Measure-dependent failure of the ball property on nonrigid product spaces

Establish that, for countable products of Polish spaces lacking Hilbertian geometry or the rigidity of a vector-space structure, there always exists a Borel probability measure whose centered metric outer measure is strictly larger than its cylinder outer measure on some Borel set, thereby preventing a purely geometric or topological characterization of the ball property for the space as a whole.

Background

The paper studies when the collection of open or closed metric balls uniquely determines Borel probability measures. For countable products of Polish spaces, cylinder sets determine measures through finite-dimensional marginals, whereas metric balls generate a potentially different outer-measure approximation. Theorem 3.1 characterizes the ball property in terms of uniform bounds on projection covering numbers, showing that measure-theoretic behavior and the geometry of finite-dimensional projections jointly control whether balls determine measures.

The unresolved conjecture concerns product spaces without the geometric rigidity supplied by Hilbert embeddings or linear vector-space structure. It asserts that in such settings the failure of the ball property should be intrinsically measure-dependent: some Borel probability measure should exhibit a strict decoupling between centered metric and cylinder outer measures. Such a result would rule out a characterization based solely on the underlying geometry or topology of the space.