Let $f$ be integrable on $\mathbb{R}^d$, let $E_\alpha : = \{x \in \mathbb{R}^d:|f(x)|>\alpha\}$. For a fixed $\alpha$, $E_\alpha$ is a measurable set.
Is the function $\chi_{E_{\alpha}}(x)$ measurable as a function of $\alpha$ and $x$ on $\mathbb{R}^{d+1}$? I think there should be an easy way to prove this with Fubini's theorem.