Catálogo de la biblioteca Antonio Monteiro - INMABB

Suppose that f : [a,b] -> R is differentiable at each point of [a,b]. Is f' integrable on [a,b]? The answer to this question depends on the integral that is used. For example, the answer is no for the Riemann and Lebesgue integrals. In this century, three integrations processes have been developed that provide an affirmative answer to this question. The principal investigators of these integrals were Denjoy, Perron, and Henstock. Each of these integrals generalizes a different property of the Lebesgue integral, but it turns out that all three integrals are equivalent. In this book, the properties of the Lebesgue, Denjoy, Perron, and Henstock integrals are developed fully from their definitions. The equivalence of the last three integrals is then established. Discussions of the integration by parts formula and convergence theorems are included. In the last part of the book, we consider approximate derivatives and attempts to develop an integration process for which every approximate derivative is integrable.

Incluye referencias bibliográficas (p. 389-390) e índices.

Capítulos: 1. Lebesgue measure -- 2. Measurable functions -- 3. The Lebesgue integral -- 4. Bounded variation and absolute continuity -- 5. Darboux and Baire class one functions -- 6. Functions of generalized bounded variation -- 7. The Denjoy integral -- 8. The Perron integral -- 9. The Henstock integral -- 10. The McShane integral -- 11. Equivalence of integrals -- 12. Integration by parts -- 13. Convergence theorems -- 14. Approximate derivatives -- 15. The Khintchine integral -- 16. The approximately continuous Henstock integral -- 17. The approximately continuous Perron integral.

MR, 95m:26010