Using Brownian motion the
following results are established:
Harmonic measure and Keldysh measure are always singular with respect
to area measure in the plane. More generally, this holds for the distribution
of the first exit point for Brownian motion of a given Borel set.
If U is open and K ⊂ ∂U is compact, then K has harmonic measure 0 w.r.t. U
if ∂U satisfies a certain metric density condition at each point of K and, in
addition, K satisfies one of the following two conditions:
K has zero length and is lying on a straight line or
K has α-dimensional Hausdorff measure zero, for some α < 1∕2.
