This is a book on geometric measure theory. The main theme is the study of the geometric structure of general Borel sets and Borel measures in the euclidean n-space Rn. There will be emphasis on “small irregular” sets having Lebesgue measure zero but being quite different from smooth curves and surfaces. Examples are Cantor-type sets, nonrectifiable curves having tangent nowhere, etc., in short, sets to which the general descriptive term fractal applies. An abundance of such sets comes from dynamical systems: Julia-sets for rational functions of one complex variable, etc. Very general curve – and surface-like objects are also studied extensively. These are rectifiable sets and measures. They include smooth curves and surfaces and share many of their geometric properties when interpreted in a measure-theoretic sense. They form an optimal class possessing such properties.

Many of the basic ideas developed here originate in the pioneering work done by Besicovitch [1], [4] and [5], by Federer [1], by Marstrand [1] and by Preiss [4]. Besicovitch laid down the foundations of geometric measure theory by describing to an amazing extent the structure of the subsets of the plane having finite one-dimensional Hausdorff measure (i.e. length). Federer extended Besicovitch's work to m-dimensional subsets of Rn, m being an integer, and Marstrand analysed general fractals in the plane whose Hausdorff dimension need not be an integer. Preiss solved one of the most long-standing fundamental open problems, introducing and using effectively tangent measures.

Good introductory texts to the mathematical theory of fractals are the books of Edgar [1] and of Falconer [4], [16]. Closest to this text is Falconer [4].