Uniform domains with rectifiable boundaries and harmonic measure

We assume that $Ω\subset \mathbb{R}^{d+1}$, $d \geq 2$, is a uniform domain with lower $d$-Ahlfors-David regular and $d$-rectifiable boundary. We show that if $\mathcal{H}^d|_{\partial Ω}$ is locally finite, then the Hausdorff measure $\mathcal{H}^d$ is absolutely continuous with respect to the harmonic measure $ω$ on $\partial Ω$, apart from a set of $\mathcal{H}^d$-measure zero.

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