I wish to prove that if $\{f_{i}\}_{i\in\mathbb{N}}$ is a sequence of measurable functions then so is $\sup_{i\in\mathbb{N}}f_{i}$.
From another question I asked today I know that it suffices to prove that both of those conditions hold.
$f^{-1}(\{\infty\}),f^{-1}(\{-\infty\})\in S$
$f$ is measurable as a function $f^{-1}(\mathbb{R})\to\mathbb{R}$
I started with the first condition: set $\varphi(x)\triangleq \sup_{i\in\mathbb{N}}f_{i}(x) $
$ \varphi^{-1}(\infty)=\{x\in X:\,\varphi(x)=\infty\} $
and I can't think of any way of simplyfing it, $\varphi(x)=\infty$ means that either one of the $f_i$'s have it that $f_i(x)=\infty$ or that that sequence $f_i(x)$ is not bounded from above.
How can I continue to solve this question ?