# Integral of Integrable Function is Additive

## Theorem

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

Let $f, g: X \to \overline{\R}$ be $\mu$-integrable functions.

Suppose that their pointwise sum $f + g$ is well-defined.

Then:

$\displaystyle \int f + g \, \mathrm d \mu = \int f \, \mathrm d \mu + \int g \, \mathrm d \mu$