本书是一部实分析方面的经典教材，主要分三部分，第一部分为经典的实变函数论和经典的巴拿赫空间理论；第二部分为抽象空间理论，主要介绍分析中有用的拓扑空间以及近代巴拿赫空间理论；第三部分为一般的测度和积分论，即在第二部分理论基础上将经典的测度、积分论推广到一般情形。.

新增了50%的习题。
扩充了基本结果，包括给出叶果洛夫定理和乌雷松引理的证明。
介绍了博雷尔-坎特利引理、切比雪夫不等式、快速柯西序列及测度和积分所共有的连续性质，以及若干其他概念。
本书是实分析课程的优秀教材，被国外众多著名大学（如斯坦福大学、哈佛大学等）采用。全书分为三部分：第一部分为实变函数论，介绍一元实变函数的勒贝格测度和勒贝格积分；第二部分为抽象空间，介绍拓扑空间、度量空间、巴拿赫空间和希尔伯特空间；第三部分为一般测度与积分理论，介绍一般度量空间上的积分，以及拓扑、代数和动态结构的一般理论。书中不仅包含数学定理和定义，而且还提出了富有启发性的问题，以便读者更深入地理解书中内容。
第4版主要更新如下:

数学

本书是实分析课程的优秀教材，被国外众多著名大学（如斯坦福大学、哈佛大学等）采用。全书分为三部分：第一部分为实变函数论，介绍一元实变函数的Lebesgue测度和Lebesgue积分；第二部分为抽象空间，介绍拓扑空间、度量空间、Banach空间和Hilbert空间；第三部分为一般测度与积分理论，介绍一般度量空间上的积分，以及拓扑、代数和动态结构的一般理论。书中不仅包含数学定理和定义，而且还提出了富有启发性的问题，以便读者更深入地理解书中内容。
　 与上一版相比，第4版主要更新如下:
● 新增了50%的习题。
● 扩充了基本结果，包括给出Egoroff定理和Urysohn引理的证明。
● 介绍了Borel-Cantelli引理、Chebychev不等式、快速Cauchy序列以及测度和积分所共有的连续性质，以及若干其他概念。

