The Generalized Riemann Integral

✍ Scribed by Robert M. McLeod

• Publisher: Mathematical Associations of America
• Year: 1982
• Tongue: English
• Leaves: 292
• Series: Carus Mathematical Monographs 20
• Edition: Revised
• Category: Library

✦ Synopsis

The Generalized Riemann Integral is addressed to persons who already have an acquaintance with integrals they wish to extend and to the teachers of generations of students to come. The organization of the work will make it possible for the first group to extract the principal results without struggling through technical details which they may find formidable or extraneous to their purposes. The technical level starts low at the opening of each chapter. Thus readers may follow each chapter as far as they wish and then skip to the beginning of the next. To readers who do wish to see all the details of the arguments, they are given.

The generalized Riemann integral can be used to bring the full power of the integral within the reach of many who, up to now, get no glimpse of such results as monotone and dominated convergence theorems. As its name hints, the generalized Riemann integral is defined in terms of Riemann sums. The path from the definition to theorems exhibiting the full power of the integral is direct and short.

✦ Table of Contents

Cover

S Title

THE CARUS MATHEMATICAL MONOGRAPHS

List of Published Monographs

THE GENERALIZED RIEMANN INTEGRAL

Copyright

 © 1980 by The Mathematical Association of America

 Complete Set ISBN 0-88385-000-1

 Vol. 20 ISBN 0-88385-021-4

 Library of Congress Catalog Card Number 80-81043

PREFACE

LIST OF SYMBOLS

CONTENTS

INTRODUCTION

CHAPTER 1 DEFINITION OF THE GENERALIZED RIEMANN INTEGRAL

 1.1. Selecting Riemann sum

 1.2. Definition of the generalized Riemann integral

 1.3. Integration over unbounded intervals.

 1.4. The fundamental theorem of calculus

 1.5. The status of improper integral

 1.6. Multiple integrals

 1.7. Sum of a series viewed as an integral

 S1.8. The limit based on gauges

 S1.9. Proof of the fundamental theorem

 1.10. Exercises

CHAPTER 2 BASIC PROPERTIES OF THE INTEGRAL

 2.1. The integral as a function of the integrand

 2.2. The Cauchy criterio

 2.3. Integrability on subintervals

 2.4. The additivity of integrals

 2.5. Finite additivity of functions of intervals

 2.6. Continuity of integrals. Existence of primitives

 2.7. Change of variables in integrals on intervals in R

 S2.8. Limits of integrals over expanding intervals

 2.9. Exercises

CHAPTER 3 ABSOLUTE INTEGRABILITY AND CONVERGENCE THEOREMS

 3.1. Henstock's lemm

 3.2. Integrability of the absolute value of an integrable function

 3.3. Lattice operations on integrable functions

 3.4. Uniformly convergent sequences of functions

 3.5. The monotone convergence theorem

 3.6. The dominated convergence theorem

 S3.7. Proof of Henstock's lemma

 S3.8. Proof of the criterion for integrability of IfI.

 S3.9. Iterated limits

 S3.10. Proof of the monotone and dominated convergence theorems.

 3.11. Exercises.

CHAPTER 4 INTEGRATION ON SUBSETS OF INTERVALS

 4.1. Null functions and null sets

 4.2. Convergence almost everywhere

 4.3. Integration over sets which are not intervals

 4.4. Integration of continuous functions on closed, bounded sets

 4.5. Integrals on sequences of sets

 4.6. Length, area, volume, and measure

 4.7. Exercises

CHAPTER 5 MEASURABLE FUNCTIONS

 5.1. Measurable functions

 5.2. Measurability and absolute integrabili

 5.3. Operations on measurable functions

 5.4. Integrability of products

 S5.5. Approximation by step functions

 5.6. Exercises

CHAPTER 6 MULTIPLE AND ITERATED INTEGRALS

 6.1. Fubini's theorem

 6.2. Determining integrability from iterated integrals

 S6.3. Compound divisions. Compatibility theorem

 S6.4. Proof of _Fubini's theorem

 S6.5. Double series

 6.6. Exercises

CHAPTER 7 INTEGRALS OF STIELTJES TYPE

 7.1. Three versions of the Riemann-Stieltjes integral

 7.2. Basic properties of Riemann-Stieltjes integrals

 7.3. Limits, continuity, and differentiability of integrals

 7.4. Values of certain integrals

 7.5. Existence theorems for Riemann-Stieltjes integrals

 7.6. Integration by parts

 7.7. Integration of absolute values. Lattice operations

 7.8. Monotone and dominated convergence

 7.9. Change of variables

 7.10. Mean value theorems for integrals

 S7.11. Sequences of integrators

 S7.12. Line integrals.

 S7.13. Functions of bounded variation and regulated functions

 S7.14. Proof of the absolute integrability theorem

 7.15. Exercises

CHAPTER 8 COMPARISON OF INTEGRALS

 S8.1. Characterization of measurable sets

 S8.2. Lebesgue measure and integral

 S8.3. Characterization of absolute integrability using Riemann sums

 8.4 Suggestions for further study

 REFERENCES

APPENDIX SOLUTIONS OF IN-TEXT EXERCISES

 Chapter 1

 Chapter 2

 Chapter 3

 Chapter 4

 Chapter 5

 Chapter 6

 Chapter 7

INDEX

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