Convergence theorems for integrals
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"The book is well thought out, organized and written. It has all the results in measure theory that are necessary for both pure and applied mathematics research." (Mathematical Reviews, 2011)
Table of Contents
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 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.
References.
Index.
