Theories of integration. The integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane

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Series in Real Analysis - Vol. 13

Published Books in this Series

THEORIES OF INTEGRATION: The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane, Second Edition

Copyright© 2012 by World Scientific Publishing
     ISBN-13 978-981-4368-99-5
     ISBN-10 981-4368-99-7

Dedication

Preface to the First Edition

Preface to the Second Edition

Contents


Chapter 1  Introduction
     1.1 Areas
     1. 2 Exercises

Chapter 2  Riemann integral
     2.1 Riemann\'s definition
     2.2 Basic properties
     2.3 Cauchy criterion
     2.4 Darboux\'s definition
          2.4.1 Necessary and sufficient conditions for Darboux integrability
          2.4.2 Equivalence of the Riemann and Darboux definitions
          2.4.3 Lattice properties
          2.4.4 Integrable functions
          2.4.5 Additivity of the integral over intervals
     2.5 Fundamental Theorem of Calculus
          2.5.1 Integration by parts and substitution
     2.6 Characterizations of integrability
          2.6.1 Lebesgue measure zero
     2. 7 Improper integrals
     2.8 Exercises

Chapter 3  Convergence theorems and the Lebesgue integral
     3.1 Lebesgue\'s descriptive definition of the integral
     3.2 Measure
          3.2.1 Outer measure
          3.2.2 Lebesgue measure
          3.2.3 The Cantor set
     3.3 Lebesgue measure in R^n
     3.4 Measurable functions
     3.5 Lebesgue integral
          3.5.1 Integrals depending on a parameter
     3.6 Riemann and Lebesgue integrals
     3.7 Mikusinski\'s characterization of the Lebesgue integral
     3.8 Fubini\'s Theorem
          3.8.1 Convolution
     3.9 The space of Lebesgue integrable functions
     3.10 Exercises

Chapter 4  Fundamental Theorem of Calculus and the Henstock-Kurzweil integral
     4.1 Denjoy and Perron integrals
     4.2 A General Fundamental Theorem of Calculus
     4.3 Basic properties
          4.3.1 Cauchy criterion
          4.3.2 The integral as a set function
     4.4 Unbounded intervals
     4.5 Henstock\'s Lemma
     4.6 Absolute integrability
          4.6.1  Bounded variation
          4.6.2 Absolute integrability and indefinite integrals
          4.6.3 Lattice properties
     4.7 Convergence theorems
     4.8 Henstock-Kurzweil and Lebesgue integrals
     4.9 Differentiating indefinite integrals
     4.10 Characterizations of indefinite integrals
          4.10.1 Derivatives of monotone functions
          4.10.2 Indefinite Lebesgue integrals
          4.10.3 Indefinite Riemann integrals
     4.11 The space of Henstock-Kurzweil integrable functions
     4.12 Henstock-Kurzweil integrals on R^n
     4.13 Exercises

Chapter 5  Absolute integrability and the McShane integral
     5.1 Definitions
     5.2 Basic properties
     5.3 Absolute integrability
          5.3.1 Fundamental Theorem of Calculus
     5.4 Convergence theorems
     5.5 The McShane integral as a set function
     5.6 The space of McShane integrable functions
     5.7 McShane, Henstock-Kurzweil and Lebesgue integrals
     5.8 McShane integrals on R^n
     5.9 Fubini and Tonelli Theorems
     5.10 McShane, Henstock-Kurzweil and Lebesgue integrals in R^n
     5 .11 Exercises


Bibliography

Index

Back Cover


NOTICE: White pages will not be counted at the End of the Front & Back Matter.