on $E$ [*a.e.*]{} $ [\omega_{W}]$, then each $f\in H^{\infty}(G)$ has nontangential limits almost everywhere on $u(E) $ with respect to $\omega_{G}$. That is, $$f(a) = \lim_{z\rightarrow u^{-1}(a)} f\circ u(z) \hspace{.05in}
\mbox{{\em a.e.} on} \hspace{.05in} u(E)  \hspace{.05in}\mbox{ with
respect to }\hspace{.05in}  \omega_{G}.$$ So, if $\mu$ is such that $\mu \ll \omega_{G}$ on on $u(E)$, then each $f\in H^{\infty}(G)$ has nontangential limits on $u(E)$ almost everywhere with respect to $\mu$.

The Main Result
===============

In this section, we introduce our main result, Theorem \[t:main\]. Recall that the connectivity of a finitely connected domain is defined to be the number of the components of its complement.

\[t:main\] Let $K$ be a compact subset and let $\mu$ be a finite positive measure supported on $K$. If each of the components of $K^{\circ}$ is multi-nicely connected and the harmonic measures of the components of $K^{\circ}$ are mutually singular, then there exists a $Borel$ partition $\{ \Delta_{n}\}_{n=0}^{\infty} $ of $supp(\mu)$ such that $$A^{q}(K,\mu) = L^{q}(\mu | \Delta_{0})  \oplus A^{q}(K, \mu | \Delta_{1})
\oplus ... \oplus A^{q}(K, \mu | \Delta_{n}) \oplus ...$$ and for each $n \geq 1$, if $U_{n} $ denotes $ \nabla A^{q}(K,
\mu|\Delta_{n})$, then

1\) $\overline{U}_{n} \supset  \Delta_{n} $ and $A^{q}(K, \mu | \Delta_{n}) 