Conformal invariance of harmonic measure (${\omega }_{\Omega }(z,E)={\omega }_{f(\Omega )}(f(z),f(E))$)

For a conformal map extending to the relevant boundary sets, compose the target harmonic measure with the map. The planar Laplacian transforms by the positive factor $|f^{\prime }{|}^2$, so the composition is a harmonic function with the correct boundary values. Uniqueness of the Dirichlet problem gives the displayed identity on regular domains, including the smooth-circle domains in the harmonic measure of an annulus.