Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T06:19:22.012Z Has data issue: false hasContentIssue false
  • English
  • Français

The Hausdorff Dimension Distribution of Finite Measures in Euclidean Space

Published online by Cambridge University Press:  20 November 2018

Colleen D. Cutler
Colleen D. Cutler*
Affiliation:
University of Waterloo, Waterloo, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let E be a Borel set of RN. The α-outer Hausdorff measure of E has been defined to be

where

and each Bi is a closed ball. d(Bi) denotes the diameter of Bi.

It is easily seen that the same value Hα(E) is obtained if we consider coverings of E by open balls or by balls which may be either open or closed.

By dim(E) we will mean the usual Hausdorff-Besicovitch dimension of E, where

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 6 , 01 December 1986 , pp. 1459 - 1484
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Billingsley, P., Hausdorff dimension in probability theory, Illinois Journal of Mathematics 4 (1960), 187–209.Google Scholar
2. Billingsley, P., Hausdorff dimension in probability theory II, Illinois Journal of Mathematics 5 (1961), 291–298.Google Scholar
3. Billingsley, P., Ergodic theory and information (Krieger, 1978).Google Scholar
4. Cutler, C., Some measure-theoretic and topological results for measure-valued and set-valued stochastic processes, Ph.D. Thesis, Carleton University (1984).Google Scholar
5. Gács, P., Hausdorff dimension and probability distributions, Periodica Mathematica Hungarica (Budapest) 3 (1973), 59–71.Google Scholar
6. Loève, M., Probability theory II (Springer-Verlag, 1978).CrossRefGoogle Scholar
7. Rogers, C. A., Hausdorff measures (Cambridge University Press, 1970).Google Scholar
8. Rogers, C. A. and Taylor, S. J., The analysis of additive set functions in Euclidean space, Acta Mathematica Stockholm 101 (1959), 273–302.Google Scholar