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