Integral of Positive Measurable Function is Additive

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.

Then:


 * $\ds \int \paren {f + g} \rd \mu = \int f \rd \mu + \int g \rd \mu$

where:
 * $f + g$ is the pointwise sum of $f$ and $g$
 * the integral sign denotes $\mu$-integration

This can be summarized by saying that $\ds \int \cdot \rd \mu$ is (conventionally) additive.