A first course in integration

Cover......Page 1 A FIRST COURSE IN INTEGRATION......Page 4 LCCN 20550-0116......Page 5 Preface......Page 6 Contents......Page 12 1 Step Functions and Null Sets......Page 18 1.0 THE REAL LINE......Page 20 1.1 STEP FUNCTIONS AND THEIR INTEGRALS......Page 28 1.2 BASIC PROPERTIES OF THE INTEGRAL......Page 33 1.3 NULL SETS......Page 43 1.4 EXAMPLES......Page 48 1.5 CONVERGENCE ALMOST EVERYWHERE......Page 52 A. Step Functions and Their Integral......Page 56 B. Null Sets......Page 57 C. Convergence......Page 58 D. Piecewise Linear Functions......Page 60 SUMMARY......Page 63 2 The Lebesgue Integral......Page 66 2.0 THE RIEMANN INTEGRAL FOR CONTINUOUS FUNCTIONS......Page 68 2.1 LEBESGUE INTEGRABLE FUNCTIONS......Page 71 2.2 BASIC PROPERTIES OF THE LEBESGUE INTEGRAL......Page 79 2.3 CONVERGENCE THEOREMS......Page 84 2.4 APPLICATIONS......Page 90 2.5 THE DANIELL INTEGRAL......Page 95 2.6 RIEMANN INTEGRABLE FUNCTIONS......Page 102 A. Basic Properties of Lebesgue Integrable Functions......Page 110 B. Problems on Limits (Abstract)......Page 111 C. Problems on Limits (Concrete)......Page 113 D. Product of Functions......Page 114 E. Improper Riemann Integration......Page 116 G. The Daniell Integral......Page 117 SUMMARY......Page 118 LEBESGUE'S DOMINATED CONVERGENCE THEOREM......Page 119 C0MPAJUs0N THEOREM......Page 120 3 Measurability......Page 122 3.1 MEASURABLE FUNCTIONS......Page 124 3.2 GENERALIZED STEP FUNCTIONS......Page 130 3.3 MEASURABLE SETS......Page 136 EGOROFF'S THEOREM......Page 141 STEINHAUS THEOREM......Page 142 3.4 AXIOMATIC MEASURE THEORY......Page 143 3.5 OUTER MEASURE......Page 147 A. Measurable Functions and Measurable Sets......Page 152 C. Baire Classes of Functions of k Variables......Page 154 D. Composition of Measurable Functions with Functions in Gn......Page 156 E. Lusin's Theorem......Page 157 G. The Cauchy Functional Equation......Page 158 H. Vitali's Covering Theorem......Page 159 SUMMARY......Page 161 4 Integration of Functions of Several Variables......Page 164 4.1 DEFINITIONS AND FUBINI'S THEOREM......Page 166 4.2 THE LEBESGUE INTEGRAL ON R^n......Page 174 4.3 LINEAR CHANGE OF VARIABLES......Page 187 4.4 NONLINEAR CHANGE OF VARIABLES......Page 194 A. Product Integration......Page 204 B. The Lebesgue Integral for R^n......Page 205 C. Linear Change of Variables......Page 206 D. Nonlinear Change of Variables......Page 207 E. Integration on the Unimodular Group......Page 209 SUMMARY......Page 211 5 L^2 and L^p Spaces......Page 214 5.1 THE SPACE L^2......Page 216 5.2 METRIC PROPERTIES OF L^2......Page 221 5.3 LINEAR FUNCTIONALS ON L^2......Page 227 5.4 L^p SPACES......Page 232 5.5 INTEGRATION OF COMPLEXVALUED FUNCTIONS......Page 240 5.6 THE FOURIER TRANSFORMATION......Page 245 A. Strong and Weak Convergence......Page 255 B. Orthonormal Systems......Page 256 C. The Space l^2......Page 258 D. Continuous Quadratic Functionals on L^2......Page 259 E. Continuous Linear Functionals on L^p......Page 260 F. Complex L^2 and Fourier Transforms......Page 264 SUMMARY......Page 266 6 The Differentiation of Functions of Locally Bounded Variation......Page 270 6.1 FUNCTIONS OF LOCALLY BOUNDED VARIATION......Page 