Approximate tangents, harmonic measure, and domains with rectifiable boundaries

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

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