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On the existence of non-atomic measures

Published online by Cambridge University Press:  26 February 2010

J. D. Knowles
J. D. Knowles
Affiliation:
Westfield College, London, N.W.3.
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Throughout this note X will denote a completely regular Hausdorff space and ℬ the σ-algebra of Borel sets in X (see §2 for terminology). For x in X we may define the atomic Borel measure δx to be the unit mass placed at the point x. Observe, however, that the example of Dieudonné [3; §52, example 10] shows that not all atomic measures need have this form. The problem investigated in this note is the existence of finite Borel measures other than the atomic measures. In §3 we show that such a measure necessarily exists on a compact space without isolated points, though we are not able to add to the very meagre supply of examples which are known at the moment. A converse to our result has been given by Rudin [7] and, for completeness, we include a new proof of his result.

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Type
Research Article
Information
Mathematika , Volume 14 , Issue 1 , June 1967, pp. 62 - 67
Copyright
Copyright © University College London 1967

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References

1.Bourbaki, N., Intégration, Ch. I-IV (Actualités Sci. Ind. 1175 (Paris, 1965)).Google Scholar
2.Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, 1960).CrossRefGoogle Scholar
3.Halmos, P. R., Measure theory (Van Nostrand, 1950).CrossRefGoogle Scholar
4.Kelley, J. L., General topology (Van Nostrand, 1955).Google Scholar
5.Knowles, J. D., “Measures on topological spaces”, Proc. London Math. Soc. (3), 17 (1967), 139–156.CrossRefGoogle Scholar
6.Parthasarathy, K. R., Rao, R. R. and Varadhan, S. R. S., “On the category of indecomposable distributions on topological groups”, Trans. American Math. Soc., 102 (1962), 200–217.CrossRefGoogle Scholar
7.Rudin, W., “Continuous functions on compact spaces without perfect subsets”, Proc. American Math. Soc., 8 (1957), 39–42.CrossRefGoogle Scholar
8.Varadarajan, V. S., “Measures on topological spaces” (Russian), Mat. Sbornik (N.S.), 55 (97) (1961), 33–100.Google Scholar
English translation: American Math. Soc. Translations Series 2, 48 (1965), 161–228.CrossRefGoogle Scholar