（美）H.L.Royden, P. M. Fitzpatrick 著：暂无简介

Preface iii
Lebesgue Integration for Functions of Single Real Variable
Preliminaries on Sets, Mappings, and Relations
UnionsandIntersectionsofSets .............................
Equivalence Relations, the Axiom of Choice, and Zorn’s Lemma . . . . . . . . . .
The Real Numbers: Sets, Sequences, and Functions
1.1 The Field, Positivity, and Completeness Axioms . . . . . . . . . . . . . . . . . 7
1.2 TheNaturalandRationalNumbers ........................ 11
1.3 CountableandUncountableSets ......................... 13
1.4 Open Sets, Closed Sets, and Borel Sets of Real Numbers . . . . . . . . . . . . 16
1.5 SequencesofRealNumbers ............................ 20
1.6 Continuous Real-Valued Functions of a Real Variable . . . . . . . . . . . . . 25
Lebesgue Measure 29
2.1 Introduction ..................................... 29
2.2 LebesgueOuterMeasure.............................. 31
2.3 The σ-AlgebraofLebesgueMeasurableSets .. .. .. .. .. ... .. .. . 34
2.4 Outer and Inner Approximation of Lebesgue Measurable Sets . . . . . . . . 40
2.5 Countable Additivity, Continuity, and the Borel-Cantelli Lemma . . . . . . . 43
2.6 NonmeasurableSets................................. 47
2.7 The Cantor Set and the Cantor-Lebesgue Function . . . . . . . . . . . . . . . 49
Lebesgue Measurable Functions 54
3.1 Sums,Products,andCompositions ........................ 54
3.2 Sequential Pointwise Limits and Simple Approximation . . . . . . . . . . . . 60
3.3 Littlewood’s Three Principles, Egoroff’s Theorem, and Lusin’s Theorem . . . 64
Lebesgue Integration 68
4.1 TheRiemannIntegral................................ 68
4.2 The Lebesgue Integral of a Bounded Measurable Function over a Set of
FiniteMeasure.................................... 71
4.3 The Lebesgue Integral of a Measurable Nonnegative Function . . . . . . . . 79
4.4 TheGeneralLebesgueIntegral .......................... 85
4.5 Countable Additivity and Continuity of Integration . . . . . . . . . . . . . . . 90
4.6 Uniform Integrability: The Vitali Convergence Theorem . . . . . . . . . . . . 92
Lebesgue Integration: Further Topics 97
5.1 Uniform Integrability and Tightness: A General Vitali Convergence Theorem 97
5.2 ConvergenceinMeasure .............................. 99
5.3 Characterizations of Riemann and Lebesgue Integrability . . . . . . . . . . . 102
Differentiation and Integration 107
6.1 ContinuityofMonotoneFunctions ........................ 108
6.2 Differentiability of Monotone Functions: Lebesgue’s Theorem . . . . . . . . 109
6.3 Functions of Bounded Variation: Jordan’s Theorem . . . . . . . . . . . . . . 116
6.4 AbsolutelyContinuousFunctions ......................... 119
6.5 Integrating Derivatives: Differentiating Indefinite Integrals . . . . . . . . . . 124
6.6 ConvexFunctions .................................. 130
7The Lp Spaces: Completeness and Approximation 135
7.1 NormedLinearSpaces ............................... 135
7.2 The Inequalities of Young, H older, and Minkowski 139¨..............
7.3 Lp IsComplete:TheRiesz-FischerTheorem . . . . . . . . . . . . . . . . . . 144
7.4 ApproximationandSeparability.......................... 150
8The Lp Spaces: Duality and Weak Convergence 155
8.1 The Riesz Representation for the Dual of Lp, 1 ........... 155
8.2 Weak Sequential Convergence in Lp ....................... 162
8.3 WeakSequentialCompactness........................... 171
8.4 TheMinimizationofConvexFunctionals. . . . . . . . . . . . . . . . . . . . . 174
II Abstract Spaces: Metric, Topological, Banach, and Hilbert Spaces 181
Metric Spaces: General Properties 183
9.1 ExamplesofMetricSpaces ............................. 183
9.2 Open Sets, Closed Sets, and Convergent Sequences . . . . . . . . . . . . . . . 187
9.3 ContinuousMappingsBetweenMetricSpaces . . . . . . . . . . . . . . . . . . 190
9.4 CompleteMetricSpaces .............................. 193
9.5 CompactMetricSpaces ............................... 197
9.6 SeparableMetricSpaces .............................. 204
10 Metric Spaces: Three Fundamental Theorems 206
10.1TheArzela-AscoliTheorem `............................ 206
10.2TheBaireCategoryTheorem ........................... 211
10.3TheBanachContractionPrinciple......................... 215
11 Topological Spaces: General Properties 222
11.1 OpenSets,ClosedSets,Bases,andSubbases. . . . . . . . . . . . . . . . . . . 222
11.2TheSeparationProperties ............................. 227
11.3CountabilityandSeparability ........................... 228
11.4 Continuous Mappings Between Topological Spaces . . . . . . . . . . . . . . . 230
11.5CompactTopologicalSpaces............................ 233
11.6ConnectedTopologicalSpaces........................... 237
