This paper contains two results related to the question of when a measure on a metric space is determined by its value on certain subsets. The first is that two finite positive measures on a countable metric abelian group \(G\) which agree on all balls of some fixed non-zero radius agree on \(G\). The second relates to measures on a compact metric space that agree on all intersections of pairs of balls.
"Determination of measures.." Real Anal. Exchange 28 (2) 635 - 640, 2002/2003.