Integral of Positive Simple Function is Increasing

Theorem

Let $\struct {X, \Sigma, \mu}$ 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:

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

$\blacksquare$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $9.3 \ \text{(iv)}$