Measure and Integration

Preface.
Acknowledgments.
Introduction.
1 History of the Subject.
1.1 History of the Idea.
1.2 Deficiencies of the Riemann Integral.
1.3 Motivation for the Lebesgue Integral.
2 Fields, Borel Fields and Measures.
2.1 Fields, Monotone Classes, and Borel Fields.
2.2 Additive Measures.
2.3 Carathéodory Outer Measure.
2.4 E. Hopf's Extension Theorem.
3 Lebesgue Measure.
3.1 The Finite Interval [-N,N).
3.2 Measurable Sets, Borel Sets, and the Real Line.
3.3 Measure Spaces and Completions.
3.4 Semimetric Space of Measurable Sets.
3.5 Lebesgue Measure in R¯n.
3.6 Jordan Measure in R¯n.
4 Measurable Functions.
4.1 Measurable Functions.
4.2 Limits of Measurable Functions.
4.3 Simple Functions and Egoroff's Theorem.
4.4 Lusin's Theorem.
5 The Integral.
5.1 Special Simple Functions.
5.2 Extending the Domain of the Integral.
5.3 Lebesgue Dominated Convergence Theorem.
5.4 Monotone Convergence and Fatou's Theorem.
5.5 Completeness of L¯1 and the Pointwise ConvergenceLemma.
5.6 Complex Valued Functions.
6 Product Measures and Fubini's Theorem.
6.1 Product Measures.
6.2 Fubini's Theorem.
6.3 Comparison of Lebesgue and Riemann Integrals.
7 Functions of a Real Variable.
7.1 Functions of Bounded Variation.
7.2 A Fundamental Theorem for the Lebesgue Integral.
7.3 Lebesgue's Theorem and Vitali's CoveringTheorem.
7.4 Absolutely Continuous and Singular Functions.
8 General Countably Additive Set Functions.
8.1 Hahn Decomposition Theorem.
8.2 Radon-Nikodym Theorem.
8.3 Lebesgue Decomposition Theorem.
9. Examples of Dual Spaces from Measure Theory.
9.1 The Banach Space L¯p.
9.2 The Dual of a Banach Space.
9.3 The Dual Space of L¯p.
9.4 Hilbert Space, Its Dual, and L².
9.5 Riesz-Markov-Saks-Kakutani Theorem.
10 Translation Invariance in Real Analysis.
10.1 An Orthonormal Basis for L²(T).
10.2 Closed Invariant Subspaces of L²(T).
10.3 Schwartz Functions: Fourier Transform and Inversion.
10.4 Closed, Invariant Subspaces of L²(R).
10.5 Irreducibility of L²(R) Under Translations andRotations.
Appendix A: The Banach-Tarski Theorem.
A.1 The Limits to Countable Additivity.
References.
Index.