  In 1991, J. Thomson obtained a celebrated decomposition theorem for
$P^t(\mu),$ the closed subspace of $L^t(\mu)$ spanned by the analytic
polynomials, when $1 \le t < \i.$ In 2008, J. Brennan \cite{b08} generalized
Thomson's theorem to $R^t(K, \mu),$ the closed subspace of $L^t(\mu)$ spanned
by the rational functions with poles off a compact subset $K$ containing the
support of $\mu,$ when the diameters of the components of $\mathbb C\setminus
K$ are bounded below. We extend the above decomposition theorems for $R^t(K,
\mu)$ when the boundary of $K$ is not too wild.
