A refinement of the Fatou-Naim-Doob theorem

We introduce the notion of harmonic thin sets and establish a refinement of the Fatou-Naim-Doob Theorem in the axiomatic system of Brelot, with an added assumptiom which is fufilled for the classical Laplace operator in R-n. We verify this assumption for the Poisson-Szego integrals on the unit ball in C-n as well as the Weinstein equation on a halfspace in R-n. Thus, a stronger version of the Fatou-Naim-Doob Theorem is given in those cases. The assumption is expected to be verified for a large class of second order elliptic differential operators. The application of our result to sets of determination (for harmonic functions), introduced by Beurling [2], Dahlberg [9], Bonsall [3], and Hayman and Lyons [16], will follow in a separate paper.