Topics on Analysis in Metric Spaces

This book presents the main mathematical prerequisites for analysis in metric spaces. It covers abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorums, lower semicontinuity of the one-dimensional Hausdorff measure, Sobolev spaces of maps between metric spaces, and Gromov-Hausdorff theory, all developed ina general metric setting. The existence of geodesics (and more generally of minimal Steiner connections) is discussed on general metric spaces and as an application of the Gromov-Hausdorff theory, even in some cases when the ambient space is not locally compact. A brief and very general description of the theory of integration with respect to non-decreasing set functions is presented following the Di Giorgi method of using the 'cavalieri' formula as the definition of the integral. Based on lecture notes from Scuola Normale, this book presents the main mathematical prerequisites for analysis in metric spaces. Supplemented with exercises of varying difficulty it is ideal for a graduate-level short course for applied mathematicians and engineers.

Ambrosio and Tilli's (mathematics, Scuola Normale Superiore, Pisa) textbook is based on a course given by Ambrosio at the Scuola in 1998-1999, presenting the main mathematical prerequisites needed for the "analysis in metric spaces" and to understand several recent research papers on the topic. Coverage includes abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorems, and Sobolev spaces—all developed in a general metric setting; detailed discussion of the geodesic problem and Gromov-Hausdorff convergence; and an overview of the theory of integration with respect to nondecreasing set functions. For use in a one-semester, postgraduate course, or as a reference for researchers. Annotation ©2004 Book News, Inc., Portland, OR

both at the Scuola Normale Superiore, Pisa