Measure Theory and Fine Properties of Functions, Revised Edition
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.
Topics covered include 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 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 revised edition includes countless improvements in notation,
format, and clarity of exposition. Also new are several sections
describing the π-λ 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.