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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Author: Inder K. Rana
Journal: Proc. Amer. Math. Soc. 64 (1977), 97-100
MSC: Primary 22D05; Secondary 43A05
DOI: https://doi.org/10.1090/S0002-9939-1977-0466400-6
MathSciNet review: 0466400
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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 $S \cap {\mathbf {B}}$.


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Keywords: Measurable group, inner-regular measure, quasi-invariant measure
Article copyright: © Copyright 1977 American Mathematical Society