Is a set whose boundary is measure zero a Borel set?
Does any given Borel set has a measure zero boundary?
I want to give my ideas first: If $E \subseteq R^n $ is some set whose boundary has Lebesgue measure zero, then we can safely do the Riemann integral $\int _E1dx$, which means we can calculate the "volumn" of $E$. Hence I think $E$ is a Borel set.
But how about the inverse arguement?