Some duality results for equivalence couplings and total variation

Let $(\mathrm{\Omega },\mathcal{F})$ be a standard Borel space and $\mathcal{P}(\mathcal{F})$ the collection of all probability measures on $\mathcal{F}$. Let $E\subset \mathrm{\Omega }\times \mathrm{\Omega }$ be a measurable equivalence relation, that is, $E\in \mathcal{F}\otimes \mathcal{F}$ and the relation on $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ such that, for all $\mu ,\nu \in \mathcal{P}(\mathcal{F})$, $$\underset{P\in \mathrm{\Gamma }(\mu ,\nu )}{\inf }(1-P(E))=\text{norm}{\mu -\nu }_{{\mathcal{G}}_1}\text{and}\underset{P\in \mathrm{\Gamma }(\mu ,{\nu }_0)}{\min }(1-P(E))=\text{norm}{\mu -\nu }_{{\mathcal{G}}_0}\mathrm{.}$$ 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 $$\underset{P\in \mathrm{\Gamma }(\mu ,\nu )}{\inf }(1-P(E))=\text{norm}{\mu -\nu }_{\mathcal{G}(\mu ,\nu )}\mathrm{.}$$