Real Analysis

660 Pages·2008·3.213 MB·English

by  Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson

REAL ANALYSIS
SecondEdition(2008)
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BRUCKNER2 THOMSON
——————· ———————
Brian S. Thomson
JudithB. Bruckner
Andrew M.Bruckner
WWW.CLASSICALREALANALYSIS.COM
This second edition of Real Analysis contains all the material of the first edition
originally published byPrenticeHall(Pearson)in1997, withcorrections andrevisions
and in a new format. For further information on this title and others in the series visit
ourwebsite. TherearePDFfilesofallofourtextsfreelyavailablefordownloadaswell
asinstructions onhowtoordertradepaperback copies.
Thistitleisalsoavailableinatwo-volumeeditiontradepaperbackwhichsomeusers
willfindisamoreconvenient size.
OriginalCitation:RealAnalysis,AndrewM.Bruckner,JudithB.Bruckner,BrianS.Thomson,
Prentice-Hall,1997,xiv713pp.[ISBN0-13-458886-X]
CoverDesignandPhotography: DavidSprecher
RealAnalysis,
SecondEdition(2008)
ISBN:1434844129
EAN-13: 9781434844125
NewCitation:RealAnalysis,SecondEdition,,AndrewM.Bruckner,JudithB.Bruckner,Brian
S.Thomson,ClassicalRealAnalysis.com,2008,xvi642pp. [ISBN1434844129]
DatePDFfilecompiled:September16,2008
www.classicalrealanalysis.com
Contents
PREFACE pagexiii
VOLUME ONE 1
1 BackgroundandPreview 1
1.1 TheRealNumbers 2
1.1.1 Setsofrealnumbers 3
1.1.2 Opensetsandclosedsets 4
1.2 CompactSetsofRealNumbers 7
1.2.1 Cousincoveringtheorem 7
1.2.2 Heine-Borel andBolzano-Weierstrass theorems 8
1.3 Countable Sets 10
1.3.1 Theaxiomofchoice 10
1.4 Uncountable Cardinals 12
1.5 TransfiniteOrdinals 14
1.5.1 Atransfinitecoveringargument 17
1.6 Category 18
1.6.1 TheBairecategory theoremontherealline 18
1.6.2 Anillustration ofacategoryargument 18
1.7 OuterMeasureandOuterContent 20
1.8 SmallSets 22
1.8.1 Cantorsets 23
1.8.2 Expressing thereallineastheunionoftwo“small”sets 24
1.9 Measurable SetsofRealNumbers 25
1.10 Nonmeasurable Sets 29
1.10.1 Existence ofsetsofrealnumbersnotLebesguemeasurable 30
1.11 Zorn’sLemma 32
1.12 BorelSetsofRealNumbers 34
1.13 AnalyticSetsofRealNumbers 35
1.14 BoundedVariation 37
1.15 Newton’sIntegral 40
1.16 Cauchy’sIntegral 41
1.16.1 Cauchy’sextension oftheintegraltounbounded functions 42
iii
iv Contents
1.17 Riemann’sIntegral 43
1.17.1 Necessaryandsufficientconditions forRiemannintegrability 44
1.18 Volterra’sExample 45
1.19 Riemann–Stieltjes Integral 47
1.20 Lebesgue’s Integral 50
1.21 TheGeneralized RiemannIntegral 52
1.22 Additional ExercisesforChapter1 55
2 MeasureSpaces 58
2.1 One-Dimensional LebesgueMeasure 59
2.1.1 Lebesgueoutermeasure 60
2.1.2 Lebesgueinnermeasure 61
2.1.3 Lebesguemeasurable sets 61
2.2 AdditiveSetFunctions 64
2.2.1 Example:Distributions ofmass 65
2.2.2 Positiveandnegativevariations 66
2.2.3 Jordandecomposition theorem 67
2.3 MeasuresandSignedMeasures 69
2.3.1 s –algebras ofsets 70
2.3.2 Signedmeasures 70
2.3.3 Computations withsigned measures 71
2.3.4 Thes -algebra generated byafamilyofsets 72
2.4 LimitTheorems 73
2.4.1 Limsupandliminfofasequence ofsets 74
2.4.2 Monotone limitsinameasurespace 74
2.4.3 Liminfsandlimsupsinameasurespace 75
2.5 TheJordanandHahnDecomposition Theorems 76
2.5.1 JordanDecomposition 77
2.6 HahnDecomposition 78
2.7 CompleteMeasures 80
2.7.1 Thecompletion ofameasurespace 80
2.8 OuterMeasures 82
2.8.1 Measurable setswithrespect toanoutermeasure 84
