Why it is a question
THEORY-156’s note of 2026-10-29 re-derives Blackwell’s converse in three steps: the convex order of the standard measures, a martingale coupling of them, and a garbling built from the coupling. Only the last step uses the outcome space: it composes the coupling’s kernel with the conditional law of the simulated experiment’s outcome given its posterior, and that law exists as a kernel when the space is standard Borel. Blackwell (1953) claims the equivalence for any measurable space, resting on Theorem 2 of his 1951 paper, which the record holds only as quoted. Whether a Markov kernel always exists without that regularity, or only a randomisation in Le Cam’s operator sense, is open here.
What would answer it
Blackwell (1951)‘s proof of Theorem 2 read and checked, or a treatment of randomisations that settles when they are given by kernels, such as Torgersen’s book (LIT-778) or Le Cam’s. A counterexample would also answer it: two experiments on finitely many states, the second on a space that is not standard Borel, where the first does at least as well in every bounded problem and no Markov kernel turns its outcome into the second’s. Restatements in surveys do not count.