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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
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