Let \(X\subseteq\RR^n\) be Lebesgue measurable and let \(f\colon X\to[-\infty,\infty]\). The function \(f\) is measurable if for every \(\alpha \in \RR\) the set \[ S_\alpha \coloneqq \theset{ x\in X \suchthat f(x) > \alpha} \] is Lebesgue measurable.
Which sets must be Lebesgue measurable for $f$ to be a measurable function?