  For $1 \le t < \infty ,$ a compact subset $K$ of the complex plane $\mathbb
C,$ 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
$\Omega$ be a connected component of the set of analytic bounded point
evaluations for $R^t(K, \mu)$. In this paper, we examine the behavior of the
reproducing kernel of $R^t(K, \mu)$ near the boundary $\partial \Omega \cap
\mathbb T$, assuming that $\mu (\partial \Omega \cap \mathbb T ) > 0$, where
$\mathbb T$ is the unit circle.
