Basic analysis IV: measure theory and integration

Cover......Page 1
Half Title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Acknowledgments......Page 6
Table of Contents......Page 8
I Introductory Matter......Page 14
  1.1 The Analysis Courses......Page 16
   1.1.2 The Graduate Analysis Courses......Page 17
   1.1.3 More Advanced Courses......Page 20
  1.2 Teaching the Measure and Integration Course......Page 21
  1.3 Table of Contents......Page 22
  1.4 Acknowledgments......Page 24
II Classical Riemann Integration......Page 26
  2.1 Integration......Page 28
   2.1.1 The Riemann Integral as a Limit......Page 30
   2.1.2 The Fundamental Theorem of Calculus......Page 32
   2.1.3 The Cauchy Fundamental Theorem of Calculus......Page 35
   2.2.1 Removable Discontinuity......Page 37
   2.2.2 Jump Discontinuity......Page 38
 3 Bounded Variation......Page 42
  3.1 Partitions......Page 43
  3.2 Monotone......Page 44
   3.2.1 The Saltus Function......Page 49
   3.2.2 The Continuous Part of a Monotone Function......Page 51
  3.3 Bounded Variation......Page 57
  3.4 The Total Variation Function......Page 61
  3.5 Continuous Functions of Bounded Variation......Page 64
  4.1 Definition......Page 68
  4.2 Existence......Page 71
  4.3 Properties......Page 79
  4.4 Riemann Integrable?......Page 84
  4.5 More Properties......Page 86
  4.6 Fundamental Theorem......Page 90
  4.7 Substitution......Page 98
  4.8 Same Integral?......Page 101
  5.1 Limit Interchange......Page 106
  5.2 Riemann Integrable?......Page 112
  5.3 Content Zero......Page 113
III Riemann - Stieltjes Integration......Page 122
 6 The Riemann - Stieltjes Integral......Page 124
  6.1 Properties......Page 125
  6.2 Step Integrators......Page 128
  6.3 Monotone Integrators......Page 134
  6.4 Equivalence Theorem......Page 136
  6.5 Further Properties......Page 137
  6.6 Bounded Variation Integrators......Page 140
  7.1 Fundamental Theorem......Page 146
  7.2 Existence......Page 149
  7.3 Computations......Page 152
IV Abstract Measure Theory One......Page 160
 8 Measurability......Page 162
  8.1 Borel Sigma-Algebra......Page 165
  8.2 The Extended Borel Sigma-Algebra......Page 166
  8.3 Measurable Functions......Page 168
   8.3.1 Examples......Page 170
  8.4 Properties......Page 171
  8.5 Extended Real-Valued......Page 174
  8.6 Extended Properties......Page 176
  8.7 Continuous Compositions......Page 180
   8.7.2 The Approximation of Non-Negative Measurable Functions......Page 181
   8.7.3 Continuous Functions of Extended Real-Valued Measurable Functions......Page 183
 9 Abstract Integration......Page 186
  9.1 Measures......Page 187
  9.2 Properties......Page 188
  9.3 Sequences of Sets......Page 191
  9.4 Integration......Page 195
  9.5 Integration Properties......Page 199
  9.6 Equality a.e. Problems......Page 203
  9.7 Complete Measures......Page 204
   9.8.1 Monotone Convergence Theorems......Page 207
   9.8.2 Fatou\'s Lemma......Page 211
  9.9 The Absolute Continuity of a Measure......Page 213
  9.11 Extended Integrands......Page 215
  9.12 Levi\'s Theorem......Page 218
  9.13 Constructing Charges......Page 220
  9.14 Properties of Summable Functions......Page 222
  9.15 The Dominated Convergence Theorem......Page 224
  9.16 Alternative Abstract Integration Schemes......Page 227
