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On the dimension of product measures

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

H. Haase
H. Haase
Affiliation:
Sektion Mathematik der E.-M.-Arndt-Universität, F.-L.-Jahn-Str. 15a, Greifswald, DDR-2200.
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In recent papers on fractals attention has shifted from sets to measures [1, 5, 10]. Thus it seems interesting to know whether results for the dimension of sets remain valid for the dimension of measures. In the present paper we derive estimates for the dimension of product measures. Falconer [3] summarizes known results for sets and Tricot [8] gives a complete description in terms of Hausdorff and packing dimension. Let dim and Dim denote Hausdorff and packing dimension. If then

MSC classification

Secondary: 28A35: Measures and integrals in product spaces
Type
Research Article
Information
Mathematika , Volume 37 , Issue 2 , December 1990 , pp. 316 - 323
Copyright
Copyright © University College London 1990

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References

1. Barnsley, M. F.. Fractals Everywhere (Academic Press, 1988).Google Scholar
2. Eggeleston, H. G.. A correction to a paper on the dimension of cartesian product sets. Proc. Camb. Phil. Soc., 49 (1953), 437–440.CrossRefGoogle Scholar
3. Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, Cambridge, 1985).CrossRefGoogle Scholar
4. Federer, H.. Geometric Measure Theory (Springer, 1969).Google Scholar
5. Geronimo, J. S. and Hardin, D. P.. An exact formula for the measure dimension associated with a class of piecewise linear maps. Constr. Approx., 5 (1989), 89–98.CrossRefGoogle Scholar
6. Raymond, X. S. and Tricot, C.. Packing regularity of sets in n-space. Math. Proc. Camb. Phil. Soc., 103 (1988), 133–145.CrossRefGoogle Scholar
7. Taylor, S. J. and Tricot, C.. Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc., 288 (2) (1985) 679–699.CrossRefGoogle Scholar
8. Tricot, C.. Two definitions of fractional dimension. Math. Proc. Camb. Phi. Soc., 91 (1982), 57–74.CrossRefGoogle Scholar
9. Tricot, C.. Rarefaction indices. Mathematika, 27 (1980), 46–57.CrossRefGoogle Scholar
10. Withers, Wm. D.. Analysis of invariant measures in dynamical systems by Hausdorff measure. Pacific J. of Math., 124 (1987), 385–400.CrossRefGoogle Scholar