whole plane by defining their values to be zero off $\overline{\Omega}$. We claim that there exists a sequence $\{q_{n}\}$ in $A(K^{\circ})$ such that it weak-star converges to $r_{n}$ in $L^{\infty}(\omega)$.

For each $i\geq 1$, let $G_{i}$ be the bounded simply connected domain that contains $\Omega_{i}$ and whose boundary is a component of $\partial \Omega_{i}$. Let $G$ be the union of $\Omega$ with those domains $G_{i}$ that do not intersect $\Omega$. Then all but finitely many components of $G$ are simply connected domains and each component of $G$ is multi-nicely connected. By Lemma \[l:key\], $A(G)$ is boundedly pointwise dense in $H^{\infty}(G)$. Thus, there exists a bounded sequence $\{q_{m}\}$ in $A(G)$ so that it pointwise converges to $r_{n}$ on $G$. Now for given $\epsilon > 0$, let $f\in L^{1}(\omega)$. Then $$|\int_{\partial K} f(r_{n}-q_{m}) d\omega | \leq
 |\int_{\cup_{o}^{k} \partial \Omega_{i}} f(r_{n}-q_{m}) d\omega |
+ \frac{\epsilon}{2} \hspace{.05in} \mbox{for all} \hspace{.05in} m$$ whenever $k$ is sufficiently large. Observe that $\{q_{m}\}$ weak-star converges to $r_{n}$ in $L^{\infty}(\omega_{\Omega_{i}})$ for each $i$. Thus, $$|\int_{\partial K} f(r_{n}-q_{m}) d\omega | \leq \frac{\epsilon}{2}
+ \frac{\epsilon}{2} \hspace{.05in}\mbox{when}
 \hspace{.05in}m \hspace{.05in}\mbox{is sufficiently large.}$$ Hence, for each $ f\in L^{1}(\omega) $ we have $$\lim_{n\rightarrow \infty} \int_{\partial K} f(r_{n}-q_{m}) d\omega
= 0.$$ That is, $\{q_{m}\}$ weak-star 