Validity of Nielsen’s Theorem 4

Determine whether Nielsen’s Theorem 4 is true: whether there exist uncountably many probability functions on the Borel subsets of Cantor space satisfying countable additivity on conditional hitting times but not the approximation property.

Background

Nielsen’s Theorem 4 claims the existence of uncountably many probability functions with CHT but without the approximation property. Its published proof relies on nontrivial finitely additive extensions of a countably additive Borel probability that agree on all open sets.

The paper proves that such extensions cannot differ from the original countably additive probability, so the published argument fails. However, the authors do not determine whether the theorem’s standalone existence claim is true.