Let $E$ be a bounded Borel set in $\mathbb{R}^{n}$, and let $\mu$ be the $n-$dimensional measure of Lebesgue. Suppose that every relatively open set in $E$ has measure 0. Is it possible that $\mu(E)>0$?
Because if this is the case, we can take a measurable function $f$ on a neighborhood of $E$, and , then by Lusin's theorem there is a closed set $F$ on which $f$ is continuous and $\mu(E\setminus F)=0$ by hypothesis on $E$.
