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 $x\sim y$ $\iff$ $(x,y)\in E$ 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 $\mu ,\nu \in \mathcal{P}(\mathcal{F})$, $$\inf_{P\inΓ(μ,ν)}(1-P(E))=\norm{μ-ν}_{\mathcal{G}_1}\quad\text{and}\quad\min_{P\inΓ(μ,ν_0)}(1-P(E))=\norm{μ-ν}_{\mathcal{G}_0}.$$ 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 ${\mathcal{G}}_1\subset \hat{\mathcal{F}}$, where $\hat{\mathcal{F}}$ 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 $$\inf_{P\inΓ(μ,ν)}(1-P(E))=\norm{μ-ν}_{\mathcal{G}(μ,ν)}.$$

11 pages