Pointwise Upper Limit of Measurable Functions is Measurable

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

Let $\left({f_i}\right)_{i \in I}$, $f_i: X \to \overline{\R}$ be an $I$-indexed collection of $\Sigma$-measurable functions.

Then the pointwise upper limit $\displaystyle \limsup_{i \in I} f_i: X \to \overline{\R}$ is also $\Sigma$-measurable.