a component of $\partial \Omega_{i}$. Clearly, $G_{i}$ is also nicely connected. Now let $\Omega$ be the union of those $G_{i}$’s for which $G_{i} \cap G_{j} = \emptyset$ (different $G_{i}$’s are either disjoint or one contains other). Set $ G= \Omega \bigcup  G_{j}$. Then each component of $G$ is equal to some $G_{i}$. Let $\{G_{i_{k}}\}$ be the collection of all the components of $G$. Then our hypothesis on $K$ implies that the harmonic measures of the components of $G$ are mutually singular. It follows from [@davi] that $A(G)$ is a Dirichlet algebra on $\partial G$. Hence, every point in $\partial G$ is a peak point for $A(G)$ and every trivial Gleason part of $A(G)$ consists of a single point. Therefore, $\{G_{i_{k}}\}$ is the collection of all the nontrivial Gleason parts of $A(G)$.

Let $\eta \perp A(G)$. By the [*Abstract F. and M. Riesz Theorem*]{} [@ga], $\eta = \sum_{m\geq 0} \eta_{m}$, where each $\eta_{m}
\perp A(G)$, $\eta_{m} \ll v_{m}$ for a representing measure $ v_{m}$ at some point $a_{m}$ in $\overline{G}$, $v_{m}$’s are mutually singular. Let $a \in \partial G$. Then $a$ is a peak point. Let $f\in A(G)$ be a peak function for $a$. Then $f^{n}(z) \rightarrow \chi_{\{a\}}$ pointwise, and thus 