Attainability under exact stopping-set equality

Determine whether the attainability results for stopping sets and increasing sequences of stopping sets remain valid when attainability requires the explored stopping set to satisfy the exact equality \(\widetilde Z_{\tau}(\Phi)=Z(\Phi)\), rather than only equality up to \(\mu\)-null sets together with equality of the observed point configurations.

Background

Definition 4.1 defines attainability using the weaker requirement $\mu(\widetilde Z_{\tau}(\Phi)\triangle Z(\Phi))=0$, together with $\Phi\cap\widetilde Z_{\tau}(\Phi)=\Phi\cap Z(\Phi)$ almost surely. The results in Theorems 1.3 and 1.4 establish attainability in this sense for, respectively, stopping sets revealing at most one point and increasing sequences of stopping sets revealing at most one additional point at each step.

Remark 4.2 observes that requiring the stronger pointwise equality $\widetilde Z_{\tau}(\Phi)=Z(\Phi)$ would be a genuinely stronger formulation, and explicitly leaves unresolved whether the attainability results continue to hold under that requirement.