Boundary of Ahlfors n-dimensional domain measures 0

Question

A domain $\Omega\subset\mathbb R^n$ is said to be an Ahlfors $n-$dimensional domain if there exists some $c>0$ such that for every $r\in (0,1]$ and every $x\in \Omega$ $$ \mathcal L^n(\Omega\cap B(x,r))\geq c\mathcal L^n(B(x,r))$$ holds (where $\mathcal L^n$ stands for Lebesgue's measure in $\mathbb R^n$). I'm asked to prove, possibly using Lebesgue's differentiation theorem, that $\mathcal L^n(\partial \Omega)=0$. I'd appreciate some hint to solve this. Thanks!