目录
第一部分 一元实变量函数的Lebesgue积分
第0章 集合、映射与关系的预备知识3
0.1 集合的并与交3
0.2 集合间的映射4
0.3 等价关系、选择公理以及Zorn引理5
第1章 实数集：集合、序列与函数7
1.1 域、正性以及完备性公理7
1.2 自然数与有理数11
1.3 可数集与不可数集13
1.4 实数的开集、闭集和Borel集16
1.5 实数序列20
1.6 实变量的连续实值函数25
第2章 Lebesgue测度29
2.1 引言29
2.2 Lebesgue外测度31
2.3 Lebesgue可测集的代数34
2.4 Lebesgue可测集的外逼近和内逼近40
2.5 可数可加性、连续性以及Borel-Cantelli引理43
2.6 不可测集47
2.7 Cantor集和Cantor-Lebesgue函数49
第3章 Lebesgue可测函数54
3.1 和、积与复合54
3.2 序列的逐点极限与简单逼近60
3.3 Littlewood的三个原理、Egoroff定理以及Lusin定理64
第4章 Lebesgue积分68
4.1 Riemann积分68
4.2 有限测度集上的有界可测函数的
Lebesgue积分71
4.3 非负可测函数的Lebesgue积分79
4.4 一般的Lebesgue积分85
4.5 积分的可数可加性与连续性90
4.6 一致可积性：Vitali收敛定理92
第5章 Lebesgue积分：深入课题97
5.1 一致可积性和紧性：一般的Vitali收敛定理97
5.2 依测度收敛99
5.3 Riemann可积与Lebesgue可积的刻画102
第6章 微分与积分107
6.1 单调函数的连续性108
6.2 单调函数的可微性：Lebesgue定理109
6.3 有界变差函数：Jordan定理116
6.4 绝对连续函数119
6.5 导数的积分：微分不定积分124
6.6 凸函数130
第7章 Lp空间：完备性与逼近135
7.1 赋范线性空间135
7.2 Young、H鰈der与Minkowski不等式139
7.3 Lp是完备的：Riesz-Fischer定理144
7.4 逼近与可分性150
第8章 Lp空间：对偶与弱收敛155
8.1 关于Lp（1≤p＜∞）的对偶的Riesz表示定理155
8.2 Lp中的弱序列收敛162
8.3 弱序列紧性171
8.4 凸泛函的最小化174
第二部分 抽象空间：度量空间、
拓扑空间、Banach空间
和Hilbert空间
第9章 度量空间：一般性质183
9.1 度量空间的例子183
9.2 开集、闭集以及收敛序列187
9.3 度量空间之间的连续映射190
9.4 完备度量空间193
9.5 紧度量空间197
9.6 可分度量空间204
第10章 度量空间：三个基本定理206
10.1 Arzelà-Ascoli定理206
10.2 Baire范畴定理211
10.3 Banach压缩原理215
第11章 拓扑空间：一般性质222
11.1 开集、闭集、基和子基222
11.2 分离性质227
11.3 可数性与可分性228
11.4 拓扑空间之间的连续映射230
11.5 紧拓扑空间233
11.6 连通的拓扑空间237
第12章 拓扑空间：三个基本定理239
12.1 Urysohn引理和Tietze延拓定理239
12.2 Tychonoff乘积定理244
12.3 Stone-Weierstrass定理247
第13章 Banach空间之间的连续线性算子253
13.1 赋范线性空间253
13.2 线性算子256
13.3 紧性丧失：无穷维赋范线性空间259
13.4 开映射与闭图像定理263
13.5 一致有界原理268
第14章 赋范线性空间的对偶271
14.1 线性泛函、有界线性泛函以及弱拓扑271
14.2 Hahn-Banach定理277
14.3 自反Banach空间与弱序列
收敛性282
14.4 局部凸拓扑向量空间286
14.5 凸集的分离与Mazur定理290
14.6 Krein-Milman定理295
第15章 重新得到紧性：弱拓扑298
15.1 Helly定理的Alaoglu推广298
15.2 自反性与弱紧性：Kakutani定理300
15.3 紧性与弱序列紧性：Eberlein-mulian定理302
15.4 弱拓扑的度量化305
第16章 Hilbert空间上的连续线性算子308
16.1 内积和正交性309
16.2 对偶空间和弱序列收敛313
16.3 Bessel不等式与规范正交基316
16.4 线性算子的伴随与对称性319
16.5 紧算子324
16.6 Hilbert-Schmidt定理326
16.7 Riesz-Schauder定理：Fredholm算子的刻画329
第三部分 测度与积分：一般理论
第17章 一般测度空间：性质与构造337
17.1 测度与可测集337
17.2 带号测度：Hahn与Jordan分解342
17.3 外测度诱导的Carathéodory测度346
17.4 外测度的构造349
17.5 将预测度延拓为测度：Carathéodory-Hahn定理352
第18章 一般测度空间上的积分359
18.1 可测函数359
18.2 非负可测函数的积分365
18.3 一般可测函数的积分372
18.4 Radon-Nikodym定理381
18.5 Nikodym度量空间：Vitali-Hahn-Saks定理388
第19章 一般的Lp空间：完备性、对偶性和弱收敛性394
19.1 Lp(X, )（1≤p≤∞）的完备性394
19.2 关于Lp(X, )（1≤p<∞）的对偶的Riesz表示定理399
19.3 关于L∞(X, )的对偶的Kantorovitch表示定理404
