The Riemann Integral

8 1. The Riemann Integral
It follows that
m∑ n∑ ∑q k n∑
U(f;Q) = M l|J ′ l | = M l|J ′ l | ≤ M k |I k | = U(f;P)
l=1 k=1l=p k k=1
Similarly,
∑q k
∑q k
m ′ l|J l | ≥ m k |J l | = m k |I k |,
l=p k l=p k
and
n∑ ∑q k n∑
L(f;Q) = m ′ l|J l | ≥ m k |I k | = L(f;P),
k=1 l=p k k=1
which proves the result.
□
It follows from this theorem that all lower sums are less than or equal to all
upper sums, not just the lower and upper sums associated with the same partition.
Proposition 1.12. If f : [a,b] → R is bounded and P, Q are partitions of [a,b],
then
L(f;P) ≤ U(f;Q).
Proof. Let R be a common refinement of P and Q. Then, by Theorem 1.11,
L(f;P) ≤ L(f;R), U(f;R) ≤ U(f;Q).
It follows that
L(f;P) ≤ L(f;R) ≤ U(f;R) ≤ U(f;Q).
□
An immediate consequence of this result is that the lower integral is always less
than or equal to the upper integral.
Proposition 1.13. If f : [a,b] → R is bounded, then
Proof. Let
A = {L(f;P) : P ∈ Π},
L(f) ≤ U(f).
B = {U(f;P) : P ∈ Π}.
From Proposition 1.12, a ≤ b for every a ∈ A and b ∈ B, so Proposition 2.9 implies
that supA ≤ infB, or L(f) ≤ U(f).
□
1.4. The Cauchy criterion for integrability
The following theorem gives a criterion for integrability that is analogous to the
Cauchy condition for the convergence of a sequence.
Theorem 1.14. A bounded function f : [a,b] → R is Riemann integrable if and
only if for every ǫ > 0 there exists a partition P of [a,b], which may depend on ǫ,
such that
U(f;P)−L(f;P) < ǫ.