Boundary values in $R^t(K,\mu )$-spaces and invariant subspaces

For $1\le t<\mathrm{\infty },$ a compact subset $K$ of the complex plane $\mathbb C,$andafinitepositivemeasure$\mu$supportedon$K,$ $R^t(K,\mu )$ denotes the closure in $L^t(\mu )$ of rational functions with poles off $K\mathrm{.}$ The paper examines the boundary values of functions in $R^t(K,\mu )$ for certain compact subset $K$ and extends the work of Aleman, Richter, and Sundberg on nontangential limits for the closure in $L^t(\mu )$ of analytic polynomials (Theorem A and Theorem C in \cite{ars}). We show that the Cauchy transform of an annihilating measure has some continuity properties in the sense of capacitary density. This allows us to extend Aleman, Richter, and Sundberg’s results for $R^t(K,\mu )$ and provide alternative short proofs of their theorems as special cases.