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Lower dimensional absolute continuity properties of quasiconformal mappings

Published online by Cambridge University Press:  24 October 2008

F. W. Gehring
F. W. Gehring
Affiliation:
University of Michigan, Ann Arbor, Michigan
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Extract

Let Rn denote Euclidean n-space. For p ∈ [0, ∞), the normalized Hausdorff p-dimensional outer measure of a set E ⊂ Rn is defined as

where the infimum is taken over all countable coverings of E by sets Ej with

and

(See, for example (4).) Then ℋP (E) measures the p-dimensional size of E. For example, ℋ0 (E) ≤ ∞ if and only if E is finite, in which case ℋ0(E) = card (E). Next if E is an arc or Jordan curve, then ℋ1(E) ≤ ∞ if and only if E is rectifiable, in which case ℋ1(E) = length (E). Finally if p is an integer in [1, n] and if E lies in a p-dimensional hyperplane T in Rn, then

where mp denotes the Lebesgue p-dimensional outer measure in T. Thus as p varies from 0 to n, the measures ℋp interpolate in a natural way between the counting measure and Lebesgue outer measure in Rn.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78 , Issue 1 , July 1975 , pp. 81 - 93
Copyright
Copyright © Cambridge Philosophical Society 1975

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