实分析 : 第4版

Lebesgue Integration for Functions of a Single Real Variable

Preliminaries on Sets, Mappings, and Relations

Unions and Intersections of Sets

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

1 The Real Numbers: Sets. Sequences, and Functions

The Field, Positivity, and Completeness Axioms

The Natural and Rational Numbers

Countable and Uncountable Sets

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

Sequences of Real Numbers

Continuous Real-Valued Functions of a Real Variable

2 Lebesgne Measure

Introduction

Lebesgue Outer Measure

The o'-Algebra of Lebesgue Measurable Sets

Outer and Inner Approximation of Lebesgue Measurable Sets

Countable Additivity, Continuity, and the Borel-Cantelli Lemma

Noumeasurable Sets

The Cantor Set and the Cantor Lebesgue Function

3 LebesgRe Measurable Functions

Sums, Products, and Compositions

Sequential Pointwise Limits and Simple Approximation

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

4 Lebesgue Integration

The Riemann Integral

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

Finite Measure

The Lebesgue Integral of a Measurable Nonnegative Function

The General Lebesgue Integral

Countable Additivity and Continuity of Integration

Uniform Integrability: The Vifali Convergence Theorem

viii Contents

5 Lebusgue Integration: Fm'ther Topics

Uniform Integrability and Tightness: A General Vitali Convergence Theorem

Convergence in Measure

Characterizations of Riemaun and Lebesgue Integrability

6 Differentiation and Integration

Continuity of Monotone Functions

Differentiability of Monotone Functions: Lebesgue's Theorem

Functions of Bounded Variation: Jordan's Theorem

Absolutely Continuous Functions

Integrating Derivatives: Differentiating Indefinite Integrals

Convex Function

7 The Lp Spaces: Completeness and Appro~umation

Nor/ned Linear Spaces

The Inequalities of Young, HOlder, and Minkowski

Lv Is Complete: The Riesz-Fiseher Theorem

Approximation and Separability

8 The LP Spacesc Deailty and Weak Convergence

The Riesz Representation for the Dual of

Weak Sequential Convergence in Lv

Weak Sequential Compactness

The Minimization of Convex Functionals

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

9. Metric Spaces: General Properties

Examples of Metric Spaces

Open Sets, Closed Sets, and Convergent Sequences

Continuous Mappings Between Metric Spaces

Complete Metric Spaces

Compact Metric Spaces

Separable Metric Spaces

10 Metric Spaces: Three Fundamental Thanreess

The Arzelb.-Ascoli Theorem

The Baire Category Theorem

The Banaeh Contraction Principle

H Topological Spaces: General Properties

Open Sets, Closed Sets, Bases, and Subbases

The Separation Properties

Countability and Separability

Continuous Mappings Between Topological Spaces

Compact Topological Spaces

Connected Topological Spaces

12 Topological Spaces: Three Fundamental Theorems

Urysohn's Lemma and the Tietze Extension Theorem

The Tychonoff Product Theorem

The Stone-Weierstrass Theorem

13 Continuous Linear Operators Between Bausch Spaces

Normed Linear Spaces

Linear Operators

Compactness Lost: Infinite Dimensional Normod Linear Spaces

The Open Mapping and Closed Graph Theorems

The Uniform Boundedness Principle

14 Duality for Normed Iinear Spaces

Linear Ftmctionals, Bounded Linear Functionals, and Weak Topologies

The Hahn-Banach Theorem

Reflexive Banach Spaces and Weak Sequential Convergence

Locally Convex Topological Vector Spaces

The Separation of Convex Sets and Mazur's Theorem

The Krein-Miiman Theorem

15 Compactness Regained: The Weak Topology

Alaoglu's Extension of Helley's Theorem

Reflexivity and Weak Compactness: Kakutani's Theorem

Compactness and Weak Sequential Compactness: The Eberlein-mulian

Theorem

Memzability of Weak Topologies

16 Continuous Linear Operators on Hilbert Spaces

The Inner Product and Orthogonality

The Dual Space and Weak Sequential Convergence

Bessers Inequality and Orthonormal Bases

bAdjoints and Symmetry for Linear Operators

Compact Operators

The Hilbert-Schmidt Theorem

The Riesz-Schauder Theorem: Characterization of Fredholm Operators

Measure and Integration: General Theory

17 General Measure Spaces: Their Propertles and Construction

Measures and Measurable Sets

Signed Measures: The Hahn and Jordan Decompositions

The Caratheodory Measure Induced by an Outer Measure

18 Integration Oeneral Measure Spaces

19 Gengral L Spaces:Completeness,Duality and Weak Convergence

20 The Construciton of Particular Measures

21 Measure and Topbogy

22 Invariant Measures

Bibiiography

index

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