   9.16.1 Properties of the Darboux Integral......Page 231
 10 The Lp Spaces......Page 236
  10.1 The General Lp Spaces......Page 240
  10.2 The World of Counting Measure......Page 251
  10.3 Essentially Bounded Functions......Page 253
  10.4 The Hilbert Space L2......Page 260
V Constructing Measures......Page 262
  11.1 Measures from Outer Measure......Page 264
  11.2 The Properties of the Outer Measure......Page 267
  11.3 Measures Induced by Outer Measures......Page 270
  11.4 Measures from Metric Outer Measures......Page 271
  11.5 Constructing Outer Measure......Page 277
  12.1 Outer Measure......Page 286
  12.2 Lebesgue Outer Measure is a Metric Outer Measure......Page 296
  12.3 Lebesgue Measure is Regular......Page 300
   12.4.1 Approximating Measurable Sets......Page 301
   12.4.2 Approximating Measurable Functions......Page 305
  12.5 The Summable Functions are Separable......Page 307
  12.6 The Existence of Non-Lebesgue Measurable Sets......Page 308
  12.7 Metric Spaces......Page 310
  13.1 Generalized......Page 314
  13.2 Representation......Page 317
  13.3 The Cantor Functions......Page 318
  13.4 Consequences......Page 320
 14 Lebesgue - Stieltjes Measure......Page 322
  14.1 Lebesgue - Stieltjes Outer Measure and Measure......Page 324
  14.2 Approximation Results......Page 330
  14.3 Properties......Page 331
VI Abstract Measure Theory Two......Page 336
 15 Convergence Modes......Page 338
  15.1 Extracting Subsequences......Page 341
  15.2 Egoroff\'s Theorem......Page 349
  15.3 Vitali\'s Theorem......Page 352
  15.4 Summary......Page 357
  16.1 Jordan Decomposition......Page 362
  16.2 Hahn Decomposition......Page 367
  16.3 Variation......Page 369
  16.4 Absolute Continuity......Page 372
  16.5 Radon-Nikodym......Page 375
  16.6 Lebesgue Decomposition......Page 383
 17 Connections to Riemann Integration......Page 388
  18.1 The Riemann Setting......Page 392
   18.1.1 Fubini on a Rectangle......Page 393
  18.2 The Lebesgue Setting......Page 400
  19.1 Absolutely Continuous Functions......Page 410
  19.2 LS and AC......Page 411
  19.3 Bounded Variation Derivatives......Page 414
  19.4 Measure Estimates......Page 417
  19.5 Extending the Fundamental Theorem of Calculus......Page 421
  19.6 Charges Induced by Absolutely Continuous Functions......Page 427
VII Summing It All Up......Page 430
 20 Summing It All Up......Page 432
VIII References......Page 436
IX Detailed Index......Page 440
X Appendix: Undergraduate Analysis Background Check......Page 456
 A Undergraduate Analysis Part One......Page 458
   A-1.1 Exam 1......Page 461
   A-1.2 Exam 2......Page 462
   A-1.3 Exam 3......Page 463
   A-1.4 Final......Page 464
 B Undergraduate Analysis Part Two......Page 466
   B-1.1 Exam 1......Page 470
   B-1.2 Exam 2......Page 471
   B-1.3 Exam 3......Page 472
   B-1.4 Final......Page 473
XI Appendix: Linear Analysis Background Check......Page 476
 C Linear Analysis......Page 478
   C-1.1 Exam 1......Page 480
   C-1.2 Exam 2......Page 481
   C-1.3 Exam 3......Page 482
   C-1.4 Final......Page 483
XII Appendix: Preliminary Examination Check......Page 488
   D-1.1 Exam 1......Page 490
   D-1.2 Exam 2......Page 492
   D-1.3 Exam 3......Page 494
   D-1.4 Exam 4......Page 495
   D-1.5 Exam 5......Page 496
   D-1.6 Exam 6......Page 497
   D-1.7 Exam 7......Page 498
   D-1.8 Exam 8......Page 500