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Scaling Hausdorff measures

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

R. Daniel Mauldin  and
S. C. Williams
R. Daniel Mauldin
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203-5116, U.S.A..
S. C. Williams
Affiliation:
Department of Mathematics, Utah State University, Logan, Utah 84322, U.S.A.
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Extract

In this note, we investigate those Hausdorff measures which obey a simple scaling law. Consider a continuous increasing function θ defined on with θ(0)= 0 and let be the corresponding Hausdorff measure. We say that obeys an order α scaling law provided whenever K⊂ and c> 0, then

or, equivalently, if T is a similarity map of with similarity ratio c:

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 36 , Issue 2 , December 1989 , pp. 325 - 333
Copyright
Copyright © University College London 1989

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References

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