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Approximation properties of measures generated by continuous set functions

Published online by Cambridge University Press:  26 February 2010

M. Sion  and
D. Sjerve
M. Sion
Affiliation:
The University of British Columbia, Vancouver 8, B.C., Canada
D. Sjerve
Affiliation:
The University of British Columbia, Vancouver 8, B.C., Canada
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Extract

Let X be a metric space and τ a non-negative function on the subsets of X. By the well-known Carathéodory process, we generate outer measures μδ(τ), for δ > 0, and (see §3). When, for every A ⊂ X, τA = (diamA)s for s ≥ 0, μ(τ) is the Hausdorff s-dimensional measure, and, if τA = h(diam A) for a monotone continuous function h with h(0) = 0, μ(τ) is the Hausdorff h-measure. In both of these cases, μ(τ) has been extensively studied.

Type
Research Article
Information
Mathematika , Volume 9 , Issue 2 , December 1962 , pp. 145 - 156
Copyright
Copyright © University College London 1962

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References

1. Besicovitch, A. S., “Concentrated and rarified sets of points”. Acta Math., 62 (1934), 289–300.Google Scholar
2. Besicovitch, A. S., “On the definition of tangents to sets of infinite linear measure”. Proc. Cambridge Phil. Soc., 52 (1956), 20–29.CrossRefGoogle Scholar
3. Davies, R. O., “A property of Hausdorff measure”, Proc. Cambridge Phil. Soc., 52 (1956), 30–34.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. Kuratowski, K., Topologie, 3rd ed. (Warsaw, 1952).Google Scholar
6. Munroe, M. E., Introduction to measure and integration, (Addison-Wesley, 1959).Google Scholar
7. Rogers, C. A., “Sets non-σ-finite for Hausdorff measures”, Mathematika, 9 (1962), 95–103.Google Scholar
8. Sion, M., “On eapacitability and measurability”, Annales Inst. Fourier, 13 (1963), 83–99.Google Scholar