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

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## H.L. Royden:

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

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

- ISBN 0-02-946620-2.

### Subject Matter

### 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

- 1 Set Theory

**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

- 2 The Real Number System

- 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

- 3 Lebesgue Measure

- 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

- 4 The Lebesgue Integral

- 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

- 5 Differentiation and Integration

- 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

- 6 The Classical Banach 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

- 7 Metric Spaces

- 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

- 8 Topological Spaces

- 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

- 9 Compact and Locally Compact Spaces

- 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

- 10 Banach Spaces

**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

- 11 Measure and Integration

- 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

- 12 Measure and Outer 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 $C \left({X}\right)$

- 13 Measure and Topology

- 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

- 14 Invariant Measures

- 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 $L^p$

- 15 Mappings of Measure Spaces

- 16 The Daniell Integral
- 1 Introduction
- 2 The Extension Theorem
- 3 Uniqueness
- 4 Measurability and measure

- 16 The Daniell Integral

- Bibliography
- Index of Symbols
- Subject Index