On a characterization of locally compact groups of second category, assuming the continuum hypothesis
Inder K. Rana
Proc. Amer. Math. Soc. 64 (1977), 97-100
Primary 22D05; Secondary 43A05
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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 -algebra of subsets of G such that (G, B) is a measurable group. For a probability measure P on (G, B), write for the measure defined by . The aim of this paper is to prove the following: if on (G, B) there exists an inner-regular probability measure P such that for every , where is some -finite measure on (G, B), then G is locally compact. Further if S denotes the -algebra generated by the topology of G and m denotes a Haar measure on G, then for every on the -algebra .
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