If $D$ is a domain in the complex plane bounded by a Jordan curve $J$, what's the relation between the harmonic measure and the Poisson kernel on the boundary?
More specifically, if $z_0 \in D$ and $dP_{z_0}$ is the harmonic measure on $J$ with respect to $z_0$, is it true that $dP_{z_0}(z) = P(z_0, z)dz$ (maybe with some normalization)?
Poisson kernel is the density of harmonic measure with respect to the Lebesgue measure. Indeed, if $E$ is a subset of the boundary, and $\omega(\cdot, E)$ is a harmonic function with boundary values $\chi_E $, then $$ \omega(z,E) = \int_{\partial D} \chi_E(\zeta)P(z,\zeta)\,|d\zeta| = \int_{E} P(z,\zeta)\,|d\zeta| $$ which precisely expresses the idea of $P$ being the density of the measure $\omega(z,\cdot)$.