Notes: | The book is a concise introduction to analysis in metric spaces but most topics make a good foundation also for convex and fractal geometry. The exposition covers abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorems and Sobolev spaces; all these topics are developed in a general metric setting. One chapter is devoted to the minimal connection problem. It includes both the classical problem of the existence of geodesics in finitely compact metric spaces (due to Busemann) and the abstract Steiner problem (the solution of which is based on the Gromov embedding theorem). The last chapter contains a very general description of the theory of integration with respect to a nondecreasing set of functions. The strictly presented material is enlarged by numerous remarks and also by the end-of-chapter exercises. (Zentralblatt) |