Integral of Positive Measurable Function is Monotone

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f, g: X \to \overline \R$, $f, g \in \MM_{\overline \R}^+$ be positive measurable functions.

Suppose that $f \le g$, where $\le$ denotes pointwise inequality.

Then:


 * $\ds \int f \rd \mu \le \int g \rd \mu$

where the integral sign denotes $\mu$-integration.

This can be summarized by saying that $\ds \int \cdot \rd \mu$ is monotone.