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The recognition of certain Hausdorff like measures

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Ronald Maude
Ronald Maude
Affiliation:
Department of Pure Mathematics, The University, Leeds. LS2 9JT
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Extract

For a metric space <Ω, ρ> and a ‘measure function’ h, the Hausdorff measure mh on Ω is denned by applying Method II to the premeasure defined by τ(E) = h(d(E)), E ⊆ Ω, where

with d(Φ) = 0, is the diameter of E. The set function mh is then a metric outer measure. There are many variations on this definition producing measures also associated with the name Hausdorff. Here we are concerned with those measures which arise when there is a restriction on the sets E for which τ is defined. Such measures arise, for example, as net measures, Rogers [1]. Also we might find it useful to have τ defined only on disks, or only on squares, or only on rectangles with a given relation between vertical and horizontal sides.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 29 , Issue 2 , December 1982 , pp. 260 - 263
Copyright
Copyright © University College London 1982

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References

1.Rogers, C. A.. Hausdorg Measures (C.U.P., 1970).Google Scholar