Based on lecture notes from the 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. The book covers abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorems, lower semicontinuity of the one-dimensional Hausdorff measure, Sobolev spaces of maps between metric spaces, and Gromov-Hausdorff theory, all developed in a general metric setting. The existence of geodesics (and more generally of minimal Steiner connections) is discussed in 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 Cavalieri's formula as the definition of the integral.

This book is a concise introduction to analysis in metric spaces.. * Irmina Herburt, Zentralblatt Math, Vol 1080 *
The book presents a well-chosen selection of what to say to students in the field, when time is limited, with suggestions for further studies. The enjoyable expositions of the subject is supplemented by exercises of various levels. * EMS Newsletter *
... a particularly elegant and original approach. * Mathematical Reviews *