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

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

Published $1988$, Macmillan Publishing Company

ISBN 0-02-946620-2.

### 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
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