Book:H.L. Royden/Real Analysis/Third Edition

H.L. Royden: Real Analysis (3rd Edition)

Published $\text {1988}$, Macmillan Publishing Company

ISBN 0-02-946620-2

Subject Matter

• Real Analysis

Contents

Prologue to the Student

1 Set Theory

1 Introduction

2 Functions

3 Unions, intersections, and complements

4 Algebras of sets

5 The axiom of choice and infinite direct products

6 Countable sets

7 Relations and equivalences

8 Partial orderings and the maximal principle

9 Well ordering and the countable ordinals

Part One THEORY OF FUNCTIONS OF A REAL VARIABLE

2 The Real Number System

1 Axioms for the real numbers

2 The natural and rational numbers as subsets of $\mathbf R$

3 The extended real numbers

4 Sequences of real numbers

5 Open and closed sets of real numbers

6 Continuous functions

7 Borel sets

3 Lebesgue Measure

1 Introduction

2 Outer measure

3 Measurable sets and Lebesgue measure

*4 A nonmeasurable set

5 Measurable functions

6 Littlewood's three principles

4 The Lebesgue Integral

1 The Riemann integral

2 The Lebesgue integral of a bounded function over a set of finite measure

3 The integral of a nonnegative function

4 The general Lebesgue integral

*5 Convergence in measure

5 Differentiation and Integration

1 Differentiation of monotone functions

2 Functions of bounded variation

3 Differentiation of an integral

4 Absolute continuity

5 Convex functions

6 The Classical Banach Spaces

1 The $L^p$ spaces

2 The Minkowski and Hölder inequalities

3 Convergence and completeness

4 Approximation in $L^p$

5 Bounded linear functionals on the $L^p$ spaces

Part Two ABSTRACT SPACES

7 Metric Spaces

1 Introduction

2 Open and closed sets

3 Continuous functions and homeomorphisms

4 Convergence and completeness

5 Uniform continuity and uniformity

6 Subspaces

7 Compact metric spaces

8 Baire category

9 Absolute $G_\delta$'s

10 The Ascoli–Arzelá Theorem

8 Topological Spaces

1 Fundamental notions

2 Bases and countability

3 The separation axioms and continuous real-valued functions

4 Connectedness

5 Products and direct unions of topological spaces

*6 Topological and uniform properties

*7 Nets

9 Compact and Locally Compact Spaces

1 Compact spaces

2 Countable compactness and the Bolzano–Weierstrass property

3 Products of compact spaces

4 Locally compact spaces

5 $\sigma$-compact spaces

*6 Paracompact spaces

7 Manifolds

*8 The Stone–Čech compactification

9 The Stone–Weierstrass Theorem

10 Banach Spaces

1 Introduction

2 Linear operators

3 Linear functionals and the Hahn–Banach Theorem

4 The Closed Graph Theorem

5 Topological vector spaces

6 Weak topologies

7 Convexity

8 Hilbert space

Part Three GENERAL MEASURE AND INTEGRATION THEORY

11 Measure and Integration

1 Measure spaces

2 Measurable functions

3 Integration

4 General Convergence Theorems

5 Signed measures

6 The Radon–Nikodym Theorem

7 The $L^p$-spaces

12 Measure and Outer Measure

1 Outer measure and measurability

2 The Extension Theorem

3 The Lebesgue–Stieltjes integral

4 Product measures

5 Integral operators

*6 Inner measure

*7 Extension by sets of measure zero

8 Carathéodory outer measure

9 Hausdorff measure

13 Measure and Topology

1 Baire sets and Borel sets

2 The regularity of Baire and Borel measures

3 The construction of Borel measures

4 Positive linear functionals and Borel measures

5 Bounded linear functionals on $C \left({X}\right)$

14 Invariant Measures

1 Homogeneous spaces

2 Topological equicontinuity

3 The existence of invariant measures

4 Topological groups

5 Group actions and quotient spaces

6 Unicity of invariant measures

7 Groups of diffeomorphisms

15 Mappings of Measure Spaces

1 Point mappings and set mappings

2 Boolean $\sigma$-algebras

3 Measure algebras

4 Borel equivalences

5 Borel measures on complete separable metric spaces

6 Set mappings and point mappings on complete separable metric spaces

7 The isometries of $L^p$

16 The Daniell Integral

1 Introduction

2 The Extension Theorem

3 Uniqueness

4 Measurability and measure

Bibliography

Index of Symbols

Subject Index