---
abstract: 'Let $K$ be a compact subset in the complex plane and let $A(K)$ be the uniform closure of the functions continuous on $K$ and analytic on $K^{\circ}$. Let $\mu$ be a positive finite measure with its support contained in $K$. For $1 \leq q < \infty$, let $A^{q}(K,\mu)$ denote the closure of $A(K)$ in $L^{q}(\mu)$. The aim of this work is to study the structure of the space $A^{q}(K,\mu)$. We seek a necessary and sufficient condition on $K$ so that a Thomson-type structure theorem for $A^{q}(K,\mu)$ can be established. Our theorem deduces J. Thomson’s structure theorem for $P^{q}(\mu)$, the closure of polynomials in $L^{q}(\mu)$, as the special case when $K$ is a closed disk containing the support of $\mu$.[^1]'
author:
- |
    [Zhijian Qiu ]{}\
    \
    [*Department of mathematics, Southwestern Univ of Finance and Economics*]{}\
    [*Chengdu 610072, China, E-mail:qiu@swufe.edu.cn* ]{}
title: 'The Structure of the Closure of the Rational Functions in $L^{q}$($\mu$)'
---

46E30 30H05, 30E10, 46E15

Introduction {#introduction .unnumbered}
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Let $1 \leq q < \infty$ and let $\mu$ be a positive finite (regular Borel) measure with compact support in the complex plane [**C**]{}. Let $K$ be a compact subset that contains the support of $\mu$. The 