by D. Sarason in 1972 [@sar]). For an arbitrary compact subset $K$, the author shows in [@qiu1] that it is still true: $A(K)$ is dense in $L^{q}(\mu)$ if and only if $ A^{q}(K,\mu)$ has no $abpe$s. However, the corresponding result for $A^{q}(K,\mu)$ is not always possible even the set of $abpe$s for $A^{q}(K,\mu)$ is not empty. J. Conway and N. Elias give an example in [@ce] that shows the set of $abpe$s for $A^{q}(K,\mu)$ is a simply connected domain $U$, but $A^{q}(K,\mu)\cap L^{\infty}(\mu)$ can not be identified with $H^{\infty}(U)$ via the evaluation map.

For $R^{q}(K,\mu)$, the closure in $L^{q}(\mu)$ of $R(K)$ (which is the uniform closure of the rational functions with poles off $K$), the situation is worse since the set of $abpe$s may be empty even $R(K)$ is not dense in $L^{q}(\mu)$ (see [@bre1]).

Assuming the existence of $abpe$s for $R^{q}(K,\mu)$, Conway and Elias proved a structure-type theorem for $R^{q}(K,\mu)$ under additional conditions. But their result does not imply Thomson’s theorem. Then, can we have a structure theorem for $A^{q}(K,\mu)$ or $R^{q}(K,\mu)$ that is beyond the polynomial case and that also covers Thomson’s theorem? Prior our work, it was unknown whether such a structure theorem for $A^{q}(K,\mu)$ is possible. 