Q:

A question about null sets

Let $ F: R^n \rightarrow R $ be a continuous function. Let $ S= F^{-1}(0) $. Is it true that the boundary of $ S $ has null Lebesgue measure? If it is no what happen if we suppose $ F $ smooth?
Thanks

A:

There is a theorem, I think due to Borel, which states that each closed subset of $\Bbb R^n$ is $F^{—1}(\{0\})$ for some smooth function $F$. 
So the answer is no, taking $K^n$ where $K$ is a fat Cantor set. 

