By James K. Peterson
Table of Contents
I Introductory Matter 1 1 Introduction 3 1.1 TheAnalysisCourses …………………………… 3 1.1.1 SeniorLevelAnalysis ………………………. 4 1.1.2 TheGraduateAnalysisCourses ………………….. 4 1.1.3 MoreAdvancedCourses ……………………… 7 1.2 TeachingtheMeasureandIntegrationCourse . . . . . . . . . . . . . . . . . . . . 8 1.3 TableofContents …………………………….. 9 1.4 Acknowledgments…………………………….. 11 II Classical Riemann Integration 13 2 An Overview of Riemann Integration 15 2.1 Integration………………………………… 15 2.1.1 TheRiemannIntegralasaLimit ………………….. 17 2.1.2 TheFundamentalTheoremofCalculus . . . . . . . . . . . . . . . . . . . 19 2.1.3 The Cauchy Fundamental Theorem of Calculus . . . . . . . . . . . . . . . 22 2.2 HandlingJumps ……………………………… 24 2.2.1 RemovableDiscontinuity …………………….. 24 2.2.2 JumpDiscontinuity………………………… 25 3 Bounded Variation 29 3.1 Partitions ………………………………… 30 3.2 Monotone ………………………………… 31 3.2.1 TheSaltusFunction ……………………….. 36 3.2.2 TheContinuousPartofaMonotoneFunction . . . . . . . . . . . . . . . . 38 3.3 BoundedVariation ……………………………. 44 3.4 TheTotalVariationFunction ……………………….. 48 3.5 ContinuousFunctionsofBoundedVariation . . . . . . . . . . . . . . . . . . . . . 51 4 Riemann Integration 55 4.1 Definition ………………………………… 55 4.2 Existence ………………………………… 58 4.3 Properties ………………………………… 66 4.4 RiemannIntegrable? …………………………… 71 4.5 MoreProperties ……………………………… 73 4.6 FundamentalTheorem ………………………….. 77 4.7 Substitution ……………………………….. 85 4.8 SameIntegral? ……………………………… 88
5 Further Riemann Results 93 5.1 LimitInterchange …………………………….. 93 5.2 RiemannIntegrable? …………………………… 99 5.3 ContentZero ………………………………. 100 III Riemann -Stieltjes Integration 109 6 The Riemann -Stieltjes Integral 111 6.1 Properties ………………………………… 112 6.2 StepIntegrators ……………………………… 115 6.3 MonotoneIntegrators …………………………… 121 6.4 EquivalenceTheorem …………………………… 123 6.5 FurtherProperties …………………………….. 124 6.6 BoundedVariationIntegrators ………………………. 127 7 Further Riemann -Stieltjes Results 133 7.1 FundamentalTheorem ………………………….. 133 7.2 Existence ………………………………… 136 7.3 Computations ………………………………. 139 IV Abstract Measure Theory One 147 8 Measurability 149 8.1 BorelSigma-Algebra …………………………… 152 8.2 TheExtendedBorelSigma-Algebra ……………………. 153 8.3 MeasurableFunctions…………………………… 155 8.3.1 Examples…………………………….. 157 8.4 Properties ………………………………… 158 8.5 ExtendedReal-Valued…………………………… 161 8.6 ExtendedProperties……………………………. 163 8.7 ContinuousCompositions…………………………. 167 8.7.1 The Composition with Finite Measurable Functions . . . . . . . . . . . . . 168 8.7.2 The Approximation of Non-Negative Measurable Functions . . . . . . . . 168 8.7.3 Continuous Functions of Extended Real-Valued Measurable Functions . . 170 9 Abstract Integration 173 9.1 Measures ………………………………… 174 9.2 Properties ………………………………… 175 9.3 SequencesofSets …………………………….. 178 9.4 Integration………………………………… 182 9.5 IntegrationProperties …………………………… 186 9.6 Equalitya.e.Problems ………………………….. 190 9.7 CompleteMeasures ……………………………. 191 9.8 ConvergenceTheorems ………………………….. 194 9.8.1 MonotoneConvergenceTheorems…………………. 194 9.8.2 Fatou’sLemma ………………………….. 198 9.9 TheAbsoluteContinuityofaMeasure …………………… 200 9.10 SummableFunctions …………………………… 202 9.11 ExtendedIntegrands……………………………. 202 9.12 Levi’sTheorem ……………………………… 205
9.13 ConstructingCharges …………………………… 207 9.14 PropertiesofSummableFunctions …………………….. 209 9.15 TheDominatedConvergenceTheorem …………………… 211 9.16 AlternativeAbstractIntegrationSchemes . . . . . . . . . . . . . . . . . . . . . . 214 9.16.1 PropertiesoftheDarbouxIntegral …………………. 218 10 The Lp Spaces 223 10.1 The General Lp Spaces ………………………….. 227 10.2 TheWorldofCountingMeasure ……………………… 238 10.3 EssentiallyBoundedFunctions ………………………. 240 10.4 The Hilbert Space L2 … . … . … . .. . … . … . … . .. . … 247 V Constructing Measures 249 11 Building Measures 251 11.1 MeasuresfromOuterMeasure ………………………. 251 11.2 ThePropertiesoftheOuterMeasure ……………………. 254 11.3 MeasuresInducedbyOuterMeasures …………………… 257 11.4 MeasuresfromMetricOuterMeasures …………………… 258 11.5 ConstructingOuterMeasure ……………………….. 