18 1. The Regulated and Riemann Integrals
if
V =
b
a
v(x) dx v L(f) and U =
b
a
u(x) dx u U(f) ,
then every number in the set V is less than or equal to every number
in the set U. Thus, sup V inf U as claimed.
It is not difficult to see that sometimes the two sides of this in-
equality are not equal (see Exercise 1.7.7 below); but if it should
happen that
sup
b
a
v(x) dx v L(f) = inf
b
a
u(x) dx u U(f) ,
then we have only one choice for
b
a
f(x) dx; it must be this common
value.
This motivates the definition of the next vector space of functions
that can be integrated. Henceforth, we will use the more compact
notation
sup
v∈L(f )
b
a
v(x) dx instead of sup
b
a
v(x) dx v L(f)
and
inf
u∈U(f )
b
a
u(x) dx instead of inf
b
a
u(x) dx u U(f) .
Definition 1.7.2. (Riemann integral). Suppose f is a bounded
function on the interval I = [a, b]. Let U(f) denote the set of all step
functions u(x) on I such that f(x) u(x) for all x and let L(f)
denote the set of all step functions v(x) such that v(x) f(x) for all
x. The function f is said to be Riemann integrable provided
sup
v∈L(f )
b
a
v(x) dx = inf
u∈U(f )
b
a
u(x) dx .
In this case its Riemann integral
b
a
f(x) dx is defined to be this com-
mon value.
There is a simple test for when a function f is Riemann integrable.
For any ε 0 we need only find a step function u greater than f and
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