Let \(\mu\) be a finite positive Borel measure with compact support in the complex plane \(\mathbb C\). For \(1\leq p<+\infty\), let \(H^p(\mu)\) be the closed linear span in \(L^p(\mu)\) of all polynomials. Let \(X\) be a compact subset of \(\mathbb C\) such that the support of \(\mu\) lies in (and possibly is not equal to) \(X\), and let \(R^p(X,\mu)\) be the closed linear span in \(L^p(\mu)\) of all rational functions with poles outside \(X\). In 1991, \textit{J.~E.~Thomson} [Ann. Math. 133, No. 2, 477--507 (1991; Zbl 0736.41008)] showed that \(H^p(\mu)\) can be decomposed as \(L^p(\mu_0)\oplus H^p(\mu_1)\oplus H^p(\mu_2)\oplus\cdots\), where each \(\mu_j\) is the restriction of \(\mu\) to \(\Delta_j\) and \(\{\Delta_j\}_{j\geq0}\) is a Borel partition of the support of \(\mu\). In the paper under review the author extends, as far as possible, the last result to \(R^p(\mu)\), including, as a particular case, Thomson's theorem on \(H^p(\mu)\). In particular, relying heavily on a recent work of \textit{X.~Tolsa} [Acta. Math. 190, 105--149 (2003; Zbl 1060.30031)] on the semiadditivity of the analytic capacity, the next result is shown: Assume that the diameters of the components of \({\mathbb C}\setminus X\) are bounded away from zero. Then there exist a Borel partition \(\{\Delta_j\}_{j\geq0}\) of the support of \(\mu\) and matching compacta \(\{X_j\}_{j\geq0}\) with \(\Delta_j\subset X_j\) (\(j\geq0\)) such that if \(\mu_j=\mu|_{\Delta_j}\), then \(R^p(X,\mu)=L^p(\mu_0)\oplus R^p(X_1,\mu_1)\oplus R^p(X_2,\mu_2)\oplus\cdots\).