Let $f$ be a harmonic function defined on a planar disk $B \subset \mathbb{R}^2$ and continuous on $\partial B$. For which probability measures $\mu$ on $\partial B$ is it true that $$\int_{\partial B } \! f(s) \, d\mu(s) = f(x_0)$$ for some $x_0 \in \overline{B}$?
Let me say explicitly that the point of the question is that $x_0$ is allowed to depend on $f$ as well as $\mu$, since otherwise the Poisson kernel is obviously the only solution.