2.8.2 Thes -algebra ofmeasurable sets 84
2.9 MethodI 87
2.9.1 Awarning 88
2.10 RegularOuterMeasures 89
2.10.1 RegularityofMethodIoutermeasures 90
2.10.2 RegularityofLebesgueoutermeasure 91
2.10.3 Summary 92
2.11 Nonmeasurable Sets 93
2.11.1 Ulam’stheorem 94
2.12 MoreAboutMethodI 96
2.12.1 RegularityforMethodIoutermeasures 97
Contents v
2.12.2 Theidentityµ(T)=t (T)forMethodImeasures 98
2.13 Completions 99
2.14 Additional ExercisesforChapter2 102
3 MetricOuterMeasures 105
3.1 MetricSpace 105
3.1.1 Metricspaceterminology 106
3.1.2 Borelsetsinametricspace 107
3.1.3 Characterizations oftheBorelsets 108
3.2 MeasuresonMetricSpaces 110
3.2.1 MetricOuterMeasures 111
3.2.2 Measurability ofBorelsets 111
3.3 MethodII 114
3.3.1 MethodIIoutermeasuresaremetricoutermeasures 115
3.3.2 AgreementofMethodIandMethodIImeasures 116
3.4 Approximations 118
3.4.1 Approximation frominside 118
3.4.2 Approximation fromoutside 119
3.4.3 Approximation usingFs andGd sets 120
3.5 Construction ofLebesgue–Stieltjes Measures 121
3.6 Properties ofLebesgue–Stieltjes Measures 126
3.6.1 HowregularareBorelmeasures? 128
3.6.2 Acharacterization offiniteBorelmeasuresontherealline 129
3.6.3 Measuring thegrowthofacontinuous function onaset 129
3.7 Lebesgue–Stieltjes MeasuresinIRn 131
3.8 HausdorffMeasuresandHausdorffDimension 133
3.8.1 Hausdorffdimension 135
3.8.2 Hausdorffdimension ofacurve 136
3.8.3 Exceptional sets 138
3.9 MethodsIIIandIV 140
3.9.1 Constructing measuresusingfullandfinecovers 142
3.9.2 Aregularity theorem 143
3.10 Mini-VitaliTheorem 145
3.10.1 Coveringlemmas 146
3.10.2 ProofoftheMini-Vitalicoveringtheorem 148
3.11 Lebesguedifferentiation theorem 149
3.11.1 Ageometrical lemma 150
3.11.2 ProofoftheLebesgue differentiation theorem 151
3.12 Additional RemarksonSpecialSets 154
3.12.1 Cantorsets 154
3.12.2 Bernstein sets 155
3.12.3 Lusinsets 156
3.13 Additional ExercisesforChapter3 158
4 MeasurableFunctions 162
vi Contents
4.1 DefinitionsandBasicProperties 163
4.1.1 Combiningmeasurable functions 165
4.2 SequencesofMeasurable Functions 168
4.2.1 Convergence almosteverywhere 169
4.2.2 Convergence inmeasure 170
4.2.3 Pointwiseconvergence andconvergence inmeasure 171
4.3 Egoroff’sTheorem 173
4.3.1 Comparisons 174
4.4 Approximations bySimpleFunctions 176
4.4.1 Approximation bybounded, measurablefunctions 178
4.5 Approximation byContinuous Functions 180
4.5.1 Tietzeextension theorem 181
4.5.2 Lusin’stheorem 181
4.5.3 Furtherdiscussion 183
4.6 Additional ExercisesforChapter4 184
5 Integration 188
5.1 Introduction 189
5.1.1 ScopeoftheConceptofIntegral 189
5.1.2 TheClassofIntegrable Functions 190
5.1.3 Thefundamental theorem ofcalculusforRiemannintegrals 191
5.2 IntegralsofNonnegativeFunctions 193
5.2.1 Theintegralofanonnegative simplefunction 194
5.2.2 Theintegralofanonnegative, measurable function 195
5.3 Fatou’sLemma 197
5.3.1 Aconvergence theorem forintegrals ofnonnegative functions198
5.3.2 Properties ofintegralsofnonnegative functions 199
5.4 Integrable Functions 201
5.4.1 Properties ofintegrals 202
5.4.2 TheLebesguedominated convergence theorem 202
5.5 RiemannandLebesgue 204
5.5.1 Approximation bystepfunctions 206
5.5.2 Upperandlowerboundaries ofafunction 207
5.5.3 Lebesgue’s characterization ofRiemannintegrability 209
5.5.4 Fundamental theoremofthecalculusforLebesgueintegrals 209
5.6 CountableAdditivity oftheIntegral 212
