# Lebesgue measure of the boundary of an open connected set in $\mathbb{R}^n$

Is it true that the ($n$-dimensional) lebesgue measure of the boundary of an open connected set in $\mathbb{R}^n$ is zero? Many thanks in advance!

Let $E \subset [0,1]$ be the fat Cantor set and consider
$$\Omega = (-1, 0) \times (0,1) \cup (-1, 1) \times (0,1)\setminus E\ .$$
This is connected and $\partial \Omega$ contains $(0,1) \times E$ which has positive Lebesque measure.