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Topics on analysis in metric spaces. (English) Zbl 1080.28001

Oxford Lecture Series in Mathematics and its Applications 25. Oxford: Oxford University Press (ISBN 0-19-852938-4/hbk). viii, 133 p. (2004).
The book is a concise introduction to analysis in metric spaces but most topics make a good foundation also for convex and fractal geometry. The exposition covers abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorems and Sobolev spaces; all these topics are developed in a general metric setting. One chapter is devoted to the minimal connection problem. It includes both the classical problem of the existence of geodesics in finitely compact metric spaces (due to Busemann) and the abstract Steiner problem (the solution of which is based on the Gromov embedding theorem). The last chapter contains a very general description of the theory of integration with respect to a nondecreasing set of functions. The strictly presented material is enlarged by numerous remarks and also by the end-of-chapter exercises.

MSC:

28-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration
28A78 Hausdorff and packing measures
28A80 Fractals
28B05 Vector-valued set functions, measures and integrals
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
31C15 Potentials and capacities on other spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49J45 Methods involving semicontinuity and convergence; relaxation
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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