19.4 Lp(X, )（1＜p＜∞）的弱序列紧性407
19.5 L1(X, )的弱序列紧性：Dunford-Pettis定理409
第20章 特定测度的构造414
20.1 乘积测度：Fubini与Tonelli定理414
20.2 欧氏空间Rn上的Lebesgue测度424
20.3 累积分布函数与Borel测度437
20.4 度量空间上的Carathéodory外测度与Hausdorff测度441
第21章 测度与拓扑446
21.1 局部紧拓扑空间447
21.2 集合分离与函数延拓452
21.3 Radon测度的构造454
21.4 Cc(X)上的正线性泛函的表示：Riesz-Markov定理457
21.5 C（X）的对偶的表示：Riesz-Kakutani表示定理462
21.6 Baire测度的正则性470
第22章 不变测度477
22.1 拓扑群：一般线性群477
22.2 Kakutani不动点定理480
22.3 紧群上的不变Borel测度：von Neumann定理485
22.4 测度保持变换与遍历性：Bogoliubov-Krilov定理488
参考文献495
Contents
I Lebesgue Integration for Functions of a Single Real Variable 1
0 Preliminaries on Sets, Mappings, and Relations 3
UnionsandIntersectionsofSets ............................. 3
Mappings Between Sets............................. 4
Equivalence Relations, the Axiom of Choice, and Zorn’s Lemma . . . . . . . . . . 5
1 The Real Numbers: Sets, Sequences, and Functions 7
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
2 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
3 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
4 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
5 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
6 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 Inde.nite 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 ≤ p < ∞ ........... 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
9 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
Contents i.
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: In.nite 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 Re.exive 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 Re.exivity and Weak Compactness: Kakutani’s Theorem . . . . . . . . . . . 300
15.3 Compactness and Weak Sequential Compactness: The Eberlein-ˇSmulian 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 Lp.X, μ., 1 ≤p ≤∞ ................... 394
19.2 The Riesz Representation Theorem for the Dual of Lp.X, μ., 1 ≤p ≤∞ . . 399
19.3 The Kantorovitch Representation Theorem for the Dual of L∞.X, μ. .... 404
19.4 Weak Sequential Compactness in Lp.X, μ., 1 <p< 1............. 407
19.5 Weak Sequential Compactness in L1.X, 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 R ......... 437
20.4 Caratheodory Outer Measures and Hausdor ′ff 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 Cc.X.:The Riesz-MarkovTheorem .................... 457
21.5 The Riesz Representation Theorem for the Dual of C.X. ........... 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
