purpose of this paper is to investigate the following problem:

> *What is the closure of A(K) in $L^{q}(\mu)$?*

This is a very difficult question to get a complete answer. For a given measure $\mu$, the answer depends on $K$. Let $K$ be a closed disk that contains the support of $\mu$. Then every $f$ in $A(K)$ can be uniformly approximated by polynomials, and hence $A^{q}(K,\mu) = P^{q}(\mu)$, which is the closure of polynomials in $L^{q}(\mu)$. In this case, J. Thomson proved a structure theorem for $P^{q}(\mu)$ in [@jm 1991].

Roughly speaking, Thomson’s theorem says that there exists a Borel partition $\{\Delta_{n}\}_{n=0}$ of the support of $\mu$ such that $$P^{q}(\mu) = L^{q}(\mu|\Delta_{0}) \oplus P^{q}(\mu|\Delta_{1})\oplus ...
\oplus P^{q}(\mu|\Delta_{n}) \oplus ...,$$ where each $P^{q}(\mu|\Delta_{n}) $ is identified with a space consisting of analytic functions on a simply connected domain $U_{n}$ via a so-called [*evalaution map*]{}. The union, $\cup
U_{n}$, is known as the set of analytic bounded point evaluations ($abpe$s) for $P^{q}(\mu)$. In this special case, since $A(K)$ is the uniform closure of polynomials, Thomson’s theorem shows that $A(K)$ is dense in $ L^{q}(\mu)$ if and only if $P^{q}(\mu)$ ($=
A^{q}(K,\mu)$ has no $abpe$s (this special result answered an open question raised 