Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension-one Hausdorff measure
Jonas Azzam, Steve Hofmann, José María Martell, Svitlana Mayboroda, Mihalis Mourgoglou, Xavier Tolsa, Alexander L'vovich Vol'berg · Comptes Rendus Mathématique · 2016
In the present paper, we sketch the proof of the fact that for any open connected set Ω ⊂ R n + 1 , n ≥ 1 , and any E ⊂ ∂ Ω with 0 < H n ( E ) < ∞ , absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω | E is rectifiable.