Some duality results for equivalence couplings and total variation

Let (52, F) be a measurable space and E subset of 52 x 52. Suppose that E is an element of F circle times F and the relation on 52 defined as x similar to y ? (x, y) is an element of E is reflexive, symmetric and transitive. Following [7], say that E is strongly dualizable if there is a sub-sigma-field G subset of F such that min P is an element of Gamma(mu,nu) (1- P(E)) = max|mu(A) - v(A)| A is an element of G for all probabilities mu and v on F. This paper investigates strong duality. Essentially, it is shown that E is strongly dualizable provided some mild modifications are admitted. Let G0 be the E -invariant sub-sigma-field of F. One result is that, for all probabilities mu and v on F, there is a probability v0 on F such that (1- P(E)) = max|mu(A) - v(A)|. P is an element of Gamma(mu,nu 0) A is an element of G0 v0 =von G0andmin In the other results, (52, F) is a standard Borel space and the min over Gamma(mu, v) is replaced by the inf over Gamma(mu, v) in the definition of strong duality. Then, E is strongly dualizable provided G is allowed to depend on (mu, v) or it is taken to be the universally measurable version of the E -invariant sigma-field.