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.
Author(s): Robert M. McLeod
Series: Carus Mathematical Monographs 20
Edition: Revised
Publisher: Mathematical Association of America
Year: 1982
Language: English
Pages: xiv+275
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
