Rectifiability, interior approximation and harmonic measure

@article{Akman2019RectifiabilityIA,
  title={Rectifiability, interior approximation and harmonic measure},
  author={Murat Akman and Simon Bortz and Steve Hofmann and Jos{\'e} Mar{\'i}a Martell},
  journal={Arkiv f{\"o}r Matematik},
  year={2019}
}
We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure. Namely, that $H^n$-almost all of $E$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\mathbb{R}^{n+1}\setminus E$. As a consequence, for harmonic measure in the complement of such a set $E$, we establish a non-degeneracy condition… 

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