# Relations between two definitions of harmonic measure

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 $$$$\hat{u}(a)=\int_{\partial_\infty G} u d\omega_a, \forall u \in C_{\mathbb{R}}(\partial_\infty G).$$$$ is called the harmonic measure for $$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.

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.