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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.
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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

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