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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On a characterization of locally compact groups of second category, assuming the continuum hypothesis
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by Inder K. Rana PDF
Proc. Amer. Math. Soc. 64 (1977), 97-100 Request permission

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 $S \cap {\mathbf {B}}$.
References
  • Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869
  • Kai-Wang Ng, Quasi-invariant measures in groups of second category, J. London Math. Soc. (2) 7 (1973), 171–174. MR 430145, DOI 10.1112/jlms/s2-7.1.171
  • I. K. Rana, On a characterization of standard measurable groups, Sankhyā Ser. A 39 (1977), no. 1, 94–100. MR 492189
  • Dao Xing Xia, Measure and integration theory on infinite-dimensional spaces. Abstract harmonic analysis, Pure and Applied Mathematics, Vol. 48, Academic Press, New York-London, 1972. Translated by Elmer J. Brody. MR 0310179
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • 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