  A classical question about a metric space is whether Borel measures on the
space are determined by their values on balls. We show that for any given
measure this property is stable under Gromov-Wasserstein convergence of metric
measure spaces. We then use this result to show that suitable bounded subspaces
of the space of persistence diagrams have the property that any Borel measure
is determined by its values on balls. This justifies the use of empirical ball
volumes for statistical testing in topological data analysis (TDA). Our
intended application is to deploy the statistical foundations of van Delft and
Blumberg (2023) for time series of random geometric objects in the context of
TDA invariants, specifically persistent homology and zigzag persistence.
