Some duality results for equivalence couplings and total variation

Let $(\Omega,\mathcal{F})$ be a measurable space and $E\subset\Omega\times\Omega$. Suppose that $E\in\mathcal{F}\otimes\mathcal{F}$ and the relation on $\Omega$ defined as $x\sim y$ $\Leftrightarrow$ $(x,y)\in E$ is reflexive, symmetric and transitive. Following \cite{JAFFE}, say that $E$ is strongly dualizable if there is a sub-$\sigma$-field $\mathcal{G}\subset\mathcal{F}$ such that $$\min_{P\in\Gamma(\mu,\nu)}(1-P(E))=\max_{A\in\mathcal{G}}\,\abs{\mu(A)-\nu(A)}$$ for all probabilities $\mu$ and $\nu$ 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-$\sigma$-field of $\mathcal{F}$. One result is that, for all probabilities $\mu$ and $\nu$ on $\mathcal{F}$, there is a probability $\nu_0$ on $\mathcal{F}$ such that \begin{gather*} \nu_0=\nu\text{ on }\mathcal{G}_0\quad\text{and}\quad\min_{P\in\Gamma(\mu,\nu_0)}(1-P(E))=\max_{A\in\mathcal{G}_0}\,\abs{\mu(A)-\nu(A)}. \end{gather*} In the other results, $(\Omega,\mathcal{F})$ is a standard Borel space and the min over $\Gamma(\mu,\nu)$ is replaced by the inf over $\Gamma(\mu,\nu)$ in the definition of strong duality. Then, $E$ is strongly dualizable provided $\mathcal{G}$ is allowed to depend on $(\mu,\nu)$ or it is taken to be the universally measurable version of the $E$-invariant $\sigma$-field.