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

Mean Rational Approximation for Compact Subsets with Thin Boundaries