Abstract
Let be a measurable space and . Suppose that and the relation on Ω defined as ⇔ is reflexive, symmetric and transitive. Following [7], say that E is strongly dualizable if there is a sub-σ-field such that
for all probabilities μ and ν on . This paper investigates strong duality. Essentially, it is shown that E is strongly dualizable provided some mild modifications are admitted. Let be the E-invariant sub-σ-field of . One result is that, for all probabilities μ and ν on , there is a probability on such that
In the other results, is a standard Borel space and the min over is replaced by the inf over in the definition of strong duality. Then, E is strongly dualizable provided is allowed to depend on 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