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

Question

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!

Answer 1

fThe answer is no. We can modify the argument in Comparing the Lebesgue measure of an open set and its closure.

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 Lebesgue measure.