## Table of Contents

**PART I: LEBESGUE INTEGRATION FOR FUNCTIONS OF A SINGLE REAL VARIABLE**

**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 Lebesgue Outer Measure

2.3 The *σ*-algebra of Lebesgue Measurable Sets

2.4 Outer and Inner Approximation of Lebesgue Measurable Sets

2.5 Countable Additivity and Continuity of Lebesgue Measure

2.6 Nonmeasurable Sets

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 The Riemann Integral

4.2 The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure

4.3 The Lebesgue Integral of a Measurable Nonnegative Function

4.4 The General Lebesgue Integral

4.5 Countable Additivity and Continuity of Integraion

4.6 Uniform Integrability: The Vitali Convergence Theorem

**5. Lebesgue Integration: Further Topics**

5.1 Uniform Integrability and Tightness: A General Vitali Convergence Theorem

5.2 Convergence in measure

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 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: The Riesz-Fischer Theorem

7.4 Approximation and Separability

**8. The ***L*^{Ρ} **Spaces: Duality and Weak Convergence**

8.1 The Dual Space of *L*^{Ρ}

8.2 Weak Sequential Convergence in *L*^{Ρ}

8.3 Weak Sequential Compactness

8.4 The Minimization of Convex Functionals

**PART II: ABSTRACT SPACES: METRIC, TOPOLOGICAL, AND HILBERT**

**9. Metric Spaces: General Properties**

9.1 Examples of Metric Spaces

9.2 Open Sets, Closed Sets and Convergent Sequences

9.3 Continuous Mappings Between Metric Spaces

9.4 Complete Metric Spaces

9.5 Compact Metric Spaces

9.6 Separable Metric Spaces

**10. Metric Spaces: Three Fundamental Theorems**

10.1 The Arzelà-Ascoli Theorem

10.2 The Baire Category Theorem

10.3 The Banach Contraction Principle

**11. Topological Spaces: General Properties**

11.1 Open Sets, Closed Sets, Bases and Subbases

11.2 The Separation Properties

11.3 Countability and Separability

11.4 Continuous Mappings Between Topological Spaces

11.5 Compact Topological Spaces

11.6 Connected Topological Spaces

**12. Topological Spaces: Three Fundamental Theorems**

12.1 Urysohn's Lemma and the Tietze Extension Theorem

12.2 The Tychonoff Product Theorem

12.3 The Stone-Weierstrass Theorem

**13. Continuous Linear Operators Between Banach Spaces**

13.1 Normed Linear Spaces

13.2 Linear Operators

13.3 Compactness Lost: Infinite Dimensional Normed Linear Spaces

13.4 The Open Mapping and Closed Graph Theorems

13.5 The Uniform Boundedness Principle

**14. Duality for Normed Linear Spaces**

14.1 Linear Functionals, Bounded Linear Functionals and Weak Topologies

14.2 The Hahn-Banach Theorem

14.3 Reflexive Banach Spaces and Weak Sequential Convergence

14.4 Locally Convex Topological Vector Spaces

14.5 The Separation of Convex Sets and Mazur's Theorem

14.6 The Krein-Milman Theorem

**15. Compactness Regained: The Weak Topology**

15.1 Alaoglu's Extension of Helley's Theorem

15.2 Reflexivity and Weak Compactness: Kakutani's Theorem

15.3 Compactness and Weak Sequential Compactness: The Eberlein-Šmulian Theorem

15.4 Metrizability of Weak Topologies

**16. Continuous Linear Operators on Hilbert Spaces**

16.1 The Inner Product and Orthogonality

16.2 The Dual Space and Weak Sequential Convergence

16.3 Bessel's Inequality and Orthonormal Bases

16.4 Adjoints and Symmetry for Linear Operators

16.5 Compact Operators

16.6 The Hilbert Schmidt Theorem

16.7 The Riesz-Schauder Theorem: Characterization of Fredholm Operators

**PART III: MEASURE AND INTEGRATION: GENERAL THEORY**

**17. General Measure Spaces: Their Properties and Construction**

17.1 Measures and Measurable Sets

17.2 Signed Measures: The Hahn and Jordan Decompositions

17.3 The Carathéodory Measure Induced by an Outer Measure

17.4 The Construction of Outer Measures

17.5 The Carathéodory-Hahn Theorem: The Extension of a Premeasure to a Measure

**18. Integration Over General Measure Spaces**

18.1 Measurable Functions

18.2 Integration of Nonnegative Measurable Functions

18.3 Integration of General Measurable Functions

18.4 The Radon-Nikodym Theorem

18.5 The Saks Metric Space: The Vitali-Hahn-Saks Theorem

**19. General ***L*^{Ρ} **Spaces: Completeness, Duality and Weak Convergence**

19.1 The Completeness of *L*^{Ρ} ( *Χ*, *μ*), 1 ≤ Ρ ≤ ∞

19.2 The Riesz Representation theorem for the Dual of *L*^{Ρ} ( *Χ*, *μ*), 1 ≤ Ρ ≤ ∞

19.3 The Kantorovitch Representation Theorem for the Dual of *L*^{∞} (*Χ*, *μ*)

19.4 Weak Sequential Convergence in *L*^{Ρ} (*X*, *μ*), 1 < Ρ < 1

19.5 Weak Sequential Compactness in *L*^{1} (*X*, *μ*): The Dunford-Pettis Theorem

**20. The Construction of Particular Measures**

20.1 Product Measures: The Theorems of Fubini and Tonelli

20.2 Lebesgue Measure on Euclidean Space *R*^{n}

20.3 Cumulative Distribution Functions and Borel Measures on *R*

20.4 Carathéodory Outer Measures and hausdorff Measures on a Metric Space

**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*)

21.6 Regularity Properties of Baire Measures

**22. Invariant Measures**

22.1 Topological Groups: The General Linear Group

22.2 Fixed Points of Representations: Kakutani's Theorem

22.3 Invariant Borel Measures on Compact Groups: von Neumann's Theorem

22.4 Measure Preserving Transformations and Ergodicity: the Bogoliubov-Krilov Theorem