Symmetry Properties of Average Densities and Tangent Measure Distributions of Measures on the Line

Abstract

Answering a question by Bedford and Fisher, we show that for the circular and one-sided average densities of a Radon measure μ on the line with positive lower and finite upper α-densities, the following relations hold μ-almost everywhere: [equation] and [equation] We infer the result from a more general formula, which is proved by means of a detailed study of the structure of the measure and which involves the notion of tangent measure distributions introduced by Bandt and Graf. We show that for μ-almost every pointx, the formula [equation] holds for every tangent measure distributionPof μ atxand all Borel functionsG: M(R)×R→[0,∞). HereTuν is the measure defined byTuν(E)=ν(u+E), and M(R) is the space of Radon measures with the vague topology. By this formula, the tangent measure distributions are Palm distributions and thus define α-self-similar random measures in the sense of Zähle.