$f$ not measurable, but $\lvert f\rvert$ measurable

Question

Do you know an example of a function $f\colon\mathbb{R}\to\mathbb{R}$ which is not $\mathcal{B}$-measurable but $\lvert f\rvert$ is $\mathcal{B}$-measurable?

Answer 1

Pick any non-measurable set $A$. Define: $$ f(x) = \begin{cases} 1 &: x \in A \\ -1 &: x \not\in A \end{cases} $$

$|f| = 1$ is a constant function, hence measurable. $f$ is easily seen to be non-measurable.