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Average densities and linear rectifiability of measures

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

P. Mörters
P. Mörters
Affiliation:
Universität Kaiserslautern, Fachbereich Mathematik, 67663 Kaiserslautern, Germany.
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Abstract

We show that a measure on ℝd is linearly rectifiable if, and only if, the lower l-density is positive and finite and agrees with the lower average l-density almost everywhere.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 44 , Issue 2 , December 1997 , pp. 313 - 324
Copyright
Copyright © University College London 1997

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References

Ban92.Bandt, C.. The tangent distribution for self-similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.Google Scholar
BF92.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.CrossRefGoogle Scholar
FS95.Falconer, K. J. and Springer, O. B.. Order-two density of sets and measures with non-integral dimension. Mathematika, 42 (1995), 114.Google Scholar
Gra95.Graf, S.. On Bandt's tangential distribution for self-similar measures. Mh. Math., 120 (1995), 223–246.Google Scholar
Kir88.Kirchheim, B.. Uniformly distributed measures, tangent measures and analytic varieties. In Proc. Conf. Topology and Measure V, Wissenschaftliche Beiträge der Ernst-Moritz-Arndt-Universität Greifswald (1988), 54 60.Google Scholar
Mar64.Marstrand, J. M.The (ø s)-regular subsets of n space. Trans. Amer. Math. Soc., 113 (1964), 369–392.Google Scholar
Mar96.Marstrand, J. M.. Order-two density and the strong law of large numbers. Mathematika, 43 (1996), 1–22.CrossRefGoogle Scholar
Mat95.Mattila, P.. The Geometry of Sets and Measures in Euclidean Spaces (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
Mör95Morters, P.. Tangent Measure Distributions and the Geometry of Measures. PhD thesis (University College London, 1995).Google Scholar
Mör97Morters, P.. Symmetry properties of average densities and tangent measure distributions of measures on the line. To appear in Adv. Appl. Math., 1997.Google Scholar
Mp97.Morters, P. and Preiss, D.. Tangent measure distributions of fractal measures. Preprint, 1997 (Universität Kaiserslautern).Google Scholar
O'N95.O'Neil, T. C.. A Local Version of the Projection Theorem and Other Results in Geometric Measure Theory. PhD thesis (University College London, 1995).Google Scholar
Pre87.Preiss, D., Geometry of measures in ℝn: Distribution, rectifiability and densities. Ann. Math., 125 (1987), 537–643.Google Scholar