Chapter 25 - Fractal Measures

Abstract

This chapter describes the different aspects of fractal measures. A fractal is a highly irregular set with a fine structure that is with irregular detail at arbitrary small scales. Often a fractal has some sort of self-similarity or self-affinity, perhaps in a statistical or approximate sense. A brief account of Hausdorff measures and their variants and extensions as the natural tools for studying the geometry of fractals is presented in the chapter. The definition of Hausdorff measure immediately leads to the notion of Hausdorff dimension, and many properties are then conveniently expressed in terms of dimensions rather than measures. In a sense, Hausdorff measures are measures that are distributed as uniformly as possible across fractal sets. The calculation of Hausdorff and packing measures and dimensions is explained in the chapter. A direct covering estimate can give good upper bounds for Hausdorff measures. Tangent measures have also been used in the chapter to show that integral dimensionality and rectifiability follow from even weaker density conditions, such as conditions relating to average densities of the form.