1549901369-Elements_of_Real_Analysis__Denlinger_

448 Chapter 7 • The Riemann Integral

Let c > 0. To prove that f is integrable on [a, b], it suffices to prove that
there is a partition P of [a, b] such that S(f, P) - 3-_(f, P) < c (Riemann's
condition, 7.2.14). Toward that end, let

8= €
2(b - a)
Since S has measure 0, and S 0 (f) ~ S, S 0 (f) also has measure 0. Hence,
there exists a countable collection of open intervals {Ii, I 2, · · · In, · · · } such that

00

So(!)~ LJ Ji and (49)

i=l

where l(Ii) denotes the length of h
Now, from Remark 7.9.3 (e), S 0 (f) is compact, so it can be covered by
finitely many of the open intervals Ii , say

N N
By de Morgan's law, [a, b]-LJ Ji = n ([a, b] -Ii), which is the intersection
i=l i=l
of a collection of closed intervals, say

N M

[a, b] - u Ji= n Ji, (50)

i=l i=l

where each Ji is a closed subinterval of [a, b].

Note that x E Ji ::::} x tJ_ S 0 (f) ::::} w1(x) < 8. Thus, by Lemma 7.9.6, each
Ji can be further subdivided into closed intervals Jij of length less than 8i (for
some 8i > 0), such that

K;

WJ (Jij) < 8 and Ji= LJ Jij·

j=l

(51)

Let P = {all endpoints of all the intervals Ji and Jij described above} U
{a,b}. Say P = {x o,x 1 ,·· · xn}· Then

n

S(f, P) - 3-_(f, P) = L(Mi - mi)6i

i=l