Recovering measures from approximate values on balls

TL;DR

This paper presents a method to reconstruct an approximation of a Borel measure in a metric space from a premeasure on closed balls, ensuring stability and equivalence under certain geometric and doubling conditions.

Contribution

It introduces a packing-based construction to recover measures from approximate ball values, extending to signed measures and analyzing stability.

Findings

Reconstruction of measures from premeasures on balls under geometric assumptions.

The constructed measure is equivalent to the original measure.

The process is stable with respect to initial approximation accuracy.

Abstract

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.