that $A(\overline
{\Omega}) \neq A(\Omega)$ in general.

A point $w$ in [**C**]{} is called an analytic bounded point evaluation ($abpe$) for $A^{q}(K,\mu)$ if there is a neighborhood $G$ of $w$ and $c > 0$ such that for all $\lambda\in G$ $$|f(\lambda)|\leq c \hspace{.05in} \|f\|_{L^{q}(\mu)}\hspace{.05in}
\mbox{for all} \hspace{.05in} f\in A(K).$$ So the map, $f \rightarrow f(\lambda)$, extends to a functional in $A^{q}(K,\mu)^{*}$. Thus, there is a (kernel) function $k_{\lambda}$ in $A^{q}(K,\mu)^{*}$ such that $f(\lambda) = \int f k_{\lambda}
\hspace{.05in}d\mu,\hspace{.05in} f \in A(K)$. Clearly, the set of $abpe$s is open. For each $f \in A^{q}(K,\mu)$, let $\hat{f}(\lambda) =
 \int f k_{\lambda} \hspace{.05in}d\mu$. Then $\hat{f}(\lambda)$ is analytic on the set of $abpe$s. The $abpe$s for $R^{q}(K,\mu)$ are define similarly.

We shall use $\nabla A^{q}(K,\mu)$ to denote the set of $abpe$s for $A^{q}(K,\mu)$. The following is one of the main results in [@qiu1] which our main theorem relies on:

\[t:approx\] Let $K$ be compact subset of [**C**]{} and let $\mu$ be a positive finite measure supported on $K$. Then $A(K)$ is dense in $L^{q}(\mu)$ if and only if $\nabla A^{q}(K,\mu) = \emptyset$.

For the case when $K$ is the polynomially convex hull of the support of $\mu$, the above theorem is a consequence of 