Nontangential limits in 𝒫t(μ)-spaces and the index of invariant subgroups

Let be a finite positive measure on the closed disk D in the complex plane, let 1 t < , and let P t () denote the closure of the analytic polynomials in L t (). We suppose that D is the set of analytic bounded point evaluations for P t (), and that P t () contains no nontrivial characteristic functions. It is then known that the restriction of to D must be of the form h|dz|. We prove that every function f P t () has nontangential limits at h|dz|-almost every point of D, and the resulting boundary function agrees with f as an element of L t (h|dz|).