Almost-sure convergence from approximation and CHT

Determine whether the approximation property and countable additivity on conditional hitting times together imply almost-sure convergence to the truth for every Borel event and every arbitrary finitely additive probability function satisfying the positivity condition.

Background

The paper refutes the claimed characterization of almost-uniform convergence by showing that the approximation property and CHT are not jointly sufficient for almost-uniform convergence. This invalidates the published proof of Nielsen’s corollary asserting that the same two conditions imply almost-sure convergence to the truth.

The authors establish the almost-sure conclusion only for the particular ultrafilter-based family containing their counterexample, where CHT alone suffices. Whether the implication holds for arbitrary finitely additive probability functions remains unresolved.