Rectifiability of Harmonic Measure An overview

We give some of the history behind the problem of characterizing when harmonic measure in d+1 dimensions is absolutely continuous with respect to d-dimensional Hausdorff measure, starting way back in 1916 with a result of the brother's F. and M. Riesz. This survey is by no means exhaustive, and we have omitted results concerning when Hausdorff measure is absolutely continuous with respect to harmonic measure, or the question of when harmonic measure is absolutely continuous with respect to fractional Hausdorff measures, but the results are a good starting point and provide motivation for the main result we will cover in a later video.