Suppose $f_n : X \to \Bbb R$ is a sequence of Lebesgue measurable functions on a measurable space $(X, \Sigma)$. Then the function $g(x) = \sup_n f_n(x)$ is measurable. This is because $\sup_n f_n(x) \leq a$ iff $f_k(x) < a$ for all $k$, ergo
$$
\displaystyle \{ x : \sup_n f_n(x) \leq a \} = \bigcap_{n} \{ x : f_n(x) \leq a \}
$$
which is a measurable set in $\Sigma$ by assumption of the measurability of the $f_n$’s. We can also show that $\inf f_n$, $\limsup f_n$ and $\liminf f_n$ are all measurable by similar means.