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.

翻译：我们重新审视假设检验中的一个基本问题：给定两个概率测度集合 $\mathcal{P}$ 和 $\mathcal{Q}$，何时存在一个非平凡的（即严格无偏的）检验用于区分 $\mathcal{P}$ 与 $\mathcal{Q}$？Le~Cam 证明，当 $\mathcal{P}$ 和 $\mathcal{Q}$ 具有共同的控制测度时，存在一个检验其功效超过其水平至少 $\varepsilon$，当且仅当 $\mathcal{P}$ 和 $\mathcal{Q}$ 的凸包在全变差距离上被至少 $\varepsilon$ 分离。控制测度的要求在非参数统计中经常无法满足。在一段附注中，Le~Cam 描述了一种处理更一般情形的方法，但他没有给出一个形式化的定理。本工作完成了 Le~Cam 的研究计划，提出了一个匹配的关于可检验性的充分必要条件：为了使上述定理在无假设条件下成立，必须在有界有限可加测度空间中取 $\mathcal{P}$ 和 $\mathcal{Q}$ 的凸包的闭包。我们提供了简单的阐释性例子，并详细讨论了关于紧致性和可达性的各种微妙的测度论与拓扑学要点。