1549901369-Elements_of_Real_Analysis__Denlinger_

7.2 The Riemann Integral Defined 363

Theorem 7.2.5 If P and Q are any partitions of [a, b], then S__(f, P) ::::; S(J, Q).^2

Proof. Suppose P and Q are any partitions of [a, b]. Then PU Q is a

refinement of both P and Q. Thus, by Theorem 7.2.4, S__(f, P) ::::; S__(f, PU Q),

and S(f, PU Q) ::::; S(J, Q). Also, S__(f, PU Q) ::::; S(J, PU Q) by Lemma 7.2.2.

Putting these three inequalities together, we have

S(f, P) ::::; S(f, Pu Q) ::::; S(J, Pu Q) ::::; S(J, Q).

Therefore, S__(f, P) ::::; S(J, Q). •

Definition 7.2.6 (Upper and Lower Darboux Integrals)

Let A denote the set of lower Darboux sums for f over all possible partitions
of [a, b], and B denote the set of upper Darboux sums for f over all possible
partitions of [a, b]. That is,
A= {S__(J, P) : P is a partition of [a, b]}, and
B = {S(J, P) : Pis a partition of [a, bl}.
By Theorem 7.2.5, every element of A is ::::; every element of B. Thus, the
set A is bounded above. Hence, by the completeness property of JR, A has a
least upper bound. We define

1: f =sup A= sup {S__(J, P) : P is a partition of [a, b]}.^3

This quantity is called the lower (Darboux) integral of f over [a, b].

Similarly, the set Bis bounded below, and hence, by completeness, B has
a greatest lower bound. We define

1: f =inf B = inf {S(J, P) : P is a partition of [a, b]}.^3

This quantity is called the upper (Darboux) integral of f over [a, b].

Since every element of A is::::; every element of B, Theorem 7.1.5 guarantees
that sup A ::::; inf B. That is,

l:f::::; l:J. D

We summarize these results in the following theorem.

Theorem 7.2.7 If f is any function defined and bounded on [a,b], then both

__ J.Q_J b and J.b a f exist,. and J.b _Q f ::::; J.b a f.

2. See how this differs from Lemma 7.2.2.


3. Here, the supremum (or infimum) is t a ken over all partitions P of [a,b]. Don't be misled;
   the words "is a partition" do not mean that only one partition P is being considered.