  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.
