Nonstandard analysis, fractal properties and Brownian motion

Abstract

Using Loeb measure theory, it is possible to construct Lebesgue or even Wiener measure as a hyperfinite counting measure. In this paper I explore an extension of the idea to Hausdorff measure, which yields a hyperfinite formulation of Hausdorff dimension. In cases where the problem of dimension can be interpreted as an equivalent problem on a hyperfinite time line, this can lead to a simple and intuitively satisfying proof. For instance, I shall later consider certain properties of Brownian motion, and present nonstandard proofs which are somewhat easier than the original, and also seem to obey certain statistical “rules of thumb”. I now provide a short overview of the necessary nonstandard analysis as well as the standard formulation of Hausdorff dimension, since the nonstandard version will follow the same style and notation.