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Measures not approximable or not specifiable by means of balls

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Roy O. Davies
Roy O. Davies
Affiliation:
Department of Mathematics, University of Leicester.
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Extract

Problem I was raised (oral communication) by Goffman some three years ago, and I found the example then. Problem II was raised at about the same time by Topsøe; Christensen has given an affirmative answer for spaces satisfying certain additional conditions.

It is easy to see that if the answer to problem I were affirmative then so would be that to problem II; therefore our counter-example for problem II implies the existence of one for problem I. It is also possible that another counter-example for problem I could be found by analysing the construction of Dieudonn6 in [1], which is also concerned (implicitly) with a failure of Vitali's theorem. Nevertheless our construction may be of independent interest.

MSC classification

Secondary: 28A10: Real- or complex-valued set functions
Type
Research Article
Information
Mathematika , Volume 18 , Issue 2 , December 1971 , pp. 157 - 160
Copyright
Copyright © University College London 1971

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References

1.Dieudonné, J., “Sur un théorème de Jessen ”, Fund. Math., 37 (1950), 242–248.CrossRefGoogle Scholar
2.Besicovitch, A. S., “A general form of the covering principle and relative differentiation of additive functions I, Proc. Cambridge Philos. Soc., 41 (1945), 103–110; II, Proc. Cambridge Philos. Soc., 42 (1946), 1–10, with corrections in Proc. Cambridge Philos. Soc., 43 (1947), 590.CrossRefGoogle Scholar
3.Larman, D. G., “A new theory of dimension ”, Proc. London Math. Soc. (3), 17 (1967), 168–192.Google Scholar
4.Davies, Roy O. and Rogers, C. A., “The problem of subsets of finite positive measure ”, Bull. London Math. Soc., 1 (1969), 47–54.CrossRefGoogle Scholar
5.Holmes, R. A., “Graphical representations of complete metric spaces ”, J. London Math. Soc. (2), 2 (1970), 727–735.CrossRefGoogle Scholar