Real Analysis, 5th edition

I: LEBESGUE INTEGRATION FOR FUNCTIONS OF A SINGLE REAL VARIABLE

Preliminaries on Sets, Mappings, and Relations

• Unions and Intersections of Sets
• Mappings Between Sets
• Equivalence Relations, the Axiom of Choice and Zorn's Lemma

1. The Real Numbers: Sets, Sequences and Functions
   • 1.1 The Field, Positivity and Completeness Axioms
   • 1.2 The Natural and Rational Numbers
   • 1.3 Countable and Uncountable Sets
   • 1.4 Open Sets, Closed Sets, and Borel Sets of Real Numbers
   • 1.5 Sequences of Real Numbers
   • 1.6 Continuous Real-Valued Functions of a Real Variable

2. Lebesgue Measure
   • 2.1 Introduction
   • 2.2 Outer Measure
   • 2.3 The σ-algebra of Lebesgue Measurable Sets
   • 2.4 Finer Properties of Measurable Sets
   • 2.5 Countable Additivity and Continuity of Measure, and the Borel-Cantelli Lemma
   • 2.6 Vitali's Example of a Nonmeasurable Set
   • 2.7 The Cantor Set and the Cantor-Lebesgue Function

3. Lebesgue Measurable Functions
   • 3.1 Sums, Products, and Compositions
   • 3.2 Sequential Pointwise Limits and Simple Approximation
   • 3.3 Littlewood's Three Principles, Egoroff's Theorem and Lusin's Theorem

4. Lebesgue Integration
   • 4.1 Comments on the Riemann Integral
   • 4.2 The Integral of a Bounded, Finitely Supported, Measurable Function
   • 4.3 The Integral of a Non-Negative Measurable Function
   • 4.4 The General Lebesgue Integral
   • 4.5 Countable Additivity and Continuity of Integration

5. Lebesgue Integration: Further Topics
   • 5.1 Uniform Integrability and Tightness: The Vitali Convergence Theorems
   • 5.2 Convergence in the Mean and in Measure: A Theorem of Riesz
   • 5.3 Characterizations of Riemann and Lebesgue Integrability

6. Differentiation and Integration
   • 6.1 Continuity of Monotone Functions
   • 6.2 Differentiability of Monotone Functions: Lebesgue's Theorem
   • 6.3 Functions of Bounded Variation: Jordan's Theorem
   • 6.4 Absolutely Continuous Functions
   • 6.5 Integrating Derivatives: Differentiating Indefinite Integrals
   • 6.6 Measurability: Images of Sets, Compositions of Functions
   • 6.7 Convex Functions

7. The L^Ρ Spaces: Completeness and Approximation
   • 7.1 Normed Linear Spaces
   • 7.2 The Inequalities of Young, Hölder and Minkowski
   • 7.3  L^Ρ is Complete: Rapidly Cauchy Sequences and The Riesz-Fischer Theorem
   • 7.4 Approximation and Separability

8. The  L^Ρ Spaces: Duality, Weak Convergence and Minimization
   • 8.1 Bounded Linear Functionals on a Normed Linear Space
   • 8.2 The Riesz Representation of the Dual of L^p, 1 ≤ p < ∞
   • 8.3 Weak Sequential Convergence in L^p
   • 8.4 The Minimization of Convex Functionals

II: MEASURE AND INTEGRATION: GENERAL THEORY

9. General Measure Spaces: Their Properties and Construction
   • 9.1 Measurable Sets and Measure Spaces
   • 9.2 Measures Induced by an Outer-measure
   • 9.3 The Carathéodory-Hahn Theorem

10. Particular Measures: Lebesgue Measure on Euclidean Space, Borel Measures, and Signed Measure
    • 10.1 Lebesgue Measure on Euclidean Space
    • 10.2 Lebesgue Measurability and Measure of Images of Mappings
    • 10.3 Regularity of Borel Measures on Rⁿ, and Cumulative Distribution Functions
    • 10.4 Carathéodory Outer-measures and Hausdorff Measures
    • 10.5 Signed Measures: the Hahn and Jordan Decompositions

11. Integration Over General Measure Spaces
    • 11.1 Measurable Functions: the Egoroff and Lusin Theorems
    • 11.2 Integration of Non-negative Measurable Functions: Fatou's Lemma, the Monotone Convergence Theorem and Beppo Levi's Theorem
    • 11.3 Integration of General Measurable Functions: the Dominated Convergence Theorem and the Vitali Convergence Theorem
    • 11.4 The Radon-Nikodym Theorem
    • 11.5 Product Measures: the Tonelli and Fubini Theorems
    • 11.6 Products of Lebesgue measure on Euclidean spaces: Cavalieri's Principle

