Integral of Positive Simple Function is Increasing

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $f, g: X \to \R$, $f, g \in \EE^+$ be positive simple functions.

Suppose that:
 * $f \le g$

where $\le$ denotes pointwise inequality.

Then:


 * $\map {I_\mu} f \le \map {I_\mu} g$

where $I_\mu$ denotes $\mu$-integration

This can be summarized by saying that $I_\mu$ is an increasing mapping.

Proof
Note that:


 * $g - f \ge 0$

From Scalar Multiple of Simple Function is Simple Function and Pointwise Sum of Simple Functions is Simple Function, we then have that:


 * $g - f \in \EE^+$

Write:


 * $g = f + \paren {g - f}$

From Integral of Positive Simple Function is Additive, we then have:


 * $\map {I_\mu} g = \map {I_\mu} f + \map {I_\mu} {g - f}$

Since:


 * $\map {I_\mu} {g - f} \ge 0$

we obtain:


 * $\map {I_\mu} f \le \map {I_\mu} g$