Some duality results for equivalence couplings and total variation

Abstract

Let $(\Omega,\mathcal{F})$ be a standard Borel space and $\mathcal{P}(\mathcal{F})$ the collection of all probability measures on $\mathcal{F}$. Let $E\subset\Omega\times\Omega$ be a measurable equivalence relation, that is, $E\in\mathcal{F}\otimes\mathcal{F}$ and the relation on $\Omega$ defined as $x\sim y$ $\Leftrightarrow$ $(x,y)\in E$ is reflexive, symmetric and transitive. It is shown that there are two $\sigma$-fields $\mathcal{G}_0$ and $\mathcal{G}_1$ on $\Omega$ such that, for all $\mu,\,\nu\in\mathcal{P}(\mathcal{F})$, $$\inf_{P\in\Gamma(\mu,\nu)}(1-P(E))=\norm{\mu-\nu}_{\mathcal{G}_1}\quad\text{and}\quad\min_{P\in\Gamma(\mu,\nu_0)}(1-P(E))=\norm{\mu-\nu}_{\mathcal{G}_0}.$$ Here, $\nu_0\in\mathcal{P}(\mathcal{F})$ is a suitable probability measure satisfying $\nu_0=\nu$ on $\mathcal{G}_0$. Moreover, $\mathcal{G}_0\subset\mathcal{F}$ while $\mathcal{G}_1\subset\widehat{\mathcal{F}}$, where $\widehat{\mathcal{F}}$ is the universally measurable $\sigma$-field with respect to $\mathcal{F}$. However, for all $\mu,\,\nu\in\mathcal{P}(\mathcal{F})$, there is a $\sigma$-field $\mathcal{G}(\mu,\nu)\subset\mathcal{F}$ such that $$\inf_{P\in\Gamma(\mu,\nu)}(1-P(E))=\norm{\mu-\nu}_{\mathcal{G}(\mu,\nu)}.$$