we are unable to offer any other proof that is less involved with the theory of uniform algebra. Actually, due to the nature of this problem, we believe the rational approximation theory is the right tool in study this type of problems.

Now we would like to point out the relation between this paper and some other related papers. This paper is the sequel of the author’s work [@qiu1; @comm]. Thomson’s paper consist of two parts of important results. One is to give a sufficient and necessary condition on when $\nabla P^{q}(\mu)$ is not empty and another one is to have a structure theorem for $P^{q}(\mu)$. In [@qiu1], we only study the problem of when $\nabla A^{q}(K,\mu)$ ($\nabla R^{q}(K,\mu)$) is not empty and we show it is empty if and only if $A(K)$ is dense in $L^{q}(\mu)$. In [@comm], our effort was primarily to establish the result that is a part of 4) and 5) of Theorem \[t:main\] in this paper and to solve a problem in [@ce]. In this paper, our effort is to establish a full version of Thomson’s theorem for $A^{q}(K,\mu)$ ($R^{q}(K,\mu)$) and this paper is based on [@qiu1; @comm]. The readers may notice that our theorem 