This popular textbook provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space, with emphasis upon the roles of Hausdorff measure and capacity in characterizing the fine properties of sets and functions. Measure Theory and Fine Properties of Functions, Second Edition includes many interesting items working mathematical analysts need to know, but are rarely taught. Topics covered include a review of abstract measure theory, including Besicovitch’s covering theorem, Rademacher’s theorem (on the differentiability a.e. of Lipschitz continuous functions), the area and coarea formulas, the precise structure of Sobolev and BV functions, the precise structure of sets of finite perimeter, and Aleksandrov’s theorem (on the twice differentiability a.e. of convex functions). The topics are carefully selected, and the proofs are succinct, but complete. This book provides ideal reading for mathematicians and graduate students in pure and applied