5.7 AbsoluteContinuity 215
5.7.1 Absolutely continuous functions 215
5.7.2 Acharacterization ofabsolutely continuous functions 216
5.7.3 Absolutecontinuity andLebesgue-Stieljtes measures 217
5.8 Radon–Nikodym Theorem 220
5.8.1 MotivatingtheproofoftheRadon–Nikodym theorem 220
5.8.2 TheproofoftheRadon–Nikodym theorem 221
5.8.3 TheVitali-Lebesgue theorem 223
Contents vii
5.8.4 Properties ofRadon–Nikodym derivatives 224
5.8.5 TheLebesguedecomposition 225
5.9 Convergence Theorems 227
5.9.1 Convergence inthemean 227
5.9.2 Amoreilluminating proofusingtherectangle principle 228
5.9.3 Comparison ofconvergence conditions 229
5.9.4 Dominatedconvergence anduniformabsolute continuity 231
5.10 RelationstoOtherIntegrals 232
5.10.1 TheCauchyprocessandLebesgueintegration 233
5.10.2 Thegeneralized RiemannintegralandLebesgueintegration 234
5.11 Integration ofComplexFunctions 237
5.12 Additional ExercisesforChapter5 240
6 Fubini’sTheorem 245
6.1 ProductMeasures 246
6.1.1 Themeasureofrectangles 247
6.1.2 PreliminaryversionoftheFubinitheorem 251
6.2 Fubini’sTheorem 254
6.3 Tonelli’sTheorem 255
6.4 Additional ExercisesforChapter6 257
7 Differentiation 259
7.1 TheVitaliCoveringTheorem 259
7.1.1 Growthproperties ofrealfunctions 260
7.1.2 TheVitalicoveringtheorem 261
7.1.3 Proofofthegrowthlema 261
7.1.4 ElementaryproofoftheVitalitheorem 263
7.1.5 Banach’sproofoftheVitalitheorem 265
7.2 Lebesgue’s Differentiation Theorem 267
7.2.1 Constructing amonotonic functionwithaninfinitederivative 268
7.2.2 Integrating aderivative 269
7.3 TheBanach–Zarecki Theorem 271
7.4 DeterminingaFunctionbyaDerivative 274
7.5 Calculating aFunctionfromaDerivative 276
7.6 TotalVariationofaFunction 282
7.6.1 Growthlemmas 286
7.6.2 VBG Functions 287
∗
7.7 Approximate ContinuityandLebesguePoints 292
7.7.1 Approximately continuous functions 293
7.7.2 Lebesgue points 295
7.8 Additional ExercisesforChapter7 297
8 Differentiation ofMeasures 304
8.1 Differentiation ofLebesgue–Stieltjes Measures 304
8.1.1 Theordinaryderivativeusingthecubebasis 305
8.1.2 Mixedpartialderivatives 306
viii Contents
8.1.3 Thestrongderivativeusingtheintervalbasis 307
8.2 TheCubeBasis 309
8.2.1 Vitali’scoveringtheorem forthecubebasis 309
8.2.2 Differentiability ofLebesgue–Stieltjes measuresonIRn 310
8.2.3 AtheoremofFubini 312
8.2.4 Thefundamental theorem ofthecalculus 313
8.3 LebesgueDecomposition Theorem 314
8.4 TheIntervalBasis 316
8.4.1 TheLebesguedensity theoremfortheintervalbasis 316
8.4.2 Approximate continuity 318
8.4.3 Differentiation oftheintegralforbounded functions 319
8.4.4 Mixedpartials 320
8.4.5 Additional remarks 321
8.5 NetStructures 323
8.5.1 Differentiation withrespecttoanetstructure 324
8.5.2 Agrowthlemma 324
8.5.3 AnanalogofdelaValléePoussin’s theoremfornetstructures325
8.5.4 Furtherremarks 327
8.6 Radon–Nikodym DerivativeinaMeasureSpace 328
8.6.1 Liftings 330
8.6.2 Growthlemmas 332
8.6.3 TheRadon–Nikodym derivativeasagenuine derivative 333
8.7 Summary,Comments,andReferences 336
8.8 Additional ExercisesforChapter8 339
VOLUME TWO 341
9 MetricSpaces 341
9.1 DefinitionsandExamples 341
9.1.1 EuclideanSpace 342
9.1.2 TheDiscreteSpace 342
9.1.3 TheMinkowskiMetrics 343
9.1.4 SequenceSpaces 344
9.1.5 FunctionSpaces 346
9.1.6 SpacesofSets 348
9.2 Convergence andRelatedNotions 349
9.2.1 Metricspaceterminology 349
9.3 Continuity 352
9.3.1 Urysohn’s Lemma 354
