Pointwise Maximum 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.

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

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