12 Topological Spaces: Three Fundamental Theorems 239
12.1 Urysohn’s Lemma and the Tietze Extension Theorem . . . . . . . . . . . . . 239
12.2TheTychonoffProductTheorem ......................... 244
12.3TheStone-WeierstrassTheorem.......................... 247
13 Continuous Linear Operators Between Banach Spaces 253
13.1NormedLinearSpaces ............................... 253
13.2LinearOperators .................................. 256
13.3 Compactness Lost: Infinite Dimensional Normed Linear Spaces . . . . . . . . 259
13.4 TheOpenMappingandClosedGraphTheorems .. .. .. .. ... .. .. . 263
13.5TheUniformBoundednessPrinciple ....................... 268
14 Duality for Normed Linear Spaces 271
14.1 Linear Functionals, Bounded Linear Functionals, and Weak Topologies . . . 271
14.2TheHahn-BanachTheorem ............................ 277
14.3 Reflexive Banach Spaces and Weak Sequential Convergence . . . . . . . . . 282
14.4 LocallyConvexTopologicalVectorSpaces. . . . . . . . . . . . . . . . . . . . 286
14.5 The Separation of Convex Sets and Mazur’s Theorem . . . . . . . . . . . . . 290
14.6TheKrein-MilmanTheorem. ........................... 295
15 Compactness Regained: The Weak Topology 298
15.1 Alaoglu’sExtensionofHelley’sTheorem .. .. .. .. .. .. ... .. .. . 298
15.2 Reflexivity and Weak Compactness: Kakutani’s Theorem . . . . . . . . . . . 300
15.3 Compactness and Weak Sequential Compactness: The Eberlein-ˇ
Smulian Theorem ....................................... 302
15.4MetrizabilityofWeakTopologies ......................... 305
16 Continuous Linear Operators on Hilbert Spaces 308
16.1TheInnerProductandOrthogonality....................... 309
16.2 The Dual Space and Weak Sequential Convergence . . . . . . . . . . . . . . 313
16.3 Bessel’sInequalityandOrthonormalBases . . . . . . . . . . . . . . . . . . . 316
16.4 AdjointsandSymmetryforLinearOperators . . . . . . . . . . . . . . . . . . 319
16.5CompactOperators ................................. 324
16.6TheHilbert-SchmidtTheorem ........................... 326
16.7 The Riesz-Schauder Theorem: Characterization of Fredholm Operators . . . 329
III Measure and Integration: General Theory 335
17 General Measure Spaces: Their Properties and Construction 337
17.1MeasuresandMeasurableSets........................... 337
17.2 Signed Measures: The Hahn and Jordan Decompositions . . . . . . . . . . . 342
17.3 The Carath′346
eodory Measure Induced by an Outer Measure . . . . . . . . . . .
17.4TheConstructionofOuterMeasures ....................... 349
17.5 The Carath′eodory-Hahn Theorem: The Extension of a Premeasure to a
Measure ....................................... 352
18 Integration Over General Measure Spaces 359
18.1MeasurableFunctions................................ 359
18.2 Integration of Nonnegative Measurable Functions . . . . . . . . . . . . . . . 365
18.3 Integration of General Measurable Functions . . . . . . . . . . . . . . . . . . 372
18.4TheRadon-NikodymTheorem .......................... 381
18.5 The Nikodym Metric Space: The Vitali–Hahn–Saks Theorem . . . . . . . . . 388
19 General Lp Spaces: Completeness, Duality, and Weak Convergence 394
19.1 The Completeness of LpX,μ1 ≤≤................... 394
19.2 The Riesz Representation Theorem for the Dual of LpX,μ1 ≤≤. . 399
19.3 The Kantorovitch Representation Theorem for the Dual of L∞X,μ.... 404
19.4 Weak Sequential Compactness in LpX,μ1 19.5 Weak Sequential Compactness in L1X,μ: The Dunford-Pettis Theorem . . 409
20 The Construction of Particular Measures 414
20.1 Product Measures: The Theorems of Fubini and Tonelli . . . . . . . . . . . . 414
20.2 Lebesgue Measure on Euclidean Space Rn .................... 424
20.3 Cumulative Distribution Functions and Borel Measures on ......... 437
20.4 Caratheodory Outer Measures and Hausdorff Measures on a Metric Space ′. 441
21 Measure and Topology 446
21.1LocallyCompactTopologicalSpaces ....................... 447
21.2 SeparatingSetsandExtendingFunctions. . . . . . . . . . . . . . . . . . . . . 452
21.3TheConstructionofRadonMeasures....................... 454
21.4 The Representation of Positive Linear Functionals on CcX:The Riesz-
MarkovTheorem .................................. 457
21.5 The Riesz Representation Theorem for the Dual of CX........... 462
21.6 RegularityPropertiesofBaireMeasures . . . . . . . . . . . . . . . . . . . . . 470
22 Invariant Measures 477
22.1 Topological Groups: The General Linear Group . . . . . . . . . . . . . . . . 477
22.2Kakutani’sFixedPointTheorem ......................... 480
22.3 Invariant Borel Measures on Compact Groups: von Neumann’s Theorem . . 485
22.4 Measure Preserving Transformations and Ergodicity: The Bogoliubov-Krilov
Theorem ....................................... 488
Bibliography 495
Index 497