Let,$\mu $ and $\nu$ be two probability measures on $\Omega$ such that $|\Omega| < \infty$. let $(X,Y)$ be an optimal coupling. How to prove that the optimal coupling is not unique , by finding a counter example?
Optimal coupling of two measures
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probability
measure-theory
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1Are you looking to show that there exist $\mu$ and $\nu$ such that there are two optimal couplings, or that for every $\mu$ and $\nu$, there are two optimal couplings? – 2011-11-02
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0@Aaron I interpreted it the first way. For instance, if $\mu=\nu$ there is only one optimal coupling, the one with support on the diagonal. – 2011-11-02