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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Best filters for the general Fatou boundary limit theorem

Authors: Jürgen Bliedtner and Peter A. Loeb
Journal: Proc. Amer. Math. Soc. 123 (1995), 459-463
MSC: Primary 31B25; Secondary 31D05
MathSciNet review: 1219720
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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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Keywords: Boundary limits, Fatou Theorem, zero sets
Article copyright: © Copyright 1995 American Mathematical Society