Suppose that for any $s\in \mathbb{R}$ there is given an $M$-measurable function $f_s : X \rightarrow [-\infty, \infty]$. Suppose that $\lim_{s \to\infty} f_s(x)$ exists for all $x \in X$. Prove that $\lim_{s \to\infty} f_s$ is also $M$-measurable.
I thought about approximating $s$ with some $m\in\mathbb{Q}$ and use the fact that $f_k$ is measurable if $k \in \mathbb{N}$, since $\mathbb{N}\sim \mathbb{Q} $, and then find the limit of it. But I don't know how to describe it.
Is the idea I tried correct? If not, give me other hints. Thank you.
It might be correct, but I see no need for either $\mathbb{Q}$ or any approximation: you may just take the limit over $\mathbb{N}$ and use pointwise existence of the one over $\mathbb{R}$ to show that they are the same.
This solves the problem, since you are calculating a limit of countably many measurable functions, which is measurable.