange 26 (1) 449 - 452, 2000/2001.


Published: 2000/2001
First available in Project Euclid: 2 January 2009

zbMATH: 1011.43001
MathSciNet: MR1825526

Primary: 43A05

Keywords: inner-regular measures , Measurable group , Quasi-invariant measures

Rights: Copyright © 2000 Michigan State University Press

Vol.26 • No. 1 • 2000/2001
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