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

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

* Definition 1*The 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.