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Sets non-σ-finite for Hausdorff measures

Published online by Cambridge University Press:  26 February 2010

C. A. Rogers
C. A. Rogers
Affiliation:
University of British Columbia, Vancouver 8, B.C., Canada.
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Let ℋ be the system of all continuous increasing functions h(t), denned for t ≥ 0, with h(0) = 0 and h(t)>0 for t > 0. Let Ω be a separable metric space. Then, for each h of ℋ, we may introduce a Hausdorff measure into Ω, by taking

where d(Fi) denotes the diameter of Fi, and where the infimum is taken over all sequences {Fi} of closed sets, covering E and having diameters less than δ. We introduce a natural partial order in the system of these Hausdorff measures by writing j < h, if j, h are functions of ℋ and

Type
Research Article
Information
Mathematika , Volume 9 , Issue 2 , December 1962 , pp. 95 - 103
Copyright
Copyright © University College London 1962

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References

1. Besicovitch, A. S., “On the definition of tangents to sets of infinite linear measure”, Proc. Cambridge Phil. Soc, 52 (1956), 20–29.CrossRefGoogle Scholar
2. Rogers, C. A. and Taylor, S. J., Additive set functions in Euclidean space”, II, to be submitted to Acta Mathematica.Google Scholar
3. Besicovitch, A. S., “Concentrated and rarified sets of points”, Acta Mathematica, 62 (1934), 289–300.CrossRefGoogle Scholar
4. Davies, R. O., “Non σ-finite closed subsets of analytic sets”, Proc. Cambridge Phil. Soc, 52 (1956), 174–177.CrossRefGoogle Scholar
5. Blaschke, W., Kreis und Kugel (Leipzig, 1916).CrossRefGoogle Scholar