# What's the relationship between Borel set and set whose boundary is measure zero?

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?

• For a non-constructive answer to the first question, note that there are $2^c$ many subsets of the boundary of the unit disk in the plane, all of which have (planar) measure zero, and only $c$ many of these can be Borel sets, so most subsets of the boundary of the unit disk have the property that its union with the open unit disk is not a Borel set (non-Borel union Borel is non-Borel), and all the boundaries of these non-Borel sets have measure zero. – Dave L. Renfro Oct 15 '14 at 19:59