Let $\mu,\nu$ probability measures on two compact sets respectively $X,Y\subseteq\mathbb{R}^n$.
Let $\Pi(\mu,\nu)$ be the space of measures on $X\times Y$ whose first and second marginals are $\mu,\nu$. Then let $\gamma_k,\gamma\in\Pi(\mu,\nu)$.
We say that $\gamma_k$ weakly converges to $\gamma$ if for all $g\in C^0(X\times Y)$ $$ \int g d\gamma_k \rightarrow_k \int g d\gamma $$
My question is: can we make sure there exists a topology that induces this convergence?