Best filters for the general Fatou boundary limit theorem

Bliedtner, Jürgen; Loeb, Peter A.

doi:10.1090/S0002-9939-1995-1219720-2

Best filters for the general Fatou boundary limit theorem
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by Jürgen Bliedtner and Peter A. Loeb

Proc. Amer. Math. Soc. 123 (1995), 459-463

DOI: https://doi.org/10.1090/S0002-9939-1995-1219720-2

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Abstract:

Given a suitable normalization, there is a "best" family of filters for which the Fatou Boundary Limit Theorem holds. The normalization assigns to each positive harmonic function a set of boundary points at which that function must vanish. Known limits, such as those provided by the Lebesgue Differentiation Theorem, are used to force consistency in this assignment. The zero sets, in turn, are used in constructing the coarsest filters which produce those limits almost everywhere. This procedure is formulated in terms of a general potential theoretic setting and a general reference measure. The result is new, however, even for harmonic functions on the unit disk.

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