On the riesz charge of the lower envelope of δ-subharmonic functions

Abstract

By potential theoretic methods involving the Cartan fine topology a recent result by two of the authors is extended as follows: The Riesz charge of the lower envelope of a family of 3 or more δ-subharmonic functions (no longer supposed continuous) in the plane equals the infimum of the charges of the lower envelopes of all pairs of functions from the family. As a key to this it is shown in two different ways that the (fine) harmonic measures of any 3 pairwise disjoint finely open planar sets have Borel supports with empty intersection. One proof of this uses the Jordan curve theorem and the fact that the set of inaccessible points of the fine boundary of a fine domain is Borel and has zero harmonic measure; the other involves Carleman-Tsuji type estimates together with a fine topology version of a recent result of P. Jones and T. Wolff on harmonic measure and Hausdorff dimension.