Real Analysis 4th Edition by Halsey Royden, ISBN-13: 978-0131437470

Preface

ix

I Lebesgue Integration for Functions of a Single Real Variable

1(180)

Preliminaries on Sets, Mappings, and Relations

3(1)

Unions and Intersections of Sets

3(2)

Equivalence Relations, the Axiom of Choice, and Zorn’s Lemma

5(2)

The Real Numbers: Sets, Sequences, and Functions

7(22)

The Field, Positivity, and Completeness Axioms

7(4)

The Natural and Rational Numbers

11(2)

Countable and Uncountable Sets

13(3)

Open Sets, Closed Sets, and Borel Sets of Real Numbers

16(4)

Sequences of Real Numbers

20(5)

Continuous Real-Valued Functions of a Real Variable

25(4)

Lebesgue Measure

29(25)

Introduction

29(2)

Lebesgue Outer Measure

31(3)

The &-Algebra of Lebesgue Measurable Sets

34(6)

Outer and Inner Approximation of Lebesgue Measurable Sets

40(3)

Countable Additivity, Continuity, and the Borel-Cantelli Lemma

43(4)

Nonmeasurable Sets

47(2)

The Cantor Set and the Cantor-Lebesgue Function

49(5)

Lebesgue Measurable Functions

54(14)

Sums, Products, and Compositions

54(6)

Sequential Pointwise Limits and Simple Approximation

60(4)

Littlewood’s Three Principles, Egoroff’s Theorem, and Lusin’s Theorem

64(4)

Lebesgue Integration

68(29)

The Riemann Integral

68(3)

The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure

71(8)

The Lebesgue Integral of a Measurable Nonnegative Function

79(6)

The General Lebesgue Integral

85(5)

Countable Additivity and Continuity of Integration

90(2)

Uniform Integrability: The Vitali Convergence Theorem

92(5)

Lebesgue Integration: Further Topics

97(10)

Uniform Integrability and Tightness: A General Vitali Convergence Theorem

97(2)

Convergence in Measure

99(3)

Characterizations of Riemann and Lebesgue Integrability

102(5)

Differentiation and Integration

107(28)

Continuity of Monotone Functions

108(1)

Differentiability of Monotone Functions: Lebesgue’s Theorem

109(7)

Functions of Bounded Variation: Jordan’s Theorem

116(3)

Absolutely Continuous Functions

119(5)

Integrating Derivatives: Differentiating Indefinite Integrals

124(6)

Convex Functions

130(5)

The Lp Spaces: Completeness and Approximation

135(20)

Normed Linear Spaces

135(4)

The Inequalities of Young, Holder, and Minkowski

139(5)

Lp Is Complete: The Riesz-Fischer Theorem

144(6)

Approximation and Separability

150(5)

The Lp Spaces: Duality and Weak Convergence

155(26)

The Riesz Representation for the Dual of Lp, 1 & p < &

155(7)

Weak Sequential Convergence in Lp

162(9)

Weak Sequential Compactness

171(3)

The Minimization of Convex Functionals

174(7)

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

181(154)

Metric Spaces: General Properties

183(23)

Examples of Metric Spaces

183(4)

Open Sets, Closed Sets, and Convergent Sequences

187(3)

Continuous Mappings Between Metric Spaces

190(3)

Complete Metric Spaces

193(4)

Compact Metric Spaces

197(7)

Separable Metric Spaces

204(2)

Metric Spaces: Three Fundamental Theorems

206(16)

The Arzela-Ascoli Theorem

206(5)

The Baire Category Theorem

211(4)

The Banach Contraction Principle

215(7)

Topological Spaces: General Properties

222(17)

Open Sets, Closed Sets, Bases, and Subbases

222(5)

The Separation Properties

227(1)

Countability and Separability

228(2)

Continuous Mappings Between Topological Spaces

230(3)

Compact Topological Spaces

233(4)

Connected Topological Spaces

237(2)

Topological Spaces: Three Fundamental Theorems

239(14)

