$\textbf{1.14 Theorem}$ If $f_n: X \rightarrow [-\infty, +\infty]$ is measurable, for $n = 1, 2, 3,...,$ and $g= \sup_{n \geq 1} f_{n}$ and $h = \limsup_{n \to \infty} f_{n} $, then g and h are measurable.
$\textbf{Proof}$: $g^{-1}((\alpha, + \infty]) = \bigcup_{n = 1}^{\infty} f_{n}^{-1}((\alpha, + \infty])$. Hence Theorem 1.12(c) implies that $g$ is measureble. The same result holds of course with inf in place of sup, and since
$\underline{h = \inf_{k \geq 1} \{\sup_{i \geq k} f_{i}\}}$
it follows that $h$ is measurable.
Can someone explain me how this underlined implies that $h$ is measurable?