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