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.
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