Book:Lawrence C. Evans/Measure Theory and Fine Properties of Functions

Part of the Studies in Advanced Mathematics series.

Subject Matter

 * Measure Theory

Contents

 * Preface


 * 1 General Measure Theory
 * 1.1 Measure and measurable functions
 * 1.2 Lusin's and Egoroff's Theorems
 * 1.3 Integrals and limit theorems
 * 1.4 Product measures, Fubini's theorem, Lebesgue measure
 * 1.5 Covering theorems
 * 1.6 Differentiation of Radon measures
 * 1.7 Lebesgue points; Approximate continuity
 * 1.8 Riesz Representation Theorem
 * 1.9 Weak convergence and compactness for Radon measures


 * 2 Hausdorff Measure
 * 2.1 Definitions and elementary properties; Hausdorff dimension
 * 2.2 Isodiametric Inequality: $\mathcal H^n = \mathcal L^n$
 * 2.3 Densities
 * 2.4 Hausdorff measure and elementary properties of functions


 * 3 Area and Coarea Functions
 * 3.1 Lipschitz functions, Rademacher's Theorem
 * 3.2 Linear maps and Jacobians
 * 3.3 The Area Formula
 * 3.4 The Coarea Formula


 * 4 Sobolev Functions
 * 4.1 Definitions and elementary properties
 * 4.2 Approximation
 * 4.3 Traces
 * 4.4 Extensions
 * 4.5 Sobolev inequalities
 * 4.6 Compactness
 * 4.7 Capacity
 * 4.8 Quasicontinuity; Precise representatives of Sobolev functions
 * 4.9 Differentiability on lines


 * 5 BV Functions and Sets of Finite Perimeter
 * 5.1 Definitions; Structure Theorem
 * 5.2 Approximation and compactness
 * 5.3 Traces
 * 5.4 Extensions
 * 5.5 Coarea Formula for BV Functions
 * 5.6 Isoperimetric Inequalities
 * 5.7 The reduced boundary
 * 5.8 The measure theoretic boundary; Gauss–Green Theorem
 * 5.9 Pointwise properties of BV functions
 * 5.10 Essential variation on lines
 * 5.11 A criterion for finite perimeter


 * 6 Differentiability and Approximation by $C^1$ Functions
 * 6.1 $L^p$ differentiability; Approximate differentiability
 * 6.2 Differentiability a.e. for $W^{1,p}$ ($p > n$)
 * 6.3 Convex functions
 * 6.4 Second derivatives a.e. for convex functions
 * 6.5 Whitney's Extension Theorem
 * 6.6 Approximation by $C^1$ functions


 * Bibliography
 * Notation
 * Index