Functions in \(R^2(E)\) and points of the fine interior (Q1093856)

Let \(E\subset {\mathbb{C}}\) be a set that is compact in the usual topology. Let \(m\) denote 2-dimensional Lebesgue measure. We denote by \(R_0(E)\) the algebra of rational functions with poles off \(E\). For \(p\geq 1\), let \(L^p(E)=L^p(E,dm)\). The closure of \(R_ 0(E)\) in \(L^ p(E)\) will be denoted by \(R^p(E)\). In this paper we study the behavior of functions in \(R^2(E)\) at points of the fine interior of \(E\). We prove that if \(U\subset E\) is a finely open set of bounded point evaluations for \(R^2(E)\), then there is a finely open set \(V\subset U\) such that each \(x\in V\) is a bounded point derivation of all orders for \(R^2(E)\). We also prove that if \(R^2(E)\neq L^2(E)\), there is a subset \(S\subset E\) having positive measure such that if \(x\in S\) each function in \(\cup_{p>2}R^p(E)\) is approximately continuous at \(x\). Moreover, this approximate continuity is uniform on the unit ball of a normed linear space that contains \(\cup_{p>2}R^p(E)\).