a domain that is regular for the Dirichlet problem and let $\mu$ be a measure on $\overline{G}$. The sweep of $\mu$ is the unique positive measure $\tilde{\mu}$ on $\partial G$ that satisfies $\int
_{\overline {G}}\tilde{u} \hspace{.05in}d\mu = \int_{\partial G}
 u d\tilde{\mu}, \hspace{.05in} u \in C(\partial G)$, where $\tilde{u}$ is the solution of the Dirichlet problem for $u$. A simple fact is that if $\mu$ is a measure on $\overline{G}$, then $\tilde{\mu} = \mu|\partial G + \widetilde{\mu|G}$.

\[ABS1\] Let $K$ be a compact subset such that the components of $K^{\circ}$ are multi-nicely connected and the harmonic measures of the components of $K^{\circ}$ are mutually singular. Let $\Omega$ be a component of $K^{\circ}$. If $A^{q}(K,\mu)$ is pure and if $K^{\circ}$ is dense in $K$, then $\mu|\partial \Omega
\ll\omega_{\Omega}$.

Let $\{\Omega_{j}\}_{j=0}^{\infty}$ be the collection of the components of $K^{\circ}$. Fix an integer $j\geq 0$ and let $E$ be a component of $\partial \Omega_{j}$. Let $G_{j}$ be the unique simply connected domain in the sphere [**C**]{}$_{\infty}$ that has $E$ as its boundary and contains $\Omega_{j}$. Since $\Omega_{j}$ is multi-nicely connected, $G_{j}$ must be nicely connected. For $i \neq j$, let $G_{i}$ be the bounded simply connected domain that contains $\Omega_{i}$ and whose boundary is 