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 fine properties of sets and functions. Topics covered 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 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), the Area and Coarea Formulas, the precise structure of Sobolev and BV functions, the precise structure of sets of finite perimeter, and Alexandro's Theorem (on the twice differentiability a.e. of convex functions).
Topics are carefully selected and the proofs succinct, but complete, which makes this book ideal reading for applied mathematicians and graduate students in applied mathematics.
Table of Contents
GENERAL MEASURE THEORY
Measures and Measurable Functions
Lusin's and Egoroff's Theorems
Integrals and Limit Theorems
Product Measures, Fubini's Theorem, Lebesgue Measure
Differentiation of Radon Measures
Riesz Representation Theorem
Weak Convergence and Compactness for Radon Measures
Definitions and Elementary Properties; Hausdorff Dimension
Hausdorff Measure and Elementary Properties of Functions
AREA AND COAREA FORMULAS
Lipschitz Functions, Rademacher's Theorem
Linear Maps and Jacobians
The Area Formula
The Coarea Formula
Definitions And Elementary Properties. Approximation
Traces. Extensions. Sobolev Inequalities
Quasicontinuity; Precise Representations of Sobolev Functions. Differentiability on Lines
BV FUNCTIONS AND SETS OF FINITE PERIMETER
Definitions and Structure Theorem
Approximation and Compactness
Traces. Extensions. Coarea Formula for BV Functions. Isoperimetric Inequalities.
The Reduced Boundary
The Measure Theoretic Boundary; Gauss-Green Theorem. Pointwise Properties of BV Functions
Essential Variation on Lines
A Criterion for Finite Perimeter. DIFFERENTIABILITY AND APPROXIMATION BY C1 FUNCTIONS.
Lp Differentiability a.e.; Approximate Differentiability
Differentiability A.E. for W1,P (P > N). Convex Functions
Second Derivatives a.e. for convex functions
Whitney's Extension Theorem
Approximation by C1 Functions