Uniformly distributed measures in Euclidean spaces
Bernd Kirchheim and David Preiss
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Submission date: 09. Sep. 1998
published in: Mathematica Scandinavica, 90 (2002) 1, p. 152-160
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Uniformly distributed measures naturally occur as blow-ups (or tangent measures) of measures having densities. Here we consider the local behaviour of such measures in Euclidean spaces. In particular, we show that the support of such a measure is always an analytic variety, which is even algebraic provided that the measure of the whole space is finite. As an application we obtain a simple proof of the remarkable result by Marstrand on the non-existence of densities for fractal dimensions.