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 ℝn. We verify this assumption for the Poisson–Szegö integrals on the unit ball in ℂn as well as the Weinstein equation on a halfspace in ℝn. 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.