Measure Theory and Fine Properties of Functions, Revised Edition (Hardcover)

General Measure Theory
Measures and Measurable Functions
Lusin’s and Egoroff’s Theorems
Integrals and Limit Theorems
Product Measures, Fubini’s Theorem, Lebesgue Measure
Covering Theorems
Differentiation of Radon Measures
Lebesgue Points, Approximate Continuity
Riesz Representation Theorem
Weak Convergence
References and Notes

Hausdorff Measures
Definitions and Elementary Properties
Isodiametric Inequality, Hⁿ=Lⁿ
Densities
Functions and Hausdorff Measure
References and Notes

Area and Coarea Formulas
Lipschitz Functions, Rademacher’s Theorem
Linear Maps and Jacobians
The Area Formula
The Coarea Formula
References and Notes

Sobolev Functions
Definitions and Elementary Properties
Approximation
Traces
Extensions
Sobolev Inequalities
Compactness
Capacity
Quasicontinuity; Precise Representatives of Sobolev Functions
Differentiability on Lines
References and Notes

Functions of Bounded Variation, Sets of Finite Perimeter
Definitions, Structure Theorem
Approximation and Compactness
Traces
Extensions
Coarea Formula for BV Functions
Isoperimetric Inequalities
The Reduced Boundary
Gauss-Green Theorem
Pointwise Properties of BV Functions
Essential Variation on Lines
A Criterion for Finite Perimeter
References and Notes

Differentiability, Approximation by C¹ Functions
L^p Differentiability; Approximate Differentiability
Differentiability a.e. for W^1,p (p>n)
Convex Functions
Second Derivatives a.e. for Convex Functions
Whitney’s Extension Theorem
Approximation by C¹ Functions
References and Notes

Bibliography