Approximate tangents, harmonic measure, and domains with rectifiable boundaries

Abstract: Let $\Omega \subset \mathbb{R}^{n+1}$, $n \geq 1$, be an open and connected set. Set $\mathcal{T}n$ to be the set of points $\xi \in \partial \Omega$ so that there exists an approximate tangent $n$-plane for $\partial\Omega$ at $\xi$ and $\partial\Omega$ satisfies the weak lower Ahlfors-David $n$-regularity condition at $\xi$. We first show that $\mathcal{T}_n$ can be covered by a countable union of boundaries of bounded Lipschitz domains. Then, letting $\partial^\star \Omega$ be a subset of $\mathcal{T}_n$ where $\Omega$ satisfies an appropriate thickness condition, we prove that $\partial^\star \Omega$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\Omega$. As a corollary we obtain that if $\Omega$ has locally finite perimeter, $\partial\Omega$ is weakly lower Ahlfors-David $n$-regular, and the measure-theoretic boundary coincides with the topological boundary of $\Omega$ up to a set of $\mathcal{H}^n$-measure zero, then $\partial \Omega$ can be covered, up to a set of $\mathcal{H}^n$-measure zero, by a countable union of boundaries of bounded Lipschitz domains that are contained in $\Omega$. This implies that in such domains, $\mathcal{H}^n|{\partial\Omega}$ is absolutely continuous with respect to harmonic measure.