Integral of Positive Measurable Function is Additive/Corollary

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

Let $f : X \to \overline \R$ and $g : X \to \overline \R$ be positive $\Sigma$-measurable functions. Let $A \in \Sigma$.

Then:


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

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

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

Proof
We have: