## 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

$$\underset{P\in \mathrm{\Gamma}(\mathrm{\mu},\mathrm{\nu})}{min}(1-P(E))=\underset{A\in \mathcal{G}}{max}\phantom{\rule{0.1667em}{0ex}}|\mathrm{\mu}(A)-\mathrm{\nu}(A)|$$

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

$${\mathrm{\nu}}_{0}=\mathrm{\nu}\phantom{\rule{2.5pt}{0ex}}\text{on}\phantom{\rule{2.5pt}{0ex}}{\mathcal{G}}_{0}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{P\in \mathrm{\Gamma}(\mathrm{\mu},{\mathrm{\nu}}_{0})}{min}(1-P(E))=\underset{A\in {\mathcal{G}}_{0}}{max}\phantom{\rule{0.1667em}{0ex}}|\mathrm{\mu}(A)-\mathrm{\nu}(A)|.$$

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.

## Acknowledgments

We are grateful to an anonymous referee for many comments and remarks which improved this paper.

## Citation

Luca Pratelli. Pietro Rigo. "Some duality results for equivalence couplings and total variation." Electron. Commun. Probab. 29 1 - 12, 2024. https://doi.org/10.1214/24-ECP586

## Information