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. 

It appears that the statement in your question is not always true. See https://mathoverflow.net/questions/25993/sets-with-positive-lebesgue-measure-boundary for excellent answers. For related details, see also Regular open set whose boundary has nonzero volume. and https://mathoverflow.net/questions/24264/a-question-about-the-osgood-curve.
