Thomson’s theorem. However, since it was known long before Thomson’s paper [@jm] that there is a compact $K$ and a measure $\mu$ on $K$ such that $R(K)$ is not dense in $L^{q}(\mu)$ but $R^{q}(K,\mu)$ has no $abpe$, and since $$P^{q}(\mu) \subset R^{q}(K,\mu)
 \subset A^{q}(K,\mu)$$ always holds, Theorem 1.1 was unexpected before [@qiu1]. Somehow, it was a surprise that the theorem is true for the spaces on the both sides of the inequality above, but fails for the spaces between. [*Nicely connected domains*]{}. Following Glicksburg [@glic], we call a domain $\Omega$ nicely connected if it is multi-nicely connected and if it is simply connected.

[*Harmonic measures.*]{} Let $\Omega$ be a domain in the extended plane [**C**]{}$_{\infty}$ such that it is solvable for the Dirichlet problem and $\infty$ is not in $\partial\Omega$. For $u \in C(\partial \Omega)$, let $\hat{u} =$ sup $\{f: f$ is subharmonic on $\Omega$ and $\limsup_{z\rightarrow
a} f(z) \leq u(a), a \in \partial \Omega\}$. The function $\hat{u}$ turns out to be harmonic on $\Omega$ and continuous on $\overline{\Omega}$, and the map $u\rightarrow \hat{u}(z)$ defines a positive linear functional on $C(\partial \Omega)$ with norm one, so the [*Riesz representing theorem*]{} implies that there is a probability measure $\omega_{z}$ on $\partial 