The Calculus Integral

Preface......Page 4
Note to the instructor......Page 6
Table of Contents......Page 9
I Elementary Theory of the Integral......Page 26
What you should know first......Page 28
What is an interval?......Page 29
Sequences......Page 32
Series......Page 35
Partitions......Page 36
Cousin's partitioning argument......Page 37
What is a function?......Page 38
Continuous functions......Page 39
Uniformly continuous and continuous functions......Page 40
Oscillation of a function......Page 44
Endpoint limits......Page 45
Boundedness properties......Page 49
Existence of maximum and minimum......Page 51
The Darboux property of continuous functions......Page 52
Derivatives......Page 54
Mean-value theorem......Page 57
Rolle's theorem......Page 58
Mean-Value theorem......Page 59
The Darboux property of the derivative......Page 64
Vanishing derivatives with exceptional sets......Page 65
Lipschitz functions......Page 67
The Indefinite Integral......Page 70
An indefinite integral on an interval......Page 71
Role of the finite exceptional set......Page 72
The notation f(x)dx......Page 73
Existence of indefinite integrals......Page 75
Upper functions......Page 76
The main existence theorem......Page 77
Basic properties of indefinite integrals......Page 78
Integration by parts......Page 79
Change of variable......Page 80
What is the derivative of the indefinite integral?......Page 81
Partial fractions......Page 82
Tables of integrals......Page 85
The Definite Integral......Page 86
Definition of the calculus integral......Page 87
Alternative definition of the integral......Page 88
Infinite integrals......Page 89
Simple properties of integrals......Page 90
Integrability of bounded functions......Page 93
Integrability for the unbounded case......Page 94
Products of integrable functions......Page 96
The dummy variable: what is the ``x'' in abf(x)dx?......Page 97
Definite vs. indefinite integrals......Page 98
The calculus student's notation......Page 99
Mean-value theorems for integrals......Page 100
Riemann sums......Page 102
Exact computation by Riemann sums......Page 103
Uniform Approximation by Riemann sums......Page 106
Theorem of G. A. Bliss......Page 108
Pointwise approximation by Riemann sums......Page 110
Inequalities......Page 112
Subintervals......Page 113
Change of variable......Page 114
What is the derivative of the definite integral?......Page 116
Absolute integrability......Page 117
Functions of bounded variation......Page 118
Sequences and series of integrals......Page 122
The counterexamples......Page 123
Uniform convergence......Page 129
Uniform convergence and integrals......Page 135
A defect of the calculus integral......Page 137
Uniform limits of continuous derivatives......Page 138
Uniform limits of discontinuous derivatives......Page 140
Summing inside the integral......Page 141
Monotone convergence theorem......Page 143
Integration of power series......Page 144
Applications of the integral......Page 153
Area and the method of exhaustion......Page 155
Volume......Page 158
Length of a curve......Page 161
Numerical methods......Page 163
Maple methods......Page 169
Maple and infinite integrals......Page 171
More Exercises......Page 172
Beyond the calculus integral......Page 174
Countable sets......Page 175
Cantor's theorem......Page 176
Calculus integral [countable set version]......Page 177
Sets of measure zero......Page 179
The Cantor dust......Page 181
Construction of Cantor's function......Page 185
Functions with zero variation......Page 188
Zero variation lemma......Page 190
Continuity and zero variation......Page 191
Absolute continuity......Page 192
Absolute continuity in Vitali's sense......Page 194
The integral......Page 195
Infinite integrals......Page 198
Lipschitz functions and bounded integrable functions......Page 199
Approximation by Riemann sums......Page 200
Inequalities......Page 201
Subintervals......Page 202
Change of variable......Page 203
Monotone convergence theorem......Page 204
Summation of series theorem......Page 205
Null functions......Page 206
The Henstock-Kurweil integral......Page 207
The Lebesgue integral......Page 208
The Riemann integral......Page 210
II Theory of the Integral on the Real Line......Page 212
Covering Theorems......Page 214
Partitions and subpartitions......Page 215
Prunings......Page 216
Uniformly full covers......Page 217
Cousin covering lemma......Page 221
Decomposition of full covers......Page 222
Riemann sums......Page 223
Lebesgue measure of open sets......Page 227
Sequences of measure zero sets......Page 229
Almost everywhere language......Page 233
Full null sets......Page 234
Fine null sets......Page 236
The Mini-Vitali Covering Theorem......Page 237
Covering lemmas for families of compact intervals......Page 238
Proof of the Mini-Vitali covering theorem......Page 239
Functions having zero variation......Page 242
Zero variation and zero derivatives......Page 244
Generalization of the zero derivative/variation......Page 245
Absolutely continuous functions......Page 247
Absolute continuity and derivatives......Page 248
Lebesgue differentiation theorem......Page 250
Upper and lower derivates......Page 251
Geometrical lemmas......Page 252
Proof of the Lebesgue differentiation theorem......Page 253
The Integral......Page 258
The integral and integrable functions......Page 259
Infinite integrals......Page 260
Approximation by Riemann sums......Page 261
Definition of Henstock and Kurzweil......Page 264
Upper and lower integrals......Page 265
The integral and integrable functions......Page 267
First Cauchy criterion......Page 269
Second Cauchy criterion......Page 270
