of $P^{q}(\mu)$: we make extensive use of results and technics related to uniform algebra or rational approximation: such as peak points, harmonic measures, hypo-Dirichlet algebra, multi-nicely connected domains, representing measures, pointwise bounded approximation, etc (these concepts are not needed for $P^{q}(\mu)$). So, besides Thomson’s technic and method, we need a significant part of the theories from the uniform algebras and the rational approximation to get the work done. Combination for Thomson’s technic and uniform rational approximation theory is the key to prove Theorem \[t:main\]. However, not every thing we need in uniform algebra theory is ready for us. We have to prove some results in that theory by ourself. In doing so, we first introduce the concept of multi-nicely connected domains, then we prove Proposition \[p:ABS2\] and an interesting result in uniform algebra, Lemma \[l:key\], which is crucial for us to prove Lemma \[l:key2\]. That is one of our key lemmas and it is needed to prove another key lemma, Lemma \[l:scheme\]. The rest of paper is to use these two lemmas and results in [@qiu1] and [@comm] to prove the main theorem and extend those lemmas and results that were proved for the polynomial case in [@jm]. So far, 