There is a subset $E$ of $\Bbb R$ that doesn’t satisfy $\lim\limits_{\epsilon\rightarrow 0}\mu((E+\epsilon)\backslash E)=0$ where $\mu$ is the Lebesgue measure.
I came up with $\Bbb R\backslash \Bbb Q$ and $\bigcup\limits_{n\in\Bbb N}[n,n+{1\over n}]$ but they don’t seem to work. I know that it has to have infinite measure...