Minimax risk nonattainment outside dominated settings

Construct a pair of nonempty sets of countably additive probability measures P and Q on a measurable space (Ω, F) that do not admit a common dominating measure such that the minimax risk R(P, Q) is not achieved by any bounded measurable test φ; equivalently, show that no minimax optimal test exists for some nondominated hypotheses.

Background

Theorem T_kraft establishes a necessary and sufficient condition for the existence of nontrivial tests and provides an exact value for the minimax risk via total variation distance between weak-* convex closures in the space of finitely additive measures. In the dominated setting, Proposition 1 shows that a minimax optimal test exists. Outside the dominated setting, the authors indicate that achievability of the minimax risk by a test may fail, but they currently lack a confirming example, leaving open the existence of a nondominated problem where no minimax optimal test exists.

Resolving this would clarify the extent to which Theorem T_kraft’s characterization is operational in terms of implementable tests, and would delineate the boundary between settings with and without minimax optimal tests.