Let $X$ be a Polish metric space, and let $\mu,\nu$ be two Borel probability measures on $X$, when is the product measure the only couponing of $\mu$ and $\nu$. More formally, let $$\Gamma(\mu,\nu):=\{\pi\textrm{ Borel probability measure on }X\times X: \pi_1=\mu, \pi_2=\nu\}$$ Where $\pi_1,\pi_2$ are the marginal distributions of $\pi$ on the two factors. Are there known results providins sufficient conditions for $\Gamma$ to be a singleton.

PS pf course one trivial case where this happens is when $X$ is a singleton or when both measures are Diracs.

Let $X$ be a Polish metric space, and let $\mu,\nu$ be two Borel probability measures on $X$, when is the product measure the only coupling of $\mu$ and $\nu$. More formally, let $$\Gamma(\mu,\nu):=\{\pi\textrm{ Borel probability measure on }X\times X: \pi_1=\mu, \pi_2=\nu\}$$ Where $\pi_1,\pi_2$ are the marginal distributions of $\pi$ on the two factors. Are there known results providins sufficient conditions for $\Gamma$ to be a singleton.

PS pf course one trivial case where this happens is when $X$ is a singleton or when both measures are Diracs.