444 Chapter 7 • The Riemann Integral
Unfortunately, the proof is somewhat complicated. You may even decide the
proof is indigestible, and skip it. But I recommend that you at least read the
statement of the theorem and contemplate its significance. Even without the
proof, simply knowing the statement of the theorem will enrich your under-
standing of Riemann integrability. At the point where you decide you have had
enough of the proof, go directly to the examples and applications that follow.
Theorem 7.9.1 (Lebesgue's Criterion for Riemann Integrability) A
bounded function f : [a, b] --+ JR is Riemann integrable on [a, b] if and only if th e
set of points of discontinuity off in [a, b] has measure zero.
Obviously, to understand the statement of Lebesgue's criterion one must
understand what is meant by "measure zero." This concept was defined in
Section 3.4. We repeat the definition here.
Definition 7.9.2 A set A of real numbers has measure zero if Ve > 0, A
can be covered by a countable collection of open intervals of total length less
than e. That is, A has measure zero iff Ve > 0, :J collection {In : n E N} of
= =
open intervals In = (an, bn) such that A ~ LJ In and 2:::: l(In) < e, where
n=l n=l
l(In) = length(In) = (bn - an)·
Countable sets were defined and discussed in Section 2.8. In Theorem 3.4. 20
we proved that every countable set has measure zero. Hence, "measure O" is, in
a sense, a generalization of "countable." In Section 3.4, we showed that some
uncountable sets also have measure 0. In fact, the Cantor set has measure 0,
even though it is uncountable.
Two additional concepts, treated earlier in (optional) sections of the book,
are also essential in proving Theorem 7.9.l. First, we shall need to use the
topological definition of "compact" set given in Section 3.3. We shall not repeat
that definition here, but advise you to review Definition 3.3.2 through Corollary
3.3.12. Secondly, given a bounded function f:V(f) --+JR, a nonempty set A~
V(f), and a point xo E V(f), we shall use the following concepts:
(a) the oscillation off on A: w1(A) =sup f(A) - inf f(A);
(b) the oscillation off at xo : w1(xo) = lim WJ (N 0 (x 0 ) n V(f)).
c:->O+The function w1:(0, +oo) --+ (0 , +oo) is called the saltus function of f,
and w1(x) is called the saltus of f at x. These concepts were defined and
discussed in Section 5.7. (See Definition 5.7.4 through Theorem 5.7.11.)
Now, we begin our proof of Lebesgue's criterion, in stages.