The Riemann Integral

40 1. The Riemann Integral
1.9.2. Pointwise convergence. On its own, the pointwise convergence of functions
is never sufficient to imply convergence of their integrals.
Example 1.65. For n ∈ N, define f n : [0,1] → R by
{
n if 0 < x < 1/n,
f n (x) =
0 if x = 0 or 1/n ≤ x ≤ 1.
Then f n → 0 pointwise on [0,1] but
∫ 1
0
f n = 1
for every n ∈ N. By slightly modifying these functions to
{
n 2 if 0 < x < 1/n,
f n (x) =
0 if x = 0 or 1/n ≤ x ≤ 1,
we get a sequence that converges pointwise to 0 but whose integrals diverge to ∞.
The fact that the f n are discontinuous is not important; we could replace the step
functions by continuous “tent” functions or smooth “bump” functions.
The behavior of the integral under pointwise convergence in the previous example
is unavoidable. A much worse feature of the Riemann integral is that the
pointwise limit of integrable functions needn’t be integrable at all, even if it is
bounded.
Example 1.66. Let {q k : k ∈ N} be an enumeration of the rationals in [0,1] and
define f n : [0,1] → R by
{
1 if x = q k for k ≤ n,
f n (x) =
0 otherwise.
The each f n is Riemann integrable since it differs from the zero function at finitely
manypoints. However, f n → f pointwise on [0,1] to the Dirichlet function f, which
is not Riemann integrable.
This is another place where the Lebesgue integral has better properties than
the Riemann integral. The pointwise (or pointwise almost everywhere) limit of
Lebesgue integrable functions is Lebesgue integrable. As Example 1.65 shows, we
still need conditions to ensure the convergence of the integrals, but there are quite
simple and general conditions for the Lebesgue integral (such as the monotone
convergence and dominated convergence theorems).
1.10. Improper Riemann integrals
The Riemann integral is only defined for a bounded function on a compact interval
(or a finite union of such intervals). Nevertheless, we frequently want to integrate
an unbounded function or a function on an infinite interval. One way to interpret
such an integral is as a limit of Riemann integrals; this limit is called an improper
Riemann integral.