turkmath.org

In the last 30 years, a series of amazing discoveries about harmonic measure has been made. One of the most important results is Makarov's theorem that harmonic measure on any simply connected domain is singular to $\alpha$-dimensional Hausdorff measure for $\alpha>1$ but absolutely continuous to $\alpha$-dimensional Hausdorff measure if $\alpha <1$. We talk about the concepts of harmonic measure, Hausdorff measures and equilibrium measures first. Then, we discuss realations between Hausdorff measures and harmonic measure for some infinitely connected domain.