4 1. The Regulated and Riemann Integrals

There is one additional property which we will need. It differs

from the earlier ones in that we can prove that it holds whenever the

properties above are satisfied.

Proposition 1.2.1. (Absolute value). Suppose we have defined

the integral

b

a

f(x) dx for all f in some vector space of functions V

and for all a, b ∈ I. Suppose this integral satisfies properties I-III

above and both f and |f| are in V. Then for any a, b ∈ I with a ≤ b,

b

a

f(x) dx ≤

b

a

|f(x)| dx.

If a b, then

b

a

f(x) dx ≤ −

b

a

|f(x)| dx.

Proof. Suppose first that a ≤ b. Since f(x) ≤ |f(x)| for all x we

know that

b

a

f(x) dx ≤

b

a

|f(x)| dx

by monotonicity. Likewise, −f(x) ≤ |f(x)|, so

−

b

a

f(x) dx =

b

a

−f(x) dx ≤

b

a

|f(x)| dx.

But |

b

a

f(x) dx| is either equal to

b

a

f(x) dx or to −

b

a

f(x) dx. In

either case

b

a

|f(x)| dx is greater, so

b

a

f(x) dx ≤

b

a

|f(x)| dx.

If b a, then

b

a

f(x) dx =

a

b

f(x) dx ≤

a

b

|f(x)| dx = −

b

a

|f(x)| dx.

1.3. Step Functions

The easiest functions to integrate are step functions, which we now

define.