Given two probability measures on the same measurable space $\Omega$, is their product measure the coupling with the biggest probability of $\{(x, y): \forall x, y \in\Omega, x \neq y\}$?
If not, what is the coupling with the biggest probability of $\{(x, y): \forall x, y \in\Omega, x \neq y\}$?
How is the amount of coupling quantified for the product measure? Thanks and regards!