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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!

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No: try both marginals equal to the uniform distribution on a set of size $n$, then the largest probability of the complement of the diagonal is $1$ while it is $1-1/n$ for the product distribution.

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    It is 1-1/n. $ $2012-12-13