Relations between two definitions of harmonic measure

Question

I came into two definitions of harmonic measure on a Riemann surface. The first is defined on p.180 of Riemann surfaces, 2nd by Kra and Farkas, which read as follows.

Theorem. Let $M$ be a hyperbolic Riemann surface and $K$ a compact subset with $\partial(M-K)$ regular and $M-K$ connected. Then there exists a function $\omega \in C(\overline{M-K})$ such that

1. $\omega$ is harmonic on $M-K$.
2. $\omega=1$ on $\partial (M-K)$.
3. $0 < \omega < 1$ on $M-K$.

Definition 1The smallest $\omega$ above, is called the harmonic measure of $K$.

The second definition of harmonic measure is taken from p.301, Functions of one complex variable II by Conway.

Definition 2 Let $G$ be a hyperbolic open set and $a \in G$. The unique probability measure $\omega_a$ supported on $\partial_\infty G$ and satisfying \begin{equation} \hat{u}(a)=\int_{\partial_\infty G} u d\omega_a, \forall u \in C_{\mathbb{R}}(\partial_\infty G). \end{equation} is called the harmonic measure for $G$ at $a$. For each continuous function $u \in C_{R}(\partial_\infty G)$, $\hat{u}$ means the Perron solution with respect to boundary value $u$, and $\partial_\infty G$ is the boundary of $G$ on the Riemann sphere.

Obviously, each hyperbolic open set can be viewed as a hyperbolic Riemann surface. My question is , what's the relationship between these two definitions? I think they are not equivalent, since the first definition is intrinsic, but the second is not.

Answer 1

For the case of a plane domain, the first definition is a special case of the second. Assuming that $M$ is a plane domain, take $G=M\backslash K$ in the second definition. Then, if $\omega$ is the harmonic measure from the second definition, then $u$ is the first definition is $$u=\int_{\partial{G}\cap K} d\omega.$$ In other words, $u$ is the harmonic function in $G\backslash K$ whose boundary values are $1$ on $\partial G\cap K$ and $0$ on the rest of the boundary.