  For $1\le t < \infty$, a compact subset $K\subset\mathbb C$, and a finite
positive measure $\mu$ supported on $K$, $R^t(K, \mu)$ denotes the closure in
$L^t(\mu)$ of rational functions with poles off $K$. Let $\text{abpe}(R^t(K,
\mu))$ denote the set of analytic bounded point evaluations. The objective of
this paper is to describe the structure of $R^t(K, \mu)$. In the work of
Thomson on describing the closure in $L^t(\mu)$ of analytic polynomials,
$P^t(\mu)$, the existence of analytic bounded point evaluations plays critical
roles, while $\text{abpe}(R^t(K, \mu))$ may be empty. We introduce the concept
of non-removable boundary $\mathcal F$ such that the removable set $\mathcal R
= K\setminus \mathcal F$ contains $\text{abpe}(R^t(K, \mu))$. Recent remarkable
developments in analytic capacity and Cauchy transform provide us the necessary
tools to describe $\mathcal F$ and obtain structural results for $R^t(K, \mu)$.
  Assume that $R^t(K, \mu)$ does not have a direct $L^t$ summand. Let
$H^\infty_{\mathcal R}(\mathcal L^2_{\mathcal R})$ be the weak$^*$ closure in
$L^\infty (\mathcal L^2_{\mathcal R})$ of the functions that are bounded
analytic off compact subsets of $\mathcal F$, where $\mathcal L^2_{\mathcal R}$
denotes the planar Lebesgue measure restricted to $\mathcal R$. We prove that
the identity map ($r\rightarrow r$, $r$ is a rational function with poles off
$K$) extends an isometric isomorphism and a weak$^*$ homeomorphism from $R^t(K,
\mu)\cap L^\infty(\mu )$ onto $H^\infty_{\mathcal R}(\mathcal L^2_{\mathcal R
})$.
  Consequently, we show that a decomposition theorem (Main Theorem II) of
$R^t(K, \mu)$ holds for an arbitrary compact subset $K$ and a finite positive
measure $\mu$ supported on $K$, which extends the central results regarding
$P^t(\mu)$.
