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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

1.Dvoretzky, A.. A note on Hausdorff dimension functions. Proc. Camb. Phil. Soc, 44 (1948), 13–16.Google Scholar
2.Haase, H.. Non-σ-finite sets for packing measure. Mathematika, 33 (1986), 129–136.CrossRefGoogle Scholar
3.Hausdorff, F.. Dimension und ausseres Mass. Math. Ann., 79 (1919), 157–179.Google Scholar
4.Rogers, C. A.. Hausdorff Measures (Cambridge Univ. Press, 1970).Google Scholar
5.Seneta, E.. Regularly Varying Functions. Lecture Notes in Math., vol. 508 (Springer, 1971).Google Scholar
6.Taylor, S. J.. The measure theory of random fractals. Math. Proc. Camb. Phil. Soc, 100 (1986), 383–406.CrossRefGoogle Scholar