Description
Book SynopsisThe book uses classical problems to motivate a historical development of the integration theories of Riemann, Lebesgue, Henstock-Kurzweil and McShane, showing how new theories of integration were developed to solve problems that earlier integration theories could not handle. It develops the basic properties of each integral in detail and provides comparisons of the different integrals. The chapters covering each integral are essentially independent and could be used separately in teaching a portion of an introductory real analysis course. There is a sufficient supply of exercises to make this book useful as a textbook.
Table of ContentsIntroduction: Areas; Exercises; Riemann Integral: Riemann's Definition; Basic Properties; Cauchy Criterion; Darboux's Definition; Fundamental Theorem of Calculus; Characterizations of Integrability; Improper Integrals; Exercises; Convergence Theorems and the Lebesgue Integral: Lebesgue's Descriptive Definition of the Integral; Measure; Lebesgue Measure in ℝn; Measurable Functions; Lebesgue Integral; Riemann and Lebesgue Integrals; Mikusinski's Characterization of the Lebesgue Integral; Fubini's Theorem; The Space of Lebesgue Integrable Functions; Exercises; Fundamental Theorem of Calculus and the Henstock - Kurzweil Integral: Denjoy and Perron Integrals; A General Fundamental Theorem of Calculus; Basic Properties; Unbounded Intervals; Henstock's Lemma; Absolute Integrability; Convergence Theorems; Henstock - Kurzweil and Lebesgue Integrals; Differentiating Indefinite Integrals; Characterizations of Indefinite Integrals; The Space of Henstock - Kurzweil Integrable Functions; Henstock - Kurzweil Integrals on ℝn; Exercises; Absolute Integrability and the McShane Integral: Defintions; Basic Properties; Absolute Integrability; Convergence Theorems; The McShane Integral as a Set Function; The Space of McShane Integrable Functions; McShane, Henstock - Kurzweil and Lebesgue Integrals; McShane Integrals on ℝn; Fubini and Tonelli Theorems; McShane, Henstock - Kurzweil and Lebesgue Integrals in ℝn; Exercises.
