This popular textbook provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space. The book emphasizes the roles of Hausdorff measure and capacity in characterizing the fine properties of sets and functions. This book gathers together the essentials of real analysis on Rn, with particular emphasis on integration and differentiation.

This widely popular treatment has been updated to address all needed corrections and minor edits from the previous Revised Edition. The book includes many interesting topics working mathematical analysts need to know, but ones rarely taught.

Topics covered include a quick review of abstract measure theory and includes complete proofs of many key results omitted from other books, including Besicovitch's covering theorem, Rademacher's theorem (on the differentiability a.e. of Lipschitz functions), 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).

This new edition includes countless improvements in notation, format, and clarity of exposition. Also new are several sections describing the ?-? theorem, weak compactness criteria in L1, and Young measure methods for weak convergence. In addition, the bibliography has been updated.

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 mathematics. The authors assume readers are at least fairly conversant with both Lebesgue measure and abstract measure theory. The expository style reflects this expectation. The book does not offer lengthy heuristics or motivation, but as compensation have tried to present all the technicalities of the proofs.