Calculus of Variations and Geometric Measure Theory

B. Buet - G. P. Leonardi

Recovering measures from approximate values on balls

created by leonardi on 12 Oct 2015
modified on 15 Jun 2016


Accepted Paper

Inserted: 12 oct 2015
Last Updated: 15 jun 2016

Journal: Ann. Acad. Sci. Fenn.
Volume: 41
Pages: 1--26
Year: 2016


In a metric space $(X,d)$ we reconstruct an approximation of a Borel measure $\mu$ starting from a premeasure $q$ defined on the collection of closed balls, and such that $q$ approximates the values of $\mu$ 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 $\hat\mu^{q}$ which is equivalent to $\mu$. 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.