Proof of equivalence......Page 272
Integration and order......Page 277
Change of variable......Page 278
Integration by parts......Page 279
Derivative of the integral......Page 280
Null functions......Page 281
Monotone convergence theorem......Page 282
Summing inside the integral......Page 283
Two convergence lemmas......Page 284
Equi-integrability......Page 289
Lebesgue's Integral......Page 290
Lebesgue measure......Page 291
Basic property of Lebesgue measure......Page 292
Vitali covering theorem......Page 293
Classical version of Vitali's theorem......Page 294
Proof that = * = * .......Page 296
Density theorem......Page 297
Additivity......Page 299
Properties of measurable sets......Page 302
Increasing sequences of sets......Page 304
Existence of nonmeasurable sets......Page 305
Continuous functions are measurable......Page 307
Derivatives and integrable functions are measurable......Page 308
Simple functions......Page 309
Series of simple functions......Page 310
Limits of measurable functions......Page 311
Characteristic functions of measurable sets......Page 312
Characterizations of measurable sets......Page 313
Integral of nonnegative measurable functions......Page 315
Fatou's Lemma......Page 316
Derivatives of functions of bounded variation......Page 319
Characterization of the Lebesgue integral......Page 320
McShane's Criterion......Page 321
Nonabsolutely integrable functions......Page 325
The Lebesgue integral as a set function......Page 326
Characterizations of the indefinite integral......Page 330
Integral of absolutely integrable functions......Page 332
Proofs......Page 333
Denjoy's program......Page 336
The Riemann integral......Page 337
Stieltjes integrals......Page 340
Definition of the Stieltjes integral......Page 342
Henstock's zero variation criterion......Page 345
Regulated functions......Page 346
Variation expressed as an integral......Page 349
Jordan decomposition......Page 351
Jordan decomposition theorem: differentiation......Page 352
Reducing a Stieltjes integral to an ordinary integral......Page 354
Properties of the indefinite integral......Page 357
Existence of the integral from derivative statements......Page 361
Existence of the Stieltjes integral for continuous functions......Page 362
Integration by parts......Page 363
Lebesgue-Stieltjes measure......Page 365
Mutually singular functions......Page 368
Singular functions......Page 370
Length of curves......Page 371
Formula for the length of curves......Page 372
Nonabsolutely Integrable Functions......Page 376
Variational Measures......Page 377
Full and fine variational measures......Page 378
Finite variation and -finite variation......Page 379
Kolmogorov equivalence......Page 380
Variation of continuous, increasing functions......Page 381
Variation and image measure......Page 382
Variational classifications of real functions......Page 383
Ordinary derivates and variation......Page 386
Dini derivatives and variation......Page 387
Lipschitz numbers......Page 389
Six growth lemmas......Page 391
Continuous functions with -finite variation......Page 396
Variation on compact sets......Page 397
Vitali property and differentiability......Page 399
Monotonic functions......Page 401
Functions of -finite variation......Page 402
Characterization of the Vitali property......Page 403
Characterization of -absolute continuity......Page 404
Mapping properties......Page 405
Banach-Zarecki Theorem......Page 407
Local Lebesgue integrability conditions......Page 409
Continuity of upper and lower integrals......Page 413
A characterization of the integral......Page 414
Motivation......Page 418
Quasi-Cousin covering lemma......Page 420
Estimates of integrals from derivates......Page 421
Estimates of integrals from Dini derivatives......Page 423
Some background......Page 426
Intervals and covering relations......Page 427
Measure and integral......Page 429
The fundamental lemma......Page 430
Measurable sets and measurable functions......Page 433
Measurable functions......Page 434
Notation......Page 436
General measure theory......Page 437
Iterated integrals......Page 438
Formulation of the iterated integral property......Page 440
Fubini's theorem......Page 443
Expression as a Stieltjes integral......Page 444
absolute continuity......Page 446
absolute convergence......Page 447
almost everywhere......Page 448
Baire category theorem......Page 449
bounded set......Page 451
bounded monotone sequence argument......Page 452
Cauchy sequences......Page 453
compactness argument......Page 454
component of an open set......Page 455
continuous function......Page 456
converse......Page 457
Cousin's partitioning argument......Page 458
Cousin's covering argument......Page 459
De Morgan's Laws......Page 460
Devil's staircase......Page 461
graph of a function......Page 462
Henstock-Kurzweil integral......Page 463
indirect proof......Page 464
integral test for series......Page 465
least upper bound argument......Page 466
inverse of a function......Page 467
Lebesgue integral......Page 468
limit of a function......Page 469
lower bound of a set......Page 471
meager......Page 472
measure zero......Page 473
mostly everywhere......Page 474
negations of quantified statements......Page 475
nowhere dense......Page 476
ordered pairs......Page 477
partition......Page 478
pointwise continuous function......Page 479
quantifiers......Page 480
real numbers......Page 481
Riemann sum......Page 482
Riemann integral......Page 483
series......Page 484
set notation......Page 485
sups and infs......Page 486
uniformly continuous function......Page 487
variation of a function......Page 488
Answers to exercises......Page 490