Measure and Integration: A Concise Introduction to Real Analysis

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 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 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 and Rotations.

Appendix A: The Banach-Tarski Theorem.

A.1 The Limits to Countable Additivity.

References.

Index.