This book provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space and emphasizes the roles of Hausdorff measure and the capacity in characterizing the find properties of sets and functions. Discussions include a quick review of abstract measure theory, theorems and differentiation in Mn, lower Hausdorff measures, area and coarea formulas for Lipschitz mappings and related change-of-variable formulas, and Sobolev functions and functions of bounded variation. The text includes proofs of many key results omitted from other books, including Besicovitch's covering theorem, Rademacher's theorem, and the area and coarea formulas.