Existence and uniqueness of harmonic measure on a bounded regular plane domain

For a continuous datum $\phi$ define $L(\phi ):=H_{\phi }(z)$. Each $H_{\phi }$ is harmonic on $\mathrm{\Omega }$ and has the boundary limit $\phi$ at every boundary point by [F1], so the function equal to $H_{\phi }$ on $\mathrm{\Omega }$ and to $\phi$ on $\mathrm{\partial }\mathrm{\Omega }$ is continuous on $\bar{\mathrm{\Omega }}$; by [F2] it is the unique continuous harmonic extension of $\phi$.

The map $L$ is linear: for real $\alpha ,\beta$ and continuous $\phi ,\psi$ the function $\alpha H_{\phi }+\beta H_{\psi }$ is harmonic on $\mathrm{\Omega }$ and extends continuously to the boundary with values $\alpha \phi +\beta \psi$, so it equals $H_{\alpha \phi +\beta \psi }$ by the uniqueness in step 1.1, and evaluating at $z$ gives $L(\alpha \phi +\beta \psi )=\alpha L(\phi )+\beta L(\psi )$.

The map $L$ is positive and normalized: if $\phi \ge 0$ then the constant $0$ lies in the Perron family of $\phi$ by [F2], so $U_{\phi }\ge 0$ and hence $H_{\phi }\ge U_{\phi }\ge 0$; and $H_1=1$ because the constant function $1$ is a continuous harmonic extension of the boundary datum $1$, so it equals $H_1$ by step 1.1. Consequently $\mathrm{\mid }L(\phi )\mathrm{\mid }\le {\max }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }\phi \mathrm{\mid }$ for every continuous $\phi$, by applying positivity to $\max \mathrm{\mid }\phi \mathrm{\mid }-\phi$ and $\max \mathrm{\mid }\phi \mathrm{\mid }+\phi$, and $\mathrm{\parallel }L\mathrm{\parallel }=1$.

The boundary $\mathrm{\partial }\mathrm{\Omega }$ is compact by [F1], hence a locally compact Hausdorff space on which every continuous function has compact support, so $C(\mathrm{\partial }\mathrm{\Omega })=C_0(\mathrm{\partial }\mathrm{\Omega })$; by [F3] and $\mathrm{D}\mathrm{C}$ there is a unique finite regular Borel measure $\omega$ on $\mathrm{\partial }\mathrm{\Omega }$ with $L(\phi )={\int }_{\mathrm{\partial }\mathrm{\Omega }}\phi d\omega$ for all continuous $\phi$ and $\omega (\mathrm{\partial }\mathrm{\Omega })=\mathrm{\parallel }L\mathrm{\parallel }=1$.

The measure $\omega$ of step 3.1 is a Radon Borel probability measure representing every continuous boundary datum at $z$, so it is a harmonic measure for $\mathrm{\Omega }$ at $z$ in the sense of the definition. If ${\omega }^{\mathrm{\prime }}$ were another one, then $\int \phi d{\omega }^{\mathrm{\prime }}=H_{\phi }(z)=L(\phi )$ for every continuous $\phi$, so ${\omega }^{\mathrm{\prime }}=\omega$ by the uniqueness in [F3]; hence the harmonic measure is unique.

Finally, for fixed continuous $\phi$ the function $z\mapsto {\int }_{\mathrm{\partial }\mathrm{\Omega }}\phi d{\omega }_{\mathrm{\Omega }}^z$ coincides with $H_{\phi }$ by the defining identity, so it is harmonic on $\mathrm{\Omega }$ and has the boundary values $\phi$; by step 1.1 it is the unique continuous harmonic extension. This is the only place where $\mathrm{D}\mathrm{C}$ is used, through the representation theorem [F3]; the Perron input [F1] was used as a completed theorem.