# Book:H.L. Royden/Real Analysis/Fourth Edition

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## H.L. Royden and P.M. Fitzpatrick:

## H.L. Royden and P.M. Fitzpatrick: *Real Analysis (4th Edition)*

Published $\text {2010}$, **Prentice-Hall**

- ISBN 978-0-13-143747-0.

### Subject Matter

### Contents

- Preface

- 1 Lebesgue Integration for Functions of a Single Real Variable

- Preliminaries on Sets, Mappings, and Relations
- Unions and Intersections of Sets
- Equivalence Relations, the Axiom of Choice, and Zorn's Lemma

- Preliminaries on Sets, Mappings, and Relations

- 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

- 1 The Real Numbers: Sets, Sequences, and Functions

- 2 Lebesgue Measure
- 2.1 Introduction
- 2.2 Lebesgue Outer Measure
- 2.3 The $\sigma$-Algebra of Lebesgue Measurable Sets
- 2.4 Outer and Inner Approximation of Lebesgue Measurable Sets
- 2.5 Countable Additivity, Continuity, and the Borel-Cantelli Lemma
- 2.6 Nonmeasurable Sets
- 2.7 The Cantor Set and the Cantor-Lebesgue Function

- 2 Lebesgue Measure

- 3 Lebesgue Measurable Function
- 3.1 Sums, Products, and Compositions
- 3.2 Sequential Pointwise Limits and Simple Approximations
- 3.3 Littlewood's Three Principles, Egoroff's Theorem, and Lusin's Theorem

- 3 Lebesgue Measurable Function

- 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 Integration
- 4.6 Uniform Integrability: The Vitali Convergence Theorem

- 4 Lebesgue Integration

- 5 Lebesgue Integration: Further Topics
- 5.1 Uniform Integrability and Tightness
- 5.2 Convergence in Measure
- 5.3 Characterizations of Riemann and Lebesgue Integrability

- 5 Lebesgue Integration: Further Topics

- 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

- 6 Differentiation and Integration

- 7 The $L^p$ Spaces: Completeness and Approximation
- 7.1 Normed Linear Spaces
- 7.2 The Inequalities of Young, Hölder, and Minkowski
- 7.3 $L^p$ Is Complete: The Riesz-Fischer Theorem
- 7.4 Approximation and Separability

- 7 The $L^p$ Spaces: Completeness and Approximation

- 8 The $L^p$ Spaces: Duality and Weak Convergence
- 8.1 The Riesz Representation for the Dual of $L^p$, $1 \le p < \infty$
- 8.2 Weak Sequential Convergence in $L^p$
- 8.3 Weak Sequential Compactness
- 8.4 The Minimization of Convex Functionals

- 8 The $L^p$ Spaces: Duality and Weak Convergence

- II Abstract Spaces: Metric, Topological, Banach, and Hilbert Spaces

- 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

- 9 Metric Spaces: General Properties

- 10 Metric Spaces: Three Fundamental Theorems
- 10.1 The Arzelà-Ascoli Theorem
- 10.2 The Baire Category Theorem
- 10.3 The Banach Contraction Principle

- 10 Metric Spaces: Three Fundamental Theorems

- 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

- 11 Topological Spaces: General Properties

- 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

- 12 Topological Spaces: Three Fundamental Theorems

- 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

- 13 Continuous Linear Operators Between Banach Spaces

- 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

- 14 Duality for Normed Linear Spaces

- 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

- 15 Compactness Regained: The Weak Topology

- 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

- 16 Continuous Linear Operators on Hilbert Spaces

- 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

- 17 General Measure Spaces: Their Properties and Construction

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

- 18 Integration Over General Measure Spaces

- 19 General $L^p$ Spaces: Completeness, Duality, and Weak Convergence
- 19.1 The Completeness of $L^p(X,\mu)$, $1 \le p \le \infty$
- 19.2 The Riesz Representation Theorem for the Dual of $L^p(X,\mu)$, $1 \le p \le \infty$
- 19.3 The Kantorovich Representation Theorem for the Dual of $L^\infty(X,\mu)$
- 19.4 Weak Sequential Compactness in $L^p(X,\mu)$, $1 < p < 1$
- 19.5 Weak Sequential Compactness in $L^1(X,\mu)$: The Dunford-Pettis Theorem

- 19 General $L^p$ Spaces: Completeness, Duality, and Weak Convergence

- 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

- 20 The Construction of Particular 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 $C(X)$
- 21.6 Regularity Properties of Baire Measures

- 21 Measure and Topology

- 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

- 22 Invariant Measures

- Bibliography
- Index