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

Lawrence C. Evans and Ronald F. Gariepy: Measure Theory and Fine Properties of Functions

Published $\text {1992}$, CRC Press

ISBN 978-0849371578

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