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An example concerning completion regular measures, images of measurable sets and measurable selections

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

A. G. A. G. Babiker  and
J. D. Knowles
A. G. A. G. Babiker
Affiliation:
Westfield College, London.
J. D. Knowles
Affiliation:
Westfield College, London.
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Extract

We exhibit (§2) an example of §a compact Hausdorff space supporting a Radon probability measure μ and a continuous map ø : X → I, when I is the closed unit interval, for which the image measure ø(μ) is Lebesgue measur m with the properties:

(i) there exists an open set G ⊂ X for which ø(G) is not m-measurable;

(ii) μ is a non-atomic non-completion regular measure;

(iii) the measure algebras (X, μ) and (I, m) are isomorphic but for no choice c sets B ⊂ X, B′ ⊂ I of measure zero are and homeomorphic

(iv) there exists a selection p : I → X (i.e. p(t) ∊ ø−1(t) for all t ∊ I) which i Borel m-measurable, but there is no Lusin m-measurable selection.

MSC classification

Secondary: 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets
Type
Research Article
Information
Mathematika , Volume 25 , Issue 1 , June 1978 , pp. 120 - 124
Copyright
Copyright © University College London 1978

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