12. General L^p Spaces: Completeness, Convolution, and Duality
    • 12.1 The Spaces L^p(X; μ); 1 ≤ p ≤ ∞
    • 12.2 Convolution, Smooth Approximation and a Smooth Urysohn's Lemma
    • 12.3 The Riesz Representation Theorem for the Dual of L^p(X; μ); 1 ≤ p < ∞
    • 12.4 Weak Sequential Compactness in L^p(X; μ); 1 < p < ∞
    • 12.5 The Kantorovitch Representation Theorem for the Dual of L^∞ (X; μ)

III: ABSTRACT SPACES: METRIC, TOPOLOGICAL, BANACH, AND HILBERT SPACES

13. Metric Spaces: General Properties
    • 13.1 Examples of Metric Spaces
    • 13.2 Open Sets, Closed Sets, and Convergent Sequences
    • 13.3 Continuous Mappings Between Metric Spaces
    • 13.4 Complete Metric Spaces
    • 13.5 Compact Metric Spaces
    • 13.6 Separable Metric Spaces

14. Metric Spaces: Three Fundamental Theorems and Applications
    • 14.1 The Arzelà-Ascoli Theorem
    • 14.2 The Banach Contraction Principle
    • 14.3 The Baire Category Theorem
    • 14.4 The Nikodym Metric Space: The Vitali-Hahn-Saks Theorem and the Dunford-Pettis Theorem

15. Topological Spaces: General Properties
    • 15.1 Open Sets, Closed Sets, Bases, and Subbases
    • 15.2 The Separation Properties
    • 15.3 Countability and Separability
    • 15.4 Continuous Mappings Between Topological Spaces
    • 15.5 Compact Topological Spaces
    • 15.6 Connected Topological Spaces

16. Topological Spaces: Three Fundamental Theorems
    • 16.1 Urysohn's Lemma and the Tietze Extension Theorem
    • 16.2 The Tychonoff Product Theorem
    • 16.3 The Stone-Weierstrass Theorem

17. Continuous Linear Operators Between Banach Spaces
    • 17.1 Normed Linear Spaces
    • 17.2 Linear Operators
    • 17.3 Compactness Lost: Infinite Dimensional Normed Linear Spaces
    • 17.4 The Open Mapping and Closed Graph Theorems
    • 17.5 The Uniform Boundedness Principle

18. Duality for Normed Linear Spaces
    • 18.1 Linear Functionals, Bounded Linear Functionals, and Weak Topologies
    • 18.2 The Hahn-Banach Theorem
    • 18.3 Reflexive Banach Spaces and Weak Sequential Convergence
    • 18.4 Locally Convex Topological Vector Spaces
    • 18.5 The Separation of Convex Sets and Mazur's Theorem
    • 18.6 The Krein-Milman Theorem

19. Compactness Regained: The Weak Topology
    • 19.1 Alaoglu's Extension of Helly's Theorem
    • 19.2 Reflexivity and Weak Compactness: Kakutani's Theorem
    • 19.3 Compactness and Weak Sequential Compactness: The Eberlein-Šmulian Theorem
    • 19.4 Metrizability of Weak Topologies

20. Continuous Linear Operators on Hilbert Spaces
    • 20.1 The Inner Product and Orthogonality
    • 20.2 Bessel's Inequality and Orthonormal Bases
    • 20.3 The Dual Space and Weak Sequential Convergence
    • 20.4 Symmetric Operators
    • 20.5 Compact Operators
    • 20.6 The Hilbert-Schmidt Theorem
    • 20.7 The Riesz-Schauder Theorem: Characterization of Fredholm Operators

IV: MEASURE AND TOPOLOGY: INVARIANT MEASURES

21. Measure and Topology
    • 21.1 Locally Compact Topological Spaces
    • 21.2 Separating Sets and Extending Functions
    • 21.3 The Construction of Radon Measures
    • 21.4 The Representation of Positive Linear Functionals on C_c (X): The Riesz-Markov Theorem
    • 21.5 The Riesz Representation Theorem for the Dual of C(X): The Riesz-Kakutani Theorem
    • 21.6 Regularity Properties of Baire Measures

22. Invariant Measures
    • 22.1 Topological Groups: The General Linear Group
    • 22.2 Kakutani's Fixed Point Theorem
    • 22.3 Invariant Borel Measures on Compact Groups: von Neumann's Theorem
    • 22.4 Measure Preserving Transformations and Ergodicity: The Bogoliubov-Krilov Theorem

Bibliography

Index