Infinite-family criterion for pivotal exact e-variables

Let $\mathcal{P}$ and $\mathcal{Q}$ be families of probability measures on $\mathfrak{X}$. Suppose there exist $P_0\in\mathcal{P}$ and $Q_0\in\mathcal{Q}$ such that $P\ll P_0$ for every $P\in\mathcal{P}$ and $Q\ll Q_0$ for every $Q\in\mathcal{Q}$. Assume that $(\mathcal{P},\mathcal{Q})$ is jointly atomless. Infinite-family pivotal e-variable conjecture. There exists a pivotal and exact e-variable $X$ satisfying

$$\inf_{Q\in\mathcal{Q}}\mathbb{E}^{Q}[\log X]>0$$

if and only if

$$0\notin\overline{\,\overline{\operatorname{Span}}\mathcal{P}+\overline{\operatorname{Conv}}\mathcal{Q}\,},$$

where the outer closure is taken with respect to total variation distance. This is proposed as a strengthening of the paper’s infinite-family theorem and as a characterization of when a nontrivial pivotal exact e-variable exists; the source leaves it open and suggests that simultaneous transport between infinite collections may be relevant.