This question already has an answer here:

Let $\Omega = [0,1]$. I want a Lebesgue measurable set $S$ with the following property.

$$ \ell(S \cap I) = \ell(I \backslash S)$$ for every subinterval $I$ of $[0,1]$, where $\ell(A)$ is the Lebesgue measure of $A$.

A friend recently told me that Lusin's theorem says that such a set does not exist. I don't seem to find a result I can quote (and learn from) that says the same though. Is it true that such a set does not exist?

Thanks.