Pointwise Maximum 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.

Then $\max \left({f, g}\right)$, the pointwise maximum of $f$ and $g$, is also a $\mu$-integrable function.

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