Urysohn’s Lemma and the Tietze Extension Theorem

239(5)

The Tychonoff Product Theorem

244(3)

The Stone-Weierstrass Theorem

247(6)

Continuous Linear Operators Between Banach Space

253(18)

Normed Linear Spaces

253(3)

Linear Operators

256(3)

Compactness Lost: Infinite Dimensional Normed Linear Spaces

259(4)

The Open Mapping and Closed Graph Theorems

263(5)

The Uniform Boundedness Principle

268(3)

Duality for Normed Linear Spaces

271(27)

Linear Functionals, Bounded Linear Functionals, and Weak Topologies

271(6)

The Hahn-Banach Theorem

277(5)

Reflexive Banach Spaces and Weak Sequential Convergence

282(4)

Locally Convex Topological Vector Spaces

286(4)

The Separation of Convex Sets and Mazur’s Theorem

290(5)

The Krein-Milman Theorem

295(3)

Compactness Regained: The Weak Topology

298(10)

Alaoglu’s Extension of Helley’s Theorem

298(2)

Reflexivity and Weak Compactness: Kakutani’s Theorem

300(2)

Compactness and Weak Sequential Compactness: The Eberlein-Smulian Theorem

302(3)

Metrizability of Weak Topologies

305(3)

Continuous Linear Operators on Hilbert Spaces

308(27)

The Inner Product and Orthogonality

309(4)

The Dual Space and Weak Sequential Convergence

313(3)

Bessel’s Inequality and Orthonormal Bases

316(3)

Adjoints and Symmetry for Linear Operators

319(5)

Compact Operators

324(2)

The Hilbert-Schmidt Theorem

326(3)

The Riesz-Schauder Theorem: Characterization of Fredholm Operators

329(6)

III Measure and Integration: General Theory

335(160)

General Measure Spaces: Their Properties and Construction

337(22)

Measures and Measurable Sets

337(5)

Signed Measures: The Hahn and Jordan Decompositions

342(4)

The Caratheodory Measure Induced by an Outer Measure

346(3)

The Construction of Outer Measures

349(3)

The Caratheodory-Hahn Theorem: The Extension of a Premeasure to a Measure

352(7)

Integration Over General Measure Spaces

359(35)

Measurable Functions

359(6)

Integration of Nonnegative Measurable Functions

365(7)

Integration of General Measurable Functions

372(9)

The Radon-Nikodym Theorem

381(7)

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

388(6)

General Lp Spaces: Completeness, Duality, and Weak Convergence

394(20)

The Completeness of Lp (X, &), 1 & p & &

394(5)

The Riesz Representation Theorem for the Dual of Lp (X, &), 1 & p & &

399(5)

The Kantorovitch Representation Theorem for the Dual of L & (X, &)

404(3)

Weak Sequential Compactness in Lp (X, &), 1 < p < 1

407(2)

Weak Sequential Compactness in L1 (X, &): The Dunford-Pettis Theorem

409(5)

The Construction of Particular Measures

414(32)

Product Measures: The Theorems of Fubini and Tonelli

414(10)

Lebesgue Measure on Euclidean Space Rn

424(13)

Cumulative Distribution Functions and Borel Measures on R

437(4)

Caratheodory Outer Measures and Hausdorff Measures on a Metric Space

441(5)

Measure and Topology

446(31)

Locally Compact Topological Spaces

447(5)

Separating Sets and Extending Functions

452(2)

The Construction of Radon Measures

454(3)

The Representation of Positive Linear Functionals on Cc(X): The Riesz-Markov Theorem

457(5)

The Riesz Representation Theorem for the Dual of C(X)

462(8)

Regularity Properties of Baire Measures

470(7)

Invariant Measures

477(18)

Topological Groups: The General Linear Group

477(3)

Kakutani’s Fixed Point Theorem

480(5)

Invariant Borel Measures on Compact Groups: von Neumann’s Theorem

485(3)

Measure Preserving Transformations and Ergodicity: The Bogoliubov-Krilov Theorem

488(7)

Bibliography

495(2)

Index

497