Point Sets of Harmonic Measure Zero

Abstract

As early as in Chapter II we referred to the possibility of extending the theory of harmonic measure. An immediate generalization, based on observations made at that time, results if we continue to assume that the given reference region is bounded by a finite set of Jordan arcs Γ, but take the point set α to be an arbitrary set of points on Γ instead of a set of boundary arcs. If the universal covering surface G∞ of G is once more mapped conformally onto the unit disk K, then the set α is in one-to-one correspondence with a certain set α _x of points on the circumference |x| — 1. Provided this set is measurable, we call αharmonically measurable with respect to G. Using the theory of the Poisson integral in the extended form developed by Fatou [1] on the basis of Lebesgue measure theory, as will be discussed later (Chapter VII), one could now define the harmonic measure ω(z, α, G) in exact analogy with the construction in Chapter II. However, it would lead us too far astray to go into such extensions here. We are interested principally in sets of harmonic measure zero; under the above hypotheses, such sets can be defined as follows:

The point set α is of harmonic measure zero with respect toGif the image point setα_x has (linear) measure zero, i.e., provided α_x can be covered by a sequence of arcs of arbitrarily small total length.