第一部分　一元實(shí)變量函數(shù)的Lebesgue積分
第0章　集合、映射與關(guān)系的預(yù)備知識3
0.1　集合的并與交3
0.2　集合間的映射4
0.3　等價關(guān)系、選擇公理以及Zorn引理5
第1章　實(shí)數(shù)集：集合、序列與函數(shù)7
1.1　域、正性以及完備性公理7
1.2　自然數(shù)與有理數(shù)11
1.3　可數(shù)集與不可數(shù)集13
1.4　實(shí)數(shù)的開集、閉集和Borel集16
1.5　實(shí)數(shù)序列20
1.6　實(shí)變量的連續(xù)實(shí)值函數(shù)25
第2章　Lebesgue測度29
2.1　引言29
2.2　Lebesgue外測度31
2.3　Lebesgue可測集的代數(shù)34
2.4　Lebesgue可測集的外逼近和內(nèi)逼近40
2.5　可數(shù)可加性、連續(xù)性以及Borel-Cantelli引理43
2.6　不可測集47
2.7　Cantor集和Cantor-Lebesgue函數(shù)49
第3章　Lebesgue可測函數(shù)54
3.1　和、積與復(fù)合54
3.2　序列的逐點(diǎn)極限與簡單逼近60
3.3　Littlewood的三個原理、Egoroff定理以及Lusin定理64
第4章　Lebesgue積分68
4.1　Riemann積分68
4.2　有限測度集上的有界可測函數(shù)的
　Lebesgue積分71
4.3　非負(fù)可測函數(shù)的Lebesgue積分79
4.4　一般的Lebesgue積分85
4.5　積分的可數(shù)可加性與連續(xù)性90
4.6　一致可積性：Vitali收斂定理92
第5章　Lebesgue積分：深入課題97
5.1　一致可積性和緊性：一般的Vitali收斂定理97
5.2　依測度收斂99
5.3　Riemann可積與Lebesgue可積的刻畫102
第6章　微分與積分107
6.1　單調(diào)函數(shù)的連續(xù)性108
6.2　單調(diào)函數(shù)的可微性：Lebesgue定理109
6.3　有界變差函數(shù)：Jordan定理116
6.4　絕對連續(xù)函數(shù)119
6.5　導(dǎo)數(shù)的積分：微分不定積分124
6.6　凸函數(shù)130
第7章　Lp空間：完備性與逼近135
7.1　賦范線性空間135
7.2　Young、H鰈der與Minkowski不等式139
7.3　Lp是完備的：Riesz-Fischer定理144
7.4　逼近與可分性150
第8章　Lp空間：對偶與弱收斂155
8.1　關(guān)于Lp（1≤p＜∞）的對偶的Riesz表示定理155
8.2　Lp中的弱序列收斂162
8.3　弱序列緊性171
8.4　凸泛函的最小化174
第二部分　抽象空間：度量空間、
拓?fù)淇臻g、Banach空間
和Hilbert空間
第9章　度量空間：一般性質(zhì)183
9.1　度量空間的例子183
9.2　開集、閉集以及收斂序列187
9.3　度量空間之間的連續(xù)映射190
9.4　完備度量空間193
9.5　緊度量空間197
9.6　可分度量空間204
第10章　度量空間：三個基本定理206
10.1　Arzelà-Ascoli定理206
10.2　Baire范疇定理211
10.3　Banach壓縮原理215
第11章　拓?fù)淇臻g：一般性質(zhì)222
11.1　開集、閉集、基和子基222
11.2　分離性質(zhì)227
11.3　可數(shù)性與可分性228
11.4　拓?fù)淇臻g之間的連續(xù)映射230
11.5　緊拓?fù)淇臻g233
11.6　連通的拓?fù)淇臻g237
第12章　拓?fù)淇臻g：三個基本定理239
12.1　Urysohn引理和Tietze延拓定理239
12.2　Tychonoff乘積定理244
12.3　Stone-Weierstrass定理247
第13章　Banach空間之間的連續(xù)線性算子253
13.1　賦范線性空間253
13.2　線性算子256
13.3　緊性喪失：無窮維賦范線性空間259
13.4　開映射與閉圖像定理263
13.5　一致有界原理268
第14章　賦范線性空間的對偶271
14.1　線性泛函、有界線性泛函以及弱拓?fù)?71
14.2　Hahn-Banach定理277
14.3　自反Banach空間與弱序列
　收斂性282
14.4　局部凸拓?fù)湎蛄靠臻g286
14.5　凸集的分離與Mazur定理290
14.6　Krein-Milman定理295
第15章　重新得到緊性：弱拓?fù)?98
15.1　Helly定理的Alaoglu推廣298
15.2　自反性與弱緊性：Kakutani定理300
15.3　緊性與弱序列緊性：Eberlein-mulian定理302
15.4　弱拓?fù)涞亩攘炕?05
第16章　Hilbert空間上的連續(xù)線性算子308
16.1　內(nèi)積和正交性309
16.2　對偶空間和弱序列收斂313
16.3　Bessel不等式與規(guī)范正交基316
16.4　線性算子的伴隨與對稱性319
16.5　緊算子324
16.6　Hilbert-Schmidt定理326
16.7　Riesz-Schauder定理：Fredholm算子的刻畫329
第三部分　測度與積分：一般理論
第17章　一般測度空間：性質(zhì)與構(gòu)造337
17.1　測度與可測集337
17.2　帶號測度：Hahn與Jordan分解342
17.3　外測度誘導(dǎo)的Carathéodory測度346
17.4　外測度的構(gòu)造349
17.5　將預(yù)測度延拓為測度：Carathéodory-Hahn定理352
第18章　一般測度空間上的積分359
18.1　可測函數(shù)359
18.2　非負(fù)可測函數(shù)的積分365
18.3　一般可測函數(shù)的積分372
18.4　Radon-Nikodym定理381
18.5　Nikodym度量空間：Vitali-Hahn-Saks定理388
第19章　一般的Lp空間：完備性、對偶性和弱收斂性394
19.1　Lp(X, )（1≤p≤∞）的完備性394
19.2　關(guān)于Lp(X, )（1≤p<∞）的對偶的Riesz表示定理399
19.3　關(guān)于L∞(X, )的對偶的Kantorovitch表示定理404
19.4　Lp(X, )（1＜p＜∞）的弱序列緊性407
19.5　L1(X, )的弱序列緊性：Dunford-Pettis定理409
第20章　特定測度的構(gòu)造414
20.1　乘積測度：Fubini與Tonelli定理414
20.2　歐氏空間Rn上的Lebesgue測度424
20.3　累積分布函數(shù)與Borel測度437
20.4　度量空間上的Carathéodory外測度與Hausdorff測度441
第21章　測度與拓?fù)?46
21.1　局部緊拓?fù)淇臻g447
21.2　集合分離與函數(shù)延拓452
21.3　Radon測度的構(gòu)造454
21.4　Cc(X)上的正線性泛函的表示：Riesz-Markov定理457
21.5　C（X）的對偶的表示：Riesz-Kakutani表示定理462
21.6　Baire測度的正則性470
第22章　不變測度477
22.1　拓?fù)淙海阂话憔�性群477
22.2　Kakutani不動點(diǎn)定理480
22.3　緊群上的不變Borel測度：von Neumann定理485
22.4　測度保持變換與遍歷性：Bogoliubov-Krilov定理488
參考文獻(xiàn)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
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