Pointwise Sum of Integrable Functions is Integrable Function

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 $f + g$ is also a $\mu$-integrable function.

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