## Tangent measure distributions of fractal measures

• Tangent measure distributions are a natural tool to describe the local geometry of arbitrary measures of any dimension. We show that for every measure on a Euclidean space and every s, at almost every point, all s-dimensional tangent measure distributions define statistically self-similar random measures. Consequently, the local geometry of general measures is not different from the local geometry of self-similar sets. We illustrate the strength of this result by showing how it can be used to improve recently proved relations between ordinary and average densities.

Author: Peter Mörters, David Preiss urn:nbn:de:hbz:386-kluedo-7902 Preprints (rote Reihe) des Fachbereich Mathematik (295) Preprint English 1999 1999 Technische Universität Kaiserslautern 2000/04/03 Palm distributions ; average densities ; geometric measure theory ; order-two densities; tangent measure distributions Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 28-XX MEASURE AND INTEGRATION (For analysis on manifolds, see 58-XX) / 28Axx Classical measure theory / 28A75 Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 28-XX MEASURE AND INTEGRATION (For analysis on manifolds, see 58-XX) / 28Axx Classical measure theory / 28A80 Fractals [See also 37Fxx] 60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Gxx Stochastic processes / 60G57 Random measures Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

$Rev: 13581$