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

## H.L. Royden: Real Analysis (2nd Edition)

Published $\text {1968}$, Macmillan Publishing Company

ISBN 0-02-404150-5.

### 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
8 Baire category
8 Topological Spaces
1 Fundamental notions
2 Bases and countability
3 The separation axioms and continuous real-valued functions
4 Product spaces
5 Connectedness
*6 Absolute $\mathcal G_\delta$'s
*7 Nets
9 Compact Spaces
1 Basic properties
2 Countable compactness and the Bolzano-Weierstrass property
3 Compact metric spaces
4 Products of compact spaces
4 Locally compact spaces
*6 The Stone-Čech compactification
7 The Stone-Weierstrass theorem
* The Ascoli 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 Inner measure
*6 Extension by sets of measure zero
*7 Carathéodory outer measure
13 The Daniell Integral
1 Introduction
2 The extension theorem
3 Uniqueness
4 Measurability and measure
14 Measure and Topology
1 Baire sets and Borel sets
2 Positive linear functionals and Borel measures
3 Bounded linear functionals on $C \left({X}\right)$
*4 The Borel extension of a measure
15 Mappings of Measure Spaces
1 Point mappings and set mappings
2 Measure algebras
3 Borel equivalences
4 Set mappings and point mappings on complete separable metric spaces
5 The isometries of $L^p$
Epilogue
Bibliography
Index of Symbols
Subject Index