272 6.2 DECOMPOSITION OF FUNCTIONS OF LOCALLY BOUNDED VARIATION......Page 278 6.3 PREPARATIONS FOR THE DIFFERENTIATION THEOREM......Page 283 6.4 DIFFERENTIATION OF MONOTONENON DECREASING FUNCTIONS......Page 288 6.5 DIFFERENTIATION OF INFINITE SERIES......Page 296 A. Functions of Locally Bounded Variation......Page 300 B. Convergence Theorems......Page 301 C. Functions of Bounded Variation......Page 303 D. Differentiation......Page 304 E. The Density Theorem and the Theorem on Derivates......Page 306 6.7 INTEGRATION AS INVERSE OF DIFFERENTIATION......Page 308 SUMMARY......Page 315 7 Absolutely Continuous Functions......Page 318 7.1 INTEGRATION OF THE DERIVATIVE OF A FUNCTION IN V......Page 320 7.2 THE PRIMITIVE OF A LOCALLY INTEGRABLE FUNCTION......Page 327 7.3 THE FUNDAMENTAL THEOREM FOR ABSOLUTELY CONTINUOUS FUNCTIONS......Page 332 7.4 ANOTHER CHARACTERIZATION OF ABSOLUTELY CONTINUOUS FUNCTIONS......Page 339 7.5 APPLICATIONS......Page 343 A. The Indefinite Integral......Page 349 B. Examples of Absolutely Continuous Functions......Page 351 C. Convergence of Absolutely Continuous Functions......Page 352 D. Mapping Properties of Absolutely Continuous Functions......Page 353 E. Composition of Absolutely Continuous Functions......Page 354 F. Discussion of Theorem 7.5.11......Page 355 8 Stieltj es Integrals......Page 360 8.1 DEFINITION OF THE STIELTJES INTEGRAL......Page 362 8.2 PROPERTIES OF STIELTJES INTEGRALS......Page 369 8.3 CONTINUOUS LINEAR FUNCTIONALS ON C......Page 376 8.4 OPERATIONS ON FUNCTIONALS......Page 382 8.5 THE RIESZ REPRESENTATION THEOREM......Page 387 B. Stieltjes Integrals for Piecewise Linear Functions......Page 394 C. Baire Functions......Page 396 D. Limit Operations......Page 397 E. Operations on Continuous Linear Functionals on C......Page 398 F. A Theory of Integration for Signed Stieltjes Integrals......Page 399 G. Abstract Signed Integrals......Page 401 SUMMARY......Page 402 RIEsz REPRESENTATION THEOREM......Page 405 9 The Radon-Nikodym Theorem......Page 406 9.1 ABSOLUTE CONTINUITY......Page 408 9.2 THE RADON-NIKODYM THEOREM......Page 413 9.3 THE ABSTRACT RADON-NIKODYM THEOREM......Page 421 9.4 APPLICATIONS......Page 427 A. Absolute Continuity......Page 432 B. Relations between Radon-Nikodym Derivatives......Page 433 D. Absolute Continuity for Signed Measures......Page 435 SUMMARY......Page 436 10 Applications to the Theory of Fourier Series......Page 438 10.1 TRIGONOMETRIC SERIES AND FOURIER SERIES......Page 440 10.2 CONVERGENCE TESTS......Page 447 10.3 SUMMATION BY ARITHMETIC MEANS......Page 455 10.4 FOURIER SERIES FOR FUNCTIONSIN L2(O, 2 pi)......Page 461 A. Examples of Fourier Series......Page 468 B. Trigonometric Series with Monotone-Decreasing Coefficients......Page 469 D. Summation by the Method of Arithmetic Means......Page 470 E. Fourier Series for Functions in L^2......Page 471 SUMMARY......Page 474 Bibliographical Comments and Remarks......Page 478 COMMENTS ON THE REFERENCES......Page 480 REFERENCES......Page 486 REMARKS ON THE LITERATURE......Page 489 LITERATURE......Page 490 Indexes......Page 492 THEOREMS REFERRED TO BY NAME......Page 494 SYMBOLS......Page 496 SUBJECT INDEX......Page 500