The Generalized Riemann Integral

ROBERT M. McLEOD
Volume: 20
Edition: 1
Pages: 290
https://www.jstor.org/stable/10.4169/j.ctt5hhb4z

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1. Front Matter
(pp. i-vi)
2. PREFACE
(pp. vii-viii)
Robert M. McLeod
3. LIST OF SYMBOLS
(pp. ix-x)
(pp. xi-xiv)
5. INTRODUCTION
(pp. 1-4)

The discussion of the definite integral in elementary calculus commonly starts from an area problem. Given a region under a function graph, how can its area be calculated? The sum of the areas of slender rectangles is a fairly natural approximation. A limit of such sums yields the exact area. When this process is stripped to its essentials the Riemann integral of a function over a given interval stands revealed.

Other geometric and physical quantities, such as volume and work, fit easily into the framework supplied by the concept of the Riemann integral. Moreover the link between the integral and...

6. CHAPTER 1 DEFINITION OF THE GENERALIZED RIEMANN INTEGRAL
(pp. 5-46)

The first objective in Chapter 1 is to make plain what change is being made in the Riemann definition and to indicate why it should be beneficial. Then the definition of the generalized Riemann integral is formulated. It is given first for bounded intervals of real numbers and then, in the same language, for unbounded intervals. On the basis of these formulations the fundamental theorem of calculus linking integrals and derivatives is given an appealing form. Multiple integrals are defined, and again it is possible to use the same language as that first adopted for integration on a bounded real...

7. CHAPTER 2 BASIC PROPERTIES OF THE INTEGRAL
(pp. 47-70)

The topics treated in this chapter are nearly all familiar from discussions of the Riemann integral. Where both propositions and proofs are familiar the treatment is concise with proofs omitted. In some instances the conclusions are standard but the hypotheses are less restrictive than the usual ones.

Section 2.1 treats four topics: linearity of the integral as a function of the integrand; component-by-component integration of vector functions; inequalities; and integration by parts. A few proofs are suggested as exercises.

In Section 2.2 the Cauchy criterion for sequences is used as a model for the Cauchy criterion for existence of integrals....

8. CHAPTER 3 ABSOLUTE INTEGRABILITY AND CONVERGENCE THEOREMS
(pp. 71-102)

Some of the most useful tools of integration theory will be developed in this chapter. They center on two important operations on functions. One is the formation of the absolute value. The other is the limit of a sequence of functions. The behavior of the generalized Riemann integral with respect to these operations exhibits the strength of this integral definition most vividly.

Whenfis integrable, it is important to be able to tell whether |f| is also integrable. A simple criterion is stated in Section 3.2. This criterion has implications for the calculation of the length of curves. These...

9. CHAPTER 4 INTEGRATION ON SUBSETS OF INTERVALS
(pp. 103-130)

Many problems require the notion of integration over sets in${R^p}$which are not intervals. The calculation of area, volume, moments, and work brings this idea to the attention of the calculus student. As was pointed out in Section 1.6, one way to define$\int_E f$whenfis defined initially onEis to extendfby setting it equal to zero outsideEand integrate the extension over some closed interval containingE, say${\overline R ^p}$. The obvious advantage of this method is that the facts for integration over intervals are available for the development of the properties of the...

10. CHAPTER 5 MEASURABLE FUNCTIONS
(pp. 131-148)

From the first example in Chapter 1 it has been clear that a function need not be highly regular in order to be integrable. Yet it cannot be wildly irregular. So far we have relied on two kinds of hypotheses to provide sufficient regularity to insure integrability. One type assumes integrability on certain subsets, say all bounded intervals contained in a given unbounded interval. In the other it is the relation of the function to one or more other functions which supplies the appropriate properties. There are two obvious instances of this. One is the relation of |f| tof...

11. CHAPTER 6 MULTIPLE AND ITERATED INTEGRALS
(pp. 149-176)

Two complementary aspects of integration over multidimensional intervals are treated in this chapter. First, in Section 6.1, it is assumed that$\int_1 f$exists. Conclusions are asserted concerning the existence of iterated integrals and their equality to$\int_1 f$. This is Fubini’s theorem. The proof is postponed to Section S6.4. Section 6.2 attacks the complementary question of what can be said about existence of$\int_1 f$from the known properties of an iterated integral. Some necessary conditions are developed which permit the conclusion that$\int_1 f$does not exist. Other conditions are found which allow the conclusion that$\int_1 f$exists to flow from properties...

12. CHAPTER 7 INTEGRALS OF STIELTJES TYPE
(pp. 177-230)

The Stieltjes integral is a generalization of the Riemann integral on intervals of real numbers. It employs a function a in place of the identity function in forming the measure of subintervals of [a, b]. A Riemann sum$f\Delta \alpha \left( D \right)$is made up of terms$f\left( z \right)\left( {\alpha \left( v \right) - \alpha \left( u \right)} \right)$. The resulting integral is denoted$\int_a^b {fd\alpha }$or$\int_a^b {f\left( x \right)d\alpha \left( x \right)}$. The function$\alpha$is called the integrator andfis the integrand.

The level of generality of$\int_a^b {fd\alpha }$depends on the limit process which is applied to the Riemann sum. There are three which should claim our attention. The least general of them is the limit used...

13. CHAPTER 8 COMPARISON OF INTEGRALS
(pp. 231-245)

In preceding chapters the generalized Riemann integral has been presented on its own terms with explicit reference only to the Riemann integral. Nevertheless, the theory of the Lebesgue integral has been in the background at all times. At many points it has given guidance as to what could be proved. At some points the methods of proof were drawn from the Lebesgue theory, too. The existing literature on integration centers on the Lebesgue theory. Thus it is very helpful to know how the generalized Riemann integral ties in with the Lebesgue integral in order to be able to use other...

14. REFERENCES
(pp. 245-246)
15. APPENDIX SOLUTIONS OF IN-TEXT EXERCISES
(pp. 247-268)
16. INDEX
(pp. 269-275)