Pointwise Sum of Integrable Functions is Integrable Function

Theorem
Let $\struct {X, \Sigma, \mu}$ 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 $f + g$ is also a $\mu$-integrable function.

That is, the space of $\mu$-integrable functions $\LL^1_{\overline \R}$ is closed under pointwise addition.