Recovering measures from approximate values on balls

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 ${\hatμ}^{q}$ 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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