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

To your first question: Take the example of a non borel set from this page: http://en.m.wikipedia.org/wiki/Borel_set

Then use sterograpic representation to wrap it around the open unit disc. That set (this weird boundary union disc interior) has a measure zero boundary (closure minus interior) but is not borel, otherwise the original set would be.

Your second question:

http://en.wikipedia.org/wiki/Smith–Volterra–Cantor_set is a borel set with positive measure boundary.

So the answer to both your questions is "no".

My first example satisfies the theorem, but is not Borel. So what you would like to prove is false.