Geometric and measure-theoretic characterizations of the measure-determining property of balls in metric spaces

Abstract: The question of what structural properties drive the collection of open or closed balls to uniquely determine Borel probability measures on a metric space, the ball property, has a long history in measure theory. While various sufficient conditions have been known since the 1970s, and it was shown by Preiss and Tišer (1991) that it holds for all separable Banach spaces, Davies (1971) constructed a famous counterexample that shows it can fail on compact metric spaces. It was only recently established in van Delft (2026) that metric spaces admitting an isometric Hilbertian embedding must possess the ball property. This article provides a comprehensive structural and measure-theoretic characterization of the ball property. First, we establish a necessary and sufficient condition for the ball property to fail: the collection of balls must possess sufficient combinatorial capacity, which we define as charged dyadic shattering. Second, by characterizing the ball property via $λ$-systems, we place classical vector space results within a unified context. Building on this to understand spaces that have the topological capacity but lack linear or geometric rigidity required to force ball-annihilating measures to zero, we relate the metric outer measure to the cylinder outer measure on countable products of Polish spaces, deriving necessary and sufficient measure-theoretic conditions for the ball property to hold. This allows us to derive a structural taxonomy on the decoupling of the metric outer measure from the cylinder outer measure. This is subsequently used to deconstruct Davies' counterexample. Finally, we introduce a minimal combinatorial generalization of the ball property, the $n$th order ball property, and show that such classes can fully recover measure determination in highly pathological constructions like Davies' counterexample.