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Fine and nontangential convergence on an NTA domain

The recent article by Jerison and Kenig on "Boundary behaviour of harmonic functions in nontangentially accessible domains" did not consider the relation between fine limits and nontangential limits. The results in this direction obtained by Hunt & Wheeden (5) for Lipschitz domains are extended here to NTA domains. In (6), Jerison and Kenig define a class of bounded domains D (the so-called bounded NTA domains) for which the topological boundary is the Martin boundary of D (see (6, Theorem 5.9)). Further, they show that every positive harmonic function u has a nontangential limit dcc-a.e. where « is the harmonic measure of a fixed point x0 G D. This limit is identified with dp/dec if p is a positive measure on dD that