1. Introduction. Suppose that Ω is a region (i.e. a connected open set) in ࠶n, for some fixed n ≥ 1. We define (Γ, μ) to be a Fatou pair in Ω if
(a) Γ is a continuous family of boundary curves γw in Ω, one ending at each w ∈ ∂Ω,
(b) μ is a positive finite Borel measure on ∂Ω, and
(c) the conclusion of Fatou's theorem holds with respect to Γ and μ. Let us state (a) and (c) in more detail:
(a) The map (w, t) → γw(t) is continuous, from ∂Ω × [0, 1) into Ω, and
for every w in the boundary ∂Ω of Ω.
(c) For every f ∈ H∞(Ω) (the class of all bounded holomorphic functions in Ω), the limit
exists a.e. [μ].