Rectifiable Sets, Densities and Tangent Measures
This interesting book is based on the author’s course given at the University of Zürich. It is addressed to people who are interested in geometric measure theory. The main aim of the book is to provide a self-contained proof of the celebrated Preiss theorem, which in particular characterizes k-dimensional rectifiable sets in an n-dimensional Euclidean space using existence of k-dimensional densities. The Preiss theorem (proved in 1987) is the culmination of the long deep research started by Besicovich in 1938. In its full generality, it gives a rectifiability criterion for measures in terms of upper and lower densities. The author of the book presents a simpler proof of its special case dealing with the density of measures. In this way, he succeeded in presenting a proof accessible for people who are not experts in the field. However, most of deep ideas that are used in the Preiss proof (in particular the method of tangent measures) are needed in this simpler proof and are carefully explained. The first four chapters contain an introduction to rectifiable sets and measures in Euclidean spaces, including basic properties of tangent measures and the most elementary rectifiability criterions. The fifth chapter contains the subtler Marstrand-Mattila rectifiability criterion. In the sixth chapter, the Preiss strategy is explained and its ingredients are proved in the following three chapters. The last chapter presents several open problems related to the topic.