Contents
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 TheMinimizationofConvexFunctionals174
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 [p[ 1. 407
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 SeparatingSetsandExtendingFunctions452
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
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