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

Absolute continuity between the surface measure and harmonic measure implies rectifiability