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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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