A complete characterization of testable hypotheses

We revisit a fundamental question in hypothesis testing: given two sets of probability measures $\mathcal{P}$ and $\mathcal{Q}$, when does a nontrivial (i.e.\ strictly unbiased) test for $\mathcal{P}$ against $\mathcal{Q}$ exist? Le~Cam showed that, when $\mathcal{P}$ and $\mathcal{Q}$ have a common dominating measure, a test that has power exceeding its level by more than $\varepsilon$ exists if and only if the convex hulls of $\mathcal{P}$ and $\mathcal{Q}$ are separated in total-variation distance by more than $\varepsilon$. The requirement of a dominating measure is frequently violated in nonparametric statistics. In a passing remark, Le~Cam described an approach to address more general scenarios, but he stopped short of stating a formal theorem. This work completes Le~Cam's program, by presenting a matching necessary and sufficient condition for testability: for the aforementioned theorem to hold without assumptions, one must take the closures of the convex hulls of $\mathcal{P}$ and $\mathcal{Q}$ in the space of bounded finitely additive measures. We provide simple elucidating examples, and elaborate on various subtle measure-theoretic and topological points regarding compactness and achievability.