(Theorem \[t:main\]) not only completely covers Thomson’s theorem, it also has more important consequences, such as 4) and 5) (which are not in [@jm] and are important facts. One needs to have them when applying it to operator theory. For example, see [@equi] and [@quas]). In [@ce], Conway and Elias studied the same problem as that in this paper. However, since their theorem is based on the assumption that $K$ is the closure of $\nabla R^{q}(K,\mu)$ and $R^{q}(K,\mu)$ is pure, so their result does not cover Thomson’s theorem. For a given measure $\mu$, one can not tell when their conditions are satisfied. In contrast, our paper deals with arbitrary measures just as [@jm] does.

In 1972, D. Sarason [@sar] established a structure theorem for $P^{\infty}(\mu)$, the weak star closure of polynomials in $L^{\infty}(\mu)$, which has a similar form to that of our theorem.

Preliminaries
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For a compact subset $K$ in the complex plane [**C**]{}. Let $C(K)$ denote the algebra of continuous functions on $K$. For an open subset $G$ in the sphere [**C**]{}$_{\infty}$ whose boundary does not contains $\infty$, let $A(G)$ be the closed subalgebra of $C(\overline{G})$ that consists of functions continuous on $\overline{G}$ and analytic on $G$. Notice 