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 -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 .

**[1]**Paul R. Halmos,*Measure Theory*, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR**0033869****[2]**Kai-Wang Ng,*Quasi-invariant measures in groups of second category*, J. London Math. Soc. (2)**7**(1973), 171–174. MR**0430145**, https://doi.org/10.1112/jlms/s2-7.1.171**[3]**I. K. Rana,*On a characterization of standard measurable groups*, Sankhyā Ser. A**39**(1977), no. 1, 94–100. MR**0492189****[4]**Dao Xing Xia,*Measure and integration theory on infinite-dimensional spaces. Abstract harmonic analysis*, Academic Press, New York-London, 1972. Translated by Elmer J. Brody; Pure and Applied Mathematics, Vol. 48. MR**0310179**

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DOI:
https://doi.org/10.1090/S0002-9939-1977-0466400-6

Keywords:
Measurable group,
inner-regular measure,
quasi-invariant measure

Article copyright:
© Copyright 1977
American Mathematical Society