example of function not Lebesgue Measurable with measurable level sets

Consider $V$ as in Vitali's theorem. Define the function $f: [0,1] \to [0,+\infty[$ by:

The level sets of $f$ will either be the empty set, or a singleton and thus measurable.

$f$ is not a measurable function since $f^{-1}([2,+\infty[) = V$ and $V$ is not a measurable set.