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

Author: Morters P.

Source: Advances in Applied Mathematics, Volume 21, Number 1, July 1998 , pp. 146-179(34)

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Abstract:

Answering a question by Bedford and Fisher, we show that for the circular and one-sided average densities of a Radon measure mu on the line with positive lower and finite upper alpha-densities, the following relations hold mu-almost everywhere:andWe 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 mu-almost every point x, the formulaholds for every tangent measure distribution P of mu at x and all Borel functions G: M(R) x R -< [0, [infinity]). Here Tunu is the measure defined by Tunu(E) = nu(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 alpha-self-similar random measures in the sense of Zahle. Copyright 1998 Academic Press.

Language: English

Document Type: Research article

Affiliations: Fachbereich Mathematik, Universitat Kaiserslautern, Kaiserslautern, 67663, Germany

Publication date: 1998-07-01