Recovering measures from approximate values on balls

2015/10/09 by Buet, Blanche, Leonardi, Gian Paolo
#28A78 #28C15 #49Q15 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1510.02793

Abstract

In a metric space (X,d) we reconstruct an approximation of a Borel measure μ starting from a premeasure q defined on the collection of closed balls, and such that q approximates the values of μ on these balls. More precisely, under a geometric assumption on the distance ensuring a Besicovitch covering property, and provided that there exists a Borel measure on X satisfying an asymptotic doubling-type condition, we show that a suitable packing construction produces a measure μ^q which is equivalent to μ. Moreover we show the stability of this process with respect to the accuracy of the initial approximation. We also investigate the case of signed measures.

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