A Strong Duality Principle for Equivalence Couplings and Total Variation

Abstract: We introduce and study a notion of duality for two classes of optimization problems commonly occurring in probability theory. That is, on an abstract measurable space $(\Omega,\mathcal{F})$, we consider pairs $(E,\mathcal{G})$ where $E$ is an equivalence relation on $\Omega$ and $\mathcal{G}$ is a sub-$\sigma$-algebra of $\mathcal{G}$; we say that $(E,\mathcal{F})$ satisfies "strong duality" if $E$ is $(\mathcal{F}\otimes\mathcal{F})$-measurable and if for all probability measures $\mathbb{P},\mathbb{P}'$ on $(\Omega,\mathcal{F})$ we have $$\max_{A\in\mathcal{G}}\vert \mathbb{P}(A)-\mathbb{P}'(A)\vert = \min_{\tilde{\mathbb{P}}\in\Pi(\mathbb{P},\mathbb{P}')}(1-\tilde{\mathbb{P}}(E)),$$ where $\Pi(\mathbb{P},\mathbb{P}')$ denotes the space of couplings of $\mathbb{P}$ and $\mathbb{P}'$, and where "max" and "min" assert that the supremum and infimum are in fact achieved. The results herein give wide sufficient conditions for strong duality to hold, thereby extending a form of Kantorovich duality to a class of cost functions which are irregular from the point of view of topology but regular from the point of view of descriptive set theory. The given conditions recover or strengthen classical results, and they have novel consequences in stochastic calculus, point process theory, and random sequence simulation.