# A technical question about the Lebesgue measure

Let $U$ be an open set in $\mathbb{R}^2$. How to prove that the boundary of the CLOSURE of $U$ has Lebesgue measure 0 ? Thanks.

• In fact, it's not hard to find examples where the measure of $U$ is arbitrarily small while the closure is the entire ${\bf R}^2$, just consider small balls around a countable dense subset. – tomasz Jul 19 '13 at 8:49
• @tomasz I agree; however, the boundary of $\mathbb{R}^2$ is the empty set and thus has measure zero. So, I don't think it's a counterexample to the OP's claim. – Amitesh Datta Jul 19 '13 at 9:05