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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

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