\subset C(K)$ if there is a function in $\cal B$ such that $f(a) = 1$ at $z = a$ and $|f(z)| < 1$ for $z \neq a$. [*Pure and irreducible spaces.*]{} The space $A^{q}(K,\mu)$ is called [*pure*]{} if there is no Borel subset $\Delta$ of $supp(\mu)$ such that the restriction of $A(K)$ on $\Delta$ is dense in $L^{q}(\mu|\Delta)$. An observation is that for any $A^{q}(K,\mu)$, there is a Berel partition $\{\Delta_{0},
\Delta_{1}\}$ of the support of $\mu$ such that $A^{q}(K,\mu|\Delta_{1})$ is pure and $$A^{q}(K,\mu) = L^{q}(\mu|\Delta_{0}) \oplus A^{q}(K,\mu|\Delta_{1}).$$ The space $A^{q}(K,\mu)$ is said to be [*irreducible*]{} if it contains no nontrivial characteristic functions. So an irreducible space must be pure.

[*Nontangential limits*]{}. Let $G$ be a bounded domain that is conformally equivalent to a circular domain $W$ in the plane and let $u$ be a conformal map from $W$ onto $\Omega$. Then $u$ has well-defined boundary values on $\partial
W$, which are equal to the nontangential limits of $u$ for almost every point on $\partial W$ with respect to $\omega_{W}$ (the harmonic measure of $W$). We still use $u$ to denote the boundary value function.

Now, if $E$ is a Borel subset of $\partial W$ such that $u$ is 1-1 