Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-11T00:09:13.858Z Has data issue: false hasContentIssue false
  • English
  • Français

Order-two density and the strong law of large numbers

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

J. M. Marstrand
J. M. Marstrand
Affiliation:
School of Mathematics, University Walk, Bristol. BS8 1TW
Get access

Extract

Throughout this paper we assume that k is a given positive integer. As usual, B(x, r) denotes the closed ball with centre at x∈ℝk and radius r > 0. Let μ be a Radon measure on ℝk, that is, μ is locally finite and Borel regular. For s ≥ 0, the lower and upper s–dimensional densities of μ at x are denned respectively by

and

MSC classification

Secondary: 28A78: Hausdorff and packing measures
Type
Research Article
Information
Mathematika , Volume 43 , Issue 1 , June 1996 , pp. 1 - 22
Copyright
Copyright © University College London 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bedford, T. and Fisher, A. M.. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc. (3), 64 (1992), 95–124.Google Scholar
1.Besicovitch, A. S.. On the fundamental geometrical properties of linearly measurable sets of points. Math. Annalen, 98 (1928), 422–464.CrossRefGoogle Scholar
3.Besicovitch, A. S.. On the fundamental geometrical properties of linearly measurable sets of points. II. Math. Annalen, 115 (1938), 296–329.CrossRefGoogle Scholar
4.Besicovitch, A. S.. On the fundemental geometric properties of linearly measurable sets of points. III. Math. Annalen, 116 (1939), 349–357.Google Scholar
5.Falconer, K. J.. The Geometry of Fractal Sets. Cambridge Tracts in Mathematics, 85 (Cambridge University Press, 1985).CrossRefGoogle Scholar
6.Falconer, K. J.. Wavelet transforms and order-two densities of fractals. J. Stat. Physics, 67 (1992), 781–793.CrossRefGoogle Scholar
7.Falconer, K. J. and Springer, O. B.. Order-two density of sets and measures with non-integral dimensions. Mathematika, 42 (1995), 1–14.CrossRefGoogle Scholar
8.Federer, H.. Geometric Measure Theory. Grundlehren math. Wiss., 153 (Springer, 1969).Google Scholar
9.Grimmett, G. R. and Stirzaker, D. R.. Probability and Random Processes (Clarendon Press, Oxford, 1992).Google Scholar
10.Marstrand, J. M.. The ( Ø , n)-regular subsets of n-space. Trans. Amer. Math. Soc., 113 (1964), 369–392.Google Scholar
11.Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces (Cambridge University Press, 1995).CrossRefGoogle Scholar
12.O'Neil, T. C.. A local version of the projection theorem and other results in Geometric Measure Theory. Ph.D. Thesis (University College London, 1994).Google Scholar
13.Preiss, D.. Geometry of measures in ¡n: Distribution, rectifiability and densities. Annals of Math., 125 (1987), 537–643.CrossRefGoogle Scholar
14.Springer, O. B.. Order-two Density and Self-conformal Sets. Ph.D. Thesis (University of Bristol, 1994).Google Scholar