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Nonstandard analysis, fractal properties and Brownian motion

P. Potgieter
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Abstract

In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman's lemma and show that Hausdorff dimension can be computed through a counting argument rather than by taking the infimum of a sum of certain covers. This formulation is then applied to obtain a simple proof of the doubling of the dimension of certain sets under a Brownian motion.

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P. Potgieter. 2010-05-07. Nonstandard analysis, fractal properties and Brownian motion. https://arxiv.org/abs/math/0701640

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