On a characterization of locally compact groups of second category, assuming the continuum hypothesis

Abstract:

Let G be a topological group of second category and having cardinality at most that of the continuum. Let B be some $\sigma$-algebra of subsets of G such that (G, B) is a measurable group. For a probability measure P on (G, B), write ${P_g}$ for the measure defined by ${P_g}(E) = P(gE),E \in {\mathbf {B}}$. The aim of this paper is to prove the following: if on (G, B) there exists an inner-regular probability measure P such that ${P_g} \ll \mu$ for every $g \in G$, where $\mu$ is some $\sigma$-finite measure on (G, B), then G is locally compact. Further if S denotes the $\sigma$-algebra generated by the topology of G and m denotes a Haar measure on G, then $\mu \gg m \gg {P_g}$ for every $g \in G$ on the $\sigma$-algebra $S \cap {\mathbf {B}}$.