The Generalized Riemann Integral

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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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