On a characterization of locally compact groups of second category, assuming the continuum hypothesis
HTML articles powered by AMS MathViewer
- 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
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