264 12 Lebesgue Measure 273 12.1 OuterMeasure………………………………. 273 12.2 LebesgueOuterMeasureisaMetricOuterMeasure . . . . . . . . . . . . . . . . . 283 12.3 LebesgueMeasureisRegular……………………….. 287 12.4 ApproximationResults ………………………….. 288 12.4.1 ApproximatingMeasurableSets ………………….. 288 12.4.2 ApproximatingMeasurableFunctions . . . . . . . . . . . . . . . . . . . . 292 12.5 TheSummableFunctionsareSeparable ………………….. 294 12.6 The Existence of Non-Lebesgue Measurable Sets . . . . . . . . . . . . . . . . . . 295 12.7 MetricSpaces ………………………………. 297 13 Cantor Sets 301 13.1 Generalized ……………………………….. 301 13.2 Representation………………………………. 304 13.3 TheCantorFunctions …………………………… 305 13.4 Consequences………………………………. 307 14 Lebesgue -Stieltjes Measure 309 14.1 Lebesgue-StieltjesOuterMeasureandMeasure. . . . . . . . . . . . . . . . . . . 311 14.2 ApproximationResults ………………………….. 317 14.3 Properties ………………………………… 318 VI Abstract Measure Theory Two 323 15 Convergence Modes 325 15.1 ExtractingSubsequences …………………………. 328 15.2 Egoroff’sTheorem ……………………………. 336 15.3 Vitali’sTheorem……………………………… 339 15.4 Summary ………………………………… 344 16 Decomposing Measures 349 16.1 JordanDecomposition ………………………….. 349 16.2 HahnDecomposition …………………………… 354 16.3 Variation…………………………………. 356 16.4 AbsoluteContinuity……………………………. 359 16.5 Radon-Nikodym……………………………… 362 16.6 LebesgueDecomposition …………………………. 370 17 Connections to Riemann Integration 375 18 Fubini Type Results 379 18.1 TheRiemannSetting …………………………… 379 18.1.1 FubinionaRectangle ………………………. 380 18.2 TheLebesgueSetting …………………………… 387 19 Differentiation 397 19.1 AbsolutelyContinuousFunctions……………………… 397 19.2 LSandAC………………………………… 398 19.3 BoundedVariationDerivatives ………………………. 401 19.4 MeasureEstimates ……………………………. 404 19.5 Extending the Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . 408 19.6 Charges Induced by Absolutely Continuous Functions . . . . . . . . . . . . . . . . 414 VII Summing It All Up 417 20 Summing It All Up 419 VIII References 423 IX Detailed Index 427 X Appendix: Undergraduate Analysis Background Check 443 A Undergraduate Analysis Part One 445 A-1 SampleExams………………………………. 448 A-1.1 Exam1 ……………………………… 448 A-1.2 Exam2 ……………………………… 449 A-1.3 Exam3 ……………………………… 450 A-1.4 Final ………………………………. 451 B Undergraduate Analysis Part Two 453 B-1 SampleExams………………………………. 457 B-1.1 Exam1 ……………………………… 457 B-1.2 Exam2 ……………………………… 458 B-1.3 Exam3 ……………………………… 459 B-1.4 Final ………………………………. 460 XI Appendix: Linear Analysis Background Check 463 C Linear Analysis 465 C-1 SampleExams………………………………. 467 C-1.1 Exam1 ……………………………… 467 C-1.2 Exam2 ……………………………… 468 C-1.3 Exam3 ……………………………… 469 C-1.4 Final ………………………………. 470 XII Appendix: Preliminary Examination Check 475 D The Preliminary Examination in Analysis 477 D-1 SampleExams………………………………. 477 D-1.1 Exam 1 .. . … . … . … . .. . … . … . … . .. . … 477 .. . … . … . … . .. . … . … . … . .. . … 479 .. . … . … . … . .. . … . … . … . .. . … 481 .. . … . … . … . .. . … . … . … . .. . … 482 .. . … . … . … . .. . … . … . … . .. . … 483 .. . … . … . … . .. . … . … . … . .. . … 484 .. . … . … . … . .. . … . … . … . .. . … 485 .. . … . … . … . .. . … . … . … . .. . … 487
Introduction
We believe that all students who are seriously interested in mathematics at the master’s and doctoral level should have a passion for analysis even if it is not the primary focus of their own research interests. So you should all understand that my own passion for the subject will shine though in the notes that follow! And, it goes without saying that we assume that you are all mature mathematically and eager and interested in the material! Now, the present text focuses on the topics of Measure and Integration from a very abstract point of view, but it is very helpful to place this course into its proper context. Also, for those of you who are preparing to take the qualifying examination in analysis, the overview below will help you see why all this material fits together into a very interesting web of ideas.