Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-30T19:29:58.142Z Has data issue: false hasContentIssue false
  • English
  • Français

Lebesgue measures on topological spaces

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

A. G. A. G. Babiker
A. G. A. G. Babiker
Affiliation:
Department of Mathematics, Faculty of Science, University of Khartoum, Khartoum, Sudan.
Get access

Extract

Let μ be a Borel measure on a completely regular space X, and denote by ℱ the σ-algebra of all μ*-measurable subsets of X. Suppose that, as an abstract measure space, (X, ℱ, μ) is isomorphism mod zero with the standard Lebesgue space (I, ℒ, m) via an isomorphism φ : X → I. In this note we attempt to answer the following question: Under what conditions can the isomorphism φ be chosen to be a homeomorphism mod zero? When X is compact, the existence of such a homeomorphism was established in [3, §4] under the assumption of uniform regularity of μ. Whether or not the result can be established without this assumption, was posed as an open question there. Here, we give necessary and sufficient conditions for the existence of the above homeomorphism, together with various examples showing, among other things, that the assumption of uniform regularity used in [3, §4] cannot be dropped.

MSC classification

Secondary: 28A60: Measures on Boolean rings, measure algebras
Type
Research Article
Information
Mathematika , Volume 24 , Issue 1 , June 1977 , pp. 52 - 59
Copyright
Copyright © University College London 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Babiker, A. G. A. G.. “Some measure theoretic properties of completely regular space II ”, Rend. Fisci, Mat., Accademia Nazionale deiLincei, Rome, 59 (1975,6).Google Scholar
2.Babiker, A. G. A. G.. “Uniform regularity of measures on compact spaces ”, J. Für die reine und angewandte math. (in print).Google Scholar
3.Babiker, A. G. A. G.. “On uniformly regular topological measure spaces”, Duke Math. Jour., (in print).Google Scholar
4.Berberian, S. K.. Measure and Integration (London and N. Y. 1965).Google Scholar
5.Bourbaki, N.. Elements de Mathematique, Integration, Chap. 5 (Paris 1956).Google Scholar
6.Bourbaki, N.. Elements de Mathematique, Topologie générate, Chap. 9 (Paris 1958).Google Scholar
7.Dieudonné, J.. “Sur la convergence des suites de measures de Radon ”, Anais Acad. Brasil Ci, 23 (1951), 21–38; 277–282.Google Scholar
8.Gillman, L. and Jerison, M.. Rings of continuous functions (Van Nostrand 1960).Google Scholar
9.Halmos, P. R.. Measure theory (Van Nostrand 1955).Google Scholar
10.Kateov, M.. “Measures on fully normal spaces ”, Fund. Math., 38 (1951), 73–84.CrossRefGoogle Scholar
11.Kelley, J. L.. General topology (Van Nostrand, 1955).Google Scholar
12.Knowles, J. D.. “Measures on topological spaces ”, Proc. London Math. Soc., 17 (1967), 139–156.CrossRefGoogle Scholar
13.Marczewski, E. and Sikorski, R.. “Measures in non-separable metric spaces ”, Colloq., 1 (1948), 133–139.Google Scholar
14.Moran, W.. “The additivity of measures on completely regular spaces ”, J. London Math. Soc., 43 (1968), 633–639.Google Scholar
15.Rochlin, V. A.. “On the fundamental ideas of measure theory ”, Mat. Sb. (N.S.) 25 (67) (1949), 107–150, (Russian). A.M.S. Translation, No. 71 (1952), (English).Google Scholar
16.Schwartz, L.. Radon Measures on arbitrary topological spaces and cylindrical measures (Tata Inst. and Oxford U.P., 1973).Google Scholar
17.Tarski, A.. “Über unerreichbare Kardinalzahlen ”, Fund. Math., 30 (1938), 68–89.CrossRefGoogle Scholar
18.Ulam, S.. “Sur masstheorie der allgemein mengenlehre ”, Fund. Math., 16 (1931), 140–150.Google Scholar
19.Varadarajan, V. S.. “Measures on topological spaces ”, Mat. Sb., (N.S.) 55 (97) (1961), 33–100, (Russian). A.M.S. Translations, No. 48, 14–228, (English).Google Scholar