Harmonic measure on a bounded regular plane domain

Let $\mathrm{\Omega }\subseteq \mathbb{C}$ be a bounded complex domain (A complex domain is a nonempty connected open subset of $\mathbb{C}$) every boundary point of which is regular in the sense of Barriers and regular boundary points, and let $z\in \mathrm{\Omega }$. The Euclidean boundary $\mathrm{\partial }\mathrm{\Omega }$ is closed and, $\mathrm{\Omega }$ being bounded, also bounded, hence compact (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line), and it carries the Borel $\sigma$-algebra (The Borel sigma-algebra of a topological space).

For every real continuous function $\phi :\mathrm{\partial }\mathrm{\Omega }\to \mathbb{R}$, let $H_{\phi }$ denote the regularized Perron envelope of the bounded plane Dirichlet problem with boundary datum $\phi$ (The Perron envelope and its regularization).

A harmonic measure for $\mathrm{\Omega }$ at $z$ is a Radon Borel probability measure ${\omega }_{\mathrm{\Omega }}^z$ on $\mathrm{\partial }\mathrm{\Omega }$, in the sense of Radon measure on an LCH space, such that

$H_{\phi }(z)={\int }_{\mathrm{\partial }\mathrm{\Omega }}\phi d{\omega }_{\mathrm{\Omega }}^z$ for every real continuous $\phi :\mathrm{\partial }\mathrm{\Omega }\to \mathbb{R}$.

The measure is written in the superscript slot $z$ because it is a measure attached to the point $z$; for a fixed Borel set $E\subseteq \mathrm{\partial }\mathrm{\Omega }$ the assignment $z\mapsto {\omega }_{\mathrm{\Omega }}^z(E)$ is a scalar function on $\mathrm{\Omega }$, a distinct object from the measure itself.

• Existence and uniqueness are not part of this definition. A harmonic measure for $\mathrm{\Omega }$ at $z$ is a Radon probability measure satisfying the displayed identity for all continuous data. Existence and uniqueness are proved later on this page, for every bounded regular plane domain and every $z\in \mathrm{\Omega }$; this item only fixes the object and its test identity.
• No probabilistic interpretation is used. This library defines no Brownian motion and no hitting distribution, and none is invoked: the defining property above is the totality of what "harmonic measure" means here.
• The test identity is linear and normalized. Taking $\phi \equiv 1$ in the defining identity and using that the constant function $1$ solves its own Dirichlet problem gives $\int 1d{\omega }_{\mathrm{\Omega }}^z=1$ for every candidate measure, which is why probability measures rather than arbitrary finite measures are used.