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On the selection of non-σ-finite subsets

Published online by Cambridge University Press:  26 February 2010

D. G. Larman
D. G. Larman
Affiliation:
University College, London.
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Let A be a subset of a compact metric space Ω, and suppose that A has non-σ-finite h-measure, where h is some Hausdorff function. The following problem was suggested to me by Professor C. A. Rogers:

If A is analytic, is it possible to construct 2ℵodisjoint closed subsets of A which also have non-σ-finite h-measure?

At this level of generality the problem, like others which involve selection of subsets, appears to offer some difficulty. Here we prove two results which were motivated by it.

Type
Research Article
Information
Mathematika , Volume 14 , Issue 2 , December 1967 , pp. 161 - 164
Copyright
Copyright © University College London 1967

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References

1.Larman, D. G., “A new theory of dimension”, Proc. London Math. Soc. (3), 17 (1967), 178–192.Google Scholar
2.Larman, D. G., “On Hausdorff measure in finite-dimensional compact metric spaces”, Proc. London Math. Soc. (3), 17 (1967), 193–206.Google Scholar
3.Besicovitch, A. S., “Concentrated and rarified sets of points”, Acta Mathematica, 62 (1934), 289–300.Google Scholar
4.Rogers, C. A., “Sets non-σ-finite for Hausdorff measures”, Mathematika, 9 (1962), 95–103.CrossRefGoogle Scholar
5.Besicovitch, A. S., “A theorem on s-dimensional sets of points”, Proc. Cambridge Phil. Soc., 38 (1942), 24–27.Google Scholar
6.Sion, M. and Sjerve, D., “Approximation properties of measures generated by continuous set functions”, Mathematika, 9 (1962), 145–156.Google Scholar