Topics on Analysis in Metric Spaces

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Based on lectures given at the Scuola Normale in Pisa, this book presents the main mathematical prerequisites for analysis in metric spaces. It 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 De Giorgi method of using Cavalieri's formula as the definition of the integral. Intended for a one-semester course at the postgraduate level, or as a reference book for researchers, the text is supplemented with exercises of varying difficulty.
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Table of Contents

1 Some preliminaries on measure theory 1
1.1 Outer measures 1
1.2 Signed and vector measures 9
1.3 Weak convergence of measures 14
2 Hausdorff measures and covering theorems in metric spaces 19
2.1 Hausdorff measures 19
2.2 Covering theorems 23
2.3 Relationships between Hausdorff and Lebesgue measures 28
2.4 Densities 30
3 Lipschitz functions in metric spaces 35
3.1 Definition and general properties 35
3.2 Lipschitz functions of several real variables 39
3.3 The area formula 44
3.4 The one-dimensional area formula 45
4 The geodesic problem and Gromov-Hausdorff convergence 53
4.1 Metric derivative and geodesics in metric spaces 54
4.2 Reparametrization 59
4.3 Existence of geodesics 61
4.4 The intrinsic formulation 64
4.5 Gromov-Hausdorff convergence of metric spaces 79
5 Sobolev spaces in a metric framework 89
5.1 Definition of metric Sobolev spaces 89
5.2 Doubling measures and maximal operators 92
5.3 Equivalence between classical and metric Sobolev spaces 98
5.4 Poincare and Sobolev inequalities 101
6 A quick overview on the theory of integration 113
6.1 The Archimedean integral 114
6.2 Integration with respect to nondecreasing set functions 114
6.3 Integral of extended real-valued functions 121
Bibliography 125
Index 131
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