Functions inR2(E) and points of the fine interior

Edwin Wolf
1988 Pacific Journal of Mathematics  
Let E c C be a set that is compact in the usual topology. Let m denote 2-dimensional Lebesgue measure. We denote by Ro(E) the algebra of rational functions with poles off E. For p > 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 C E is a finely open set of bounded point evaluations for R 2 (E) 9 then there is a finely open set V c U
more » ... open set V c U such that each x e V is a bounded point derivation of all orders for R 2 {E). We also prove that if R 2 {E) φ L 2 {E), there is a subset S c E having positive measure such that if x e S each function in \J 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 |J p>2 R P (E).
doi:10.2140/pjm.1988.134.393 fatcat:toyoesr4hfdyjjf63gqdoyhlq4