I need to show that the following real valued function on $\mathbb{R}$ is nonmeasurable:
$$ f(x) = \begin{cases} x &\text{$x \in E$} \\ -x &\text{$x \in [0,1] \setminus E$} \end{cases} $$
where $E$ is a non-measurable subset of $[0,1]$. How could one do this?
Best Answer
Hint:
If the function was measurable then $f^{-1}([0,1])$ would be a measurable set.