# Proving that if $\{f_{i}\}_{i\in\mathbb{N}}$ is a sequence of measurable functions then so is $\displaystyle\sup_{i\in\mathbb{N}}f_{i}$.

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.

1. $f^{-1}(\{\infty\}),f^{-1}(\{-\infty\})\in S$

2. $f$ 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 ?

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To show that $f$ is a measurable function, it actually suffices to show that the sets $\{f > a\}$ are measurable, for every $a \in \mathbb{R}$. Do you see why this is equivalent to what you wrote? This condition is significantly easier to check.