9.3.2 ProofofTietze’stheorem 354
9.4 HomeomorphismsandIsometries 356
9.5 SeparableSpaces 359
9.5.1 Examplesofseparable metricspaces 360
9.6 CompleteSpaces 361
9.6.1 Examplesofcompletemetricspaces 362
Contents ix
9.6.2 Completion ofametricspace 365
9.7 Contraction Maps 366
9.8 Applications 368
9.8.1 Picard’sTheorem 372
9.9 Compactness 374
9.9.1 Continuous functions oncompactmetricspaces 376
9.10 TotallyBoundedSpaces 377
9.11 CompactSetsinC(X) 378
9.11.1 Arzelà–Ascoli Theorem 379
9.12 Application oftheArzelà–Ascoli Theorem 382
9.13 TheStone–Weierstrass Theorem 384
9.13.1 TheWeierstrassapproximation theorem 386
9.14 TheIsoperimetric Problem 387
9.15 MoreonConvergence 390
9.16 Additional ExercisesforChapter9 393
10 BaireCategory 396
10.1 TheBanach-Mazur GameontheRealLine 396
10.2 TheBaireCategoryTheorem 398
10.2.1 Terminology forapplications oftheBairetheorem 399
10.2.2 Typicalproperties 399
10.3 TheBanach–Mazur Game 401
10.3.1 Thetypicalcontinuous function isnowheremonotonic 404
10.4 TheFirstClassesofBaireandBorel 406
10.4.1 TheidentityofB andBor 408
1 1
10.5 Properties ofBaire-1Functions 411
10.5.1 Weakconvergence ofmeasures 414
10.6 Topologically CompleteSpaces 415
10.6.1 Alexandroff’s Theorem 416
10.6.2 Mazurkiewicz’s theorem 417
10.7 Applications toFunctionSpaces 419
10.7.1 Continuous NowhereDifferentiable Functions 419
10.7.2 Differentiable, NowhereMonotonic Functions 422
10.7.3 TheSpaceofAutomorphisms 424
10.8 Additional ExercisesforChapter10 429
11 AnalyticSets 434
11.1 ProductsofMetricSpaces 434
11.2 BaireSpace 436
11.3 AnalyticSets 439
11.4 BorelSets 442
11.4.1 Projections ofclosedsets 442
11.4.2 Lusin’sseparation theorem 443
11.4.3 Continuous one-one imagesofclosedsets 444
11.5 AnAnalyticSetThatIsNotBorel 446
x Contents
11.6 Measurability ofAnalyticSets 448
11.7 TheSuslinOperation 450
11.8 AMethodtoShowaSetIsNotBorel 452
11.9 Differentiable Functions 455
11.10 Additional ExercisesforChapter11 458
12 BanachSpaces 461
12.1 NormedLinearSpaces 461
12.1.1 Metriclinearspaces 462
12.1.2 Sequencespaces 463
12.1.3 FunctionSpaces 464
12.2 Compactness 466
12.2.1 Theunitsphereinaninfinitedimensional space 467
12.2.2 Riesz’stheorem 467
12.2.3 Bestapproximation problems 468
12.3 LinearOperators 470
12.3.1 Boundedlinearoperators 471
12.3.2 Thespaceofbounded linearoperators 472
12.4 BanachAlgebras 474
12.4.1 Existenceanduniqueness ofsolutions ofanintegralequation 475
12.5 TheHahn–Banach Theorem 477
12.5.1 Banach’sversionoftheHahn–Banach theorem 478
12.5.2 Hahn’sversionoftheHahn–Banachtheorem 480
12.6 ImprovingLebesgueMeasure 481
12.6.1 Extension ofLebesguemeasuretoafinitelyadditivemeasure 482
12.6.2 TheBanach–Tarski paradox 485
12.6.3 Atranslation invariant improvementofLebesguemeasure 485
12.7 TheDualSpace 486
12.8 TheRieszRepresentation Theorem 489
12.9 SeparationofConvexSets 494
12.10 AnEmbeddingTheorem 499
12.11 UniformBoundedness Principle 501
12.11.1Convergence ofsequences ofcontinuous linearoperators 502
12.11.2Condensation ofsingularities 503
12.12 AnApplication toSummability 504
12.12.1Toeplitz’stheorem 505
12.13 TheOpenMappingTheorem 508
12.13.1Equivalence ofnormsonaBanachspace 510
12.13.2Perturbations indifferential equations 510
12.14 TheClosedGraphTheorem 511
12.15 Additional ExercisesforChapter12 514
13 TheL spaces 516
p
13.1 TheBasicInequalities 516
13.1.1 Hölder’sinequality 517