1.5. Regulated Integral 9

uniformly to f. As is typical in mathematics when we have a col-

lection of objects which behave in a way we like we make it into a

definition.

Definition 1.5.1. (Regulated function). A function f : [a, b] → R

is called regulated provided there is a sequence {fm} of step functions

which converges uniformly to f.

Another way to state this is to say a regulated function is one

which can be uniformly approximated as closely as we wish by a step

function. We can now prove that the limit of the integrals of the

approximating step functions always exists and does not depend on

the choice of approximating step functions.

Theorem 1.5.2. Suppose {fm} is a sequence of step functions on

[a, b] converging uniformly to a regulated function f. Then the se-

quence of numbers {

b

a

fm(x) dx} converges. Moreover, if {gm} is

another sequence of step functions which also converges uniformly to

f then,

lim

m→∞

b

a

fm(x) dx = lim

m→∞

b

a

gm(x) dx.

Proof. Let zm =

b

a

fm(x) dx. We will show that the sequence {zm}

is a Cauchy sequence and hence has a limit. To show this sequence

is Cauchy we must show that for any ε 0 there is an M such that

|zp − zq| ≤ ε whenever p, q ≥ M.

Since {fm} is a sequence of step functions on [a, b] converging

uniformly to f, if we are given ε 0, there is an M such that for all

x ∈ [a, b],

|f(x) − fm(x)|

ε

2(b − a)

whenever m ≥ M.

Hence, whenever p, q ≥ M,

|fp(x) − fq(x)| |fp(x) − f(x)| + |f(x) − fq(x)| (1.5.1)

ε

2(b − a)

+

ε

2(b − a)

=

ε

b − a

.