Some duality results for equivalence couplings and total variation

Abstract

Let $(\mathrm{\Omega },\mathcal{F})$ be a measurable space and $E\subset \mathrm{\Omega }\times \mathrm{\Omega }$. Suppose that $E\in \mathcal{F}\otimes \mathcal{F}$ and the relation on Ω defined as $x\sim y$⇔ $(x,y)\in E$ is reflexive, symmetric and transitive. Following [7], say that E is strongly dualizable if there is a sub-σ-field $\mathcal{G}\subset \mathcal{F}$ such that

$$\underset{P\in \mathrm{\Gamma }(\mathrm{\mu },\mathrm{\nu })}{min}(1-P(E))=\underset{A\in \mathcal{G}}{max}|\mathrm{\mu }(A)-\mathrm{\nu }(A)|$$

for all probabilities μ and ν on $\mathcal{F}$. This paper investigates strong duality. Essentially, it is shown that E is strongly dualizable provided some mild modifications are admitted. Let ${\mathcal{G}}_0$ be the E-invariant sub-σ-field of $\mathcal{F}$. One result is that, for all probabilities μ and ν on $\mathcal{F}$, there is a probability ${\mathrm{\nu }}_0$ on $\mathcal{F}$ such that

$${\mathrm{\nu }}_0=\mathrm{\nu }\text{on}{\mathcal{G}}_0\text{and}\underset{P\in \mathrm{\Gamma }(\mathrm{\mu },{\mathrm{\nu }}_0)}{min}(1-P(E))=\underset{A\in {\mathcal{G}}_0}{max}|\mathrm{\mu }(A)-\mathrm{\nu }(A)|.$$

In the other results, $(\mathrm{\Omega },\mathcal{F})$ is a standard Borel space and the min over $\mathrm{\Gamma }(\mathrm{\mu },\mathrm{\nu })$ is replaced by the inf over $\mathrm{\Gamma }(\mathrm{\mu },\mathrm{\nu })$ in the definition of strong duality. Then, E is strongly dualizable provided $\mathcal{G}$ is allowed to depend on $(\mathrm{\mu },\mathrm{\nu })$ or it is taken to be the universally measurable version of the E-invariant σ-field.