Calculus: Analytic Geometry and Calculus, with Vectors

216 Integrals

finite collection, that is, it may contain only 1 or 2 or 3 or 416 or 31,690

or some other positive integer number of intervals. The collection of

intervals may be a "countably infinite collection," that is, it may contain

a first, a second, a third, etcetera, so that to each positive integer k

there corresponds an interval Ik In each of these two cases, the collec-

tion of intervals is said to be a "countable collection." Only a most

rudimentary understanding of these matters enables us to reach the con-

clusion that if D contains only 0 or 1 or 2 or 3 or 416 or any other finite

number of points, then D must have Lebesgue measure 0. In any case,

we should have at least a hazy understanding of the fact that Lebesgue

(1875-1941) was a great French mathematician and that Theorem 4.26

implies the much simpler following theorem which we are required to

know in this course.

Theorem 4.27 If f is bounded over the interval a 5 x < b and if f is

continuous over the interval (or is discontinuous but has only a finite set of

discontinuities in the interval), then the Riemann integral in

(4.271) f a0 f (t) dt = limI f (t-) At,,

exists when a < x < b.

As we near the end of the text of our introductory section on Riemann

sums and integrals, we pause to think about our present state and future

development. We have a new symbol, namely, f a--f(t) dt. If a < b,

if a < x < b, and if f is defined over the interval a < x < b, then (depend-

ing upon a, x, and f) the symbol may be meaningless or it may be a

number. Answers to questions depend upon partitions and Riemann

sums. Partitions are so simple that our little sister can understand them

completely and be puzzled only by our great interest in them. Riemann

sums Ef(tk) Itk are less simple, but we can construct them in great pro-

fusion. Matters grow substantially more complex when we ask whether

there is a number fax f(t) dt such that to each positive number e there

corresponds a positive number S such that

(4.28)

n

k 1

f (tk)L1tk-f ax f (t) dt <E

whenever P is a partition of the interval a <_ t 5 x for which JPt < S.

We should all recognize this and admit that full comprehensions of

machinery and its applications is not quickly attained. In fact a sub-

stantial part of this textbook is devoted to promotion of understanding

of Riemann sums and their applications. We shall have plenty of oppor-

tunities to learn.