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