Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension one Hausdorff measure

TL;DR

This paper proves that in certain open sets, harmonic measure being absolutely continuous with respect to Hausdorff measure on a subset implies the measure's rectifiability, linking measure-theoretic and geometric properties.

Contribution

It establishes a link between absolute continuity of harmonic measure and its rectifiability in open sets, providing a proof sketch for this relationship.

Findings

Absolute continuity of harmonic measure implies rectifiability on subsets of the boundary.

The result applies to open connected sets in Euclidean space with finite Hausdorff measure.

The paper provides a proof sketch for the rectifiability of harmonic measure under these conditions.

Abstract

In the present paper we sketch the proof of the fact 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.