# 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 $1992$, **CRC Press**

- ISBN 978-0849371578.

Part of the **Studies in Advanced Mathematics** series.

### Subject Matter

### 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*