On the behavior of meromorphic functions at the ideal boundary of a Riemann surface

Abstract

In a former work the author established an analog of a classical theorem of Painlevé in the context of an arbitrary resolutive compactification of a Riemann surface. In the same setting, a refinement of the argument used in the above yields an elementary proof of a theorem of Riesz-Luzin-Privaloff type: If a meromorphic function f f tends to zero at each point of a subset E E of the ideal boundary and E E has positive harmonic measure, then f ≡ 0 f \equiv 0 on R R . The well-known inclusion relations U H B ⊂ O M B ∗ {U_{HB}} \subset {\mathcal {O}_{M{B^{\ast }}}} and U H D ∼ O M D ∗ {U_{HD}} \sim {\mathcal {O}_{M{D^{\ast }}}} , are then established from the point of view of the resolutivity of the Wiener and Royden compactification respectively.