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On almost discrete spaces

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Abdel Ghayoum A. Babiker
Abdel Ghayoum A. Babiker
Affiliation:
Westfield College, London, N.W.3.
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Extract

In [5] Knowles proved the following.

Every compact Hausdorff space with no isolated points admits a non-atomic measure.

This note is concerned with the converse problem in a more general set up. Here we deal with certain properties of the family of completely regular spaces admitting no continuous measures. In §3 it is shown that this family contains spaces with no isolated points, thus theorem (1.1) does not generalize to completely regular spaces. In §4 a canonical decomposition of the compact members of the above family into discrete subspaces is obtained, and it is shown that these spaces are metrizable whenever they satisfy the first axiom of countability.

MSC classification

Secondary: 28A10: Real- or complex-valued set functions
Type
Research Article
Information
Mathematika , Volume 18 , Issue 2 , December 1971 , pp. 163 - 167
Copyright
Copyright © University College London 1971

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References

1.Babiker, A., “Uniform continuity of measures on completely regular spaces ” (to appear).Google Scholar
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