Eliminating set-theoretic assumptions in finitely additive necessity examples

Construct examples analogous to Example 251212 that demonstrate the necessity of finitely additive measures in the characterization of testability, without invoking the assumption that there is no diffuse probability measure on the power set of the underlying space.

Background

Example 251212 shows that, even for a singleton alternative, considering finitely additive measures in the weak-* convex closure can change total variation distances and testability conclusions compared to restricting to countably additive measures. This construction relies on the set-theoretic assumption that there is no diffuse probability measure on the power set (consistent with ZFC under the continuum hypothesis).

The authors ask whether similar phenomena can be exhibited without relying on such assumptions, which would strengthen the necessity of finitely additive measures under standard axioms and remove dependence on additional set-theoretic hypotheses.