Measure Theory and Fine Properties of Functions, Revised Editionprovides a det ailed examination of the central assertions of measure theory in n-dimensional E uclidean space. The book emphasizes the roles of Hausdorff measure and capacity in characterizing the fine properties of sets and functions.Topics covered inc lude a quick review of abstract measure theory, theorems and differentiation in Rn, Hausdorff measures, area and coarea formulas for Lipschitz mappings and related change-of-variable formulas, and Sobolev functions as well as functions of bounded variation.The text provides complete proofs of many key results omi tted from other books, including Besicovitch's covering theorem, Rademacher's th eorem (on the differentiability a.e. of Lipschitz functions), area and coarea fo rmulas, the precise structure of Sobolev and BV functions, the precise structure of sets of finite perimeter, and Aleksandrov's theorem (on the twice differenti ability a.e. of convex functions).This revised edition includes countless impr ovements in notation, format, and clarity of exposition. Also new are several se ctions describing the p-? theorem, weak compactness criteria inL1, and You ng measure methods for weak convergence. In addition, the bibliography has been updated.Topics are carefully selected and the proofs are succinct, but compl ete. This book provides ideal reading for mathematicians and graduate students i n pure and applied mathematics.