Existence of Measures with Dominated Translates

Abstract

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 $G.$