Absolute continuity between the surface measure and harmonic measure implies rectifiability

In the present paper we prove that for any open connected set $Ω\subset{\mathbb R}^{n+1}$, $n\geq 1$, and any $E\subset \partialΩ$ with $0<{\mathcal H}^n(E)<\infty$ absolute continuity of the harmonic measure $ω$ with respect to the Hausdorff measure on $E$ implies that $ω|_E$ is rectifiable.

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