TITLE: Open set whose boundary is not a null set
QUESTION [14 upvotes]: I've just seen a theorm about a bounded set $A \subset \mathbb{R}^n$.
$$\chi_A \text{ is Riemann integrable} \Longleftrightarrow \partial A \text{ is a null set}$$
Then, I wonder if there's any open set whose boundary is not a null set. Can you give me some example for that?

REPLY [13 votes]: Consider $A$ a fat Cantor set, for example a set of measure $\frac13$. $A$ is closed, so its complement is open, and also $A$ is nowhere dense therefore its complement is dense. That is to say that $U=[0,1]\setminus A$ is open, so for the Lebesgue measure $m$ we have: $$m(\partial U)=m([0,1]\setminus U)=m(A)=\frac13.$$