Selected Topics on Analysis in Metric Spaces

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These notes originated from a course given by the first author in 1998/99 in the Scuola Normale Superiore of Pisa. The aim of the course was the presentation of the main mathematical prerequisites needed for the so-called Analysis in Metric Spaces. The exposition, though not fully self contained, covers abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorems, Sobolev spaces, all these topics are developed in a general metric setting. We also discuss in detail the geodesic problem and the Gromov-Hausdorff convergence. The last chapter contains a brief, but very general, description of the theory of integration with respect to nondecreasing set functions.