Lebesgue measure of boundary of an open set.

Question

Let $g$ be a continuous function on $\mathbb{R}^n$ and $$O=\{x\in \mathbb{R}^n:g(x)\neq 0\}.$$ Is it true that Lebesgue measure of Boundary of $O$ always zero?

Answer 1

In $\Bbb R$ let $C$ be a fat Cantor set. This is constructed in a similar way to the usual Cantor set, but the removed open intervals shrink quickly enough in length to ensure than $C$ has non-zero Lebesgue measure.

Let $O$ be the complement of $C$. The boundary of $O$ is the boundary of $C$ which is $C$ itself. The boundary of $O$ has non-zero measure.

If we define $g(x)=\text{distance}(x,C)$ then $g$ is continuous, and $O$ is the set of $x$ with $g(x)\ne0$.