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 µˆq 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.