is necessary.

If every $f$ in $H^{\infty}(K^{\circ})$ a pointwise limit of a bounded sequence in $A(K)$, then $K$ satisfies the condition of Theorem \[t:main\]. In particular, when $K$ is such that $R(K)$ is a hypo-Dirichlet algebra [@as; @ga], $K$ satisfies the condition of Theorem \[t:main\] (in this case, $R^{q}(K,\mu)=A^{q}(K,\mu)$). If the complement of $K$ has only finitely many components (note, $K^{\circ}$ may still has infinitely many components in this case), then $R(K)$ is a hypo-Dirichlet algebra and hence $K$ satisfies the hypothesis of Theorem \[t:main\]. Since a quite large of class of $R^{q}(K,\mu)$ satisfies our conditions, Theorem \[t:main\] is also a theorem for the rational $R^{q}(K,\mu)$.

Since the polynomial case is just the disk case in our setting and since a general compact subset $K$ is much more complicated than a disk in nature, one can expect the extension needs much more work. To get a structure theorem for $A^{q}(K,\mu)$ (here $K$ is an arbitrary compact subset), we need more than Thomson’s technic and method. In fact, we need a new Thomson-type approximation scheme as developed in [@qiu1] that takes all of what is used in Thomson’s paper [@jm]. In addition, we need what was not involved in the case 