Conformal invariance of harmonic measure

The map $\bar{F}$ is a bijection of the compact sets $\bar{\mathrm{\Omega }}$ and $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$ restricting to the bijection $F:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$; therefore it maps $\mathrm{\partial }\mathrm{\Omega }=\bar{\mathrm{\Omega }}\setminus \mathrm{\Omega }$ onto $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}\setminus {\mathrm{\Omega }}^{\mathrm{\prime }}=\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$, and it is a homeomorphism between the two boundaries. Consequently, for continuous $\phi :\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}\to \mathbb{R}$ the pullback $\phi \circ \bar{F}$ is continuous on $\mathrm{\partial }\mathrm{\Omega }$, and the pushforward $({\bar{F}}_{\ast }{\omega }_{\mathrm{\Omega }}^z)(E):={\omega }_{\mathrm{\Omega }}^z({\bar{F}}^{-1}(E))$ is well defined on Borel subsets of $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$.

For continuous $\phi$ on $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$ and $\psi :=\phi \circ \bar{F}$ on $\mathrm{\partial }\mathrm{\Omega }$, the function $u:=H_{{\mathrm{\Omega }}^{\mathrm{\prime }},\phi }\circ F$ is harmonic on $\mathrm{\Omega }$ by [F3], since $H_{{\mathrm{\Omega }}^{\mathrm{\prime }},\phi }$ is harmonic on ${\mathrm{\Omega }}^{\mathrm{\prime }}$, and it extends continuously to $\mathrm{\partial }\mathrm{\Omega }$ with boundary values $\psi$, because $H_{{\mathrm{\Omega }}^{\mathrm{\prime }},\phi }$ extends continuously to $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$ with values $\phi$ by [F2] and $\bar{F}$ maps $\mathrm{\partial }\mathrm{\Omega }$ onto $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$ by step 1.1.

The pushforward of step 1.1 represents the same value: by the defining property of ${\omega }_{\mathrm{\Omega }}^z$ in [F1] and the change of variables defining the pushforward, ${\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}}\phi d({\bar{F}}_{\ast }{\omega }_{\mathrm{\Omega }}^z)={\int }_{\mathrm{\partial }\mathrm{\Omega }}(\phi \circ \bar{F})d{\omega }_{\mathrm{\Omega }}^z=H_{\mathrm{\Omega },\psi }(z)\mathrm{.}$

The continuous harmonic extensions $u=H_{{\mathrm{\Omega }}^{\mathrm{\prime }},\phi }\circ F$ and $H_{\mathrm{\Omega },\psi }$ of step 2.1 have the same boundary values $\psi$ on $\mathrm{\partial }\mathrm{\Omega }$, so they coincide on $\mathrm{\Omega }$ by [F2]; at $z$ this is $H_{\mathrm{\Omega },\psi }(z)=H_{{\mathrm{\Omega }}^{\mathrm{\prime }},\phi }(F(z))\mathrm{.}$

Combining steps 3.1 and 2.2 with the defining property of ${\omega }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}^{F(z)}$ in [F1] gives, for every continuous $\phi :\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}\to \mathbb{R}$, ${\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}}\phi d({\bar{F}}_{\ast }{\omega }_{\mathrm{\Omega }}^z)=H_{{\mathrm{\Omega }}^{\mathrm{\prime }},\phi }(F(z))={\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}}\phi d{\omega }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}^{F(z)}\mathrm{.}$ Both sides are finite regular Borel measures on the compact boundary $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$, so by [F4] they coincide as measures, and in particular on every Borel subset of $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$.

Therefore ${\bar{F}}_{\ast }{\omega }_{\mathrm{\Omega }}^z={\omega }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}^{F(z)}$ on all Borel boundary sets. Dependent Choice was used only through the existence and uniqueness theorem [F1]; the transport itself is the identification of two continuous harmonic extensions with common boundary data, and no boundary behaviour of $F$ beyond the given closure homeomorphism was assumed.