Let $\Omega \subset \mathbb{R}^d$ be a region ( bounded, simply connected, open set ). What are some regularity conditions to assure the boundary $\partial\Omega$ is a set of (lebesgue-)measure zero? Is there any geometric / topological condition, which is equivalent to the statement that $\mu(\partial\Omega) = 0$?

I am particularly interested in some weak conditions, in a sense of not being too restrictive. I'm not interested in statements as strong as "if $\partial\Omega$ is a submanifold, ...".

upper porosity. Suppose that there exists a constant $0 < c < 1$ so that for all $x \in \partial\Omega$ there are arbitrarily small radii $r$ so that $B(y,cr) \subset B(x,r)\setminus \partial\Omega$ for some $y\in \mathbb{R}^n$. Then $\mathcal{L}(\partial\Omega)=0$, where $\mathcal{L}$ is the Lebesgue measure. (This follows immediately by considering density points.) If one assumes that this holds for all small enough radii, one also gets an estimate on the dimension. – Tapio Rajala Jan 23 '11 at 14:06