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 Rn.
3.6 Jordan Measure in Rn.
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 L1 and the Pointwise Convergence Lemma.
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 Covering Theorem.
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 Lp.
9.2 The Dual of a Banach Space.
9.3 The Dual Space of Lp.
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 and Rotations.
Appendix A: The Banach-Tarski Theorem.
A.1 The Limits to Countable Additivity.
- Provides an organized and logical selection of topics that prepares readers for subsequent courses in functional analysis and other core topics in pure and applied mathematics
- Serves as a concise, yet thorough, introduction to measure and integration, and the complete text can be covered in a one-semester course in real analysis.
- Presents the essentials of measure, integration, differentiation, and Lp spaces and clearly develops their interrelations
- Provides coverage of the Banach-Tarski theorem and culminates in the famous Riesz-Markov-Saks-Kakutani theorem
- Contains numerous relevant and interesting exercises as well as supplementary topical coverage intended to stimulate and motivate further learning
- Strikes a balance between both fundamental and specialized topics and successfully provides an introduction to measure and integration that does not focus too much attention on the possible embellishments or ramifications of various topics