Topics on Analysis in Metric Spaces

by Luigi Ambrosio, Paolo Tilli

Overview

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.

Product Details

ISBN-13:	9780198529385
Publisher:	Oxford University Press
Publication date:	02/26/2004
Series:	Oxford Lecture Series in Mathematics and Its Applications , #25
Edition description:	New Edition
Pages:	142
Product dimensions:	9.20(w) x 6.52(h) x 0.50(d)

About the Author

both at the Scuola Normale Superiore, Pisa

Table of Contents

1. Some preliminaries in measure theory2. Hausdorff measures and covering theorems in metric spaces3. Lipschitz functions in metric spaces4. Geodesic problem and Gromov-Hausdorff convergence5. Sobolev spaces in a metric framework6. A quick overview on the theory of integration