= Conformal invariance of harmonic measure
{title2=$\omega_\Omega(z,E)=\omega_{f(\Omega)}(f(z),f(E))$}
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 $|f'|^2$, 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>.
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