Lawrence Craig Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, 2nd revised ed., Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015, xi+309 pp., ISBN: 978-1-4822-4238-6/hbk.

Abstract

This is a new revised edition of a very successful book (published by CRC Press in 1992) dealing with measure theory in Rn and some special properties of functions, usually omitted from books dealing with abstract measure theory, but which a working mathematician analyst must to know. Among these special topics we mention: Vitali's and Besicovitch's covering theorems, Hausdor measures and capacities (for classifying classes of negligible sets for various ne properties of functions), Rademacher's Theorem on the a.e. dierentiability of Lipschitz functions (a topic of great actual interest in connection with its extension to innite dimensions), the area and coarea formulas (yielding change-of-variable rules for Lipschitz functions between Rn and Rm), the Lebesgue-Besicovitch dierentiation theorem, the precise structure of BV and Sobolev functions, Alexandrov's theorem on a.e. twice differentiability of convex functions, Whithney's extension theorem with applications to approximation of Sobolev and BV functions, etc. The book is clearly written with complete proofs, including all technicalities. One assumes that the reader is familiar with Lebesgue measure and abstract measure theory.