Let $\mu$ and $\nu$ two probabilites on $\mathbb{R}^{d}$
Let $\Pi(\mu,\nu)$ is the subset of the probability measure $\pi$ such as $$ \pi (A\times Y) =\mu(A) \text{ and } \pi(X\times B)=\nu(B) $$
Now suppose that the support of $\mu$ and $\nu$ is compact. I would like to deduce that it exists a compact $K$ such that $$ \forall \gamma \in \Pi(\mu,\nu), \text{supp}(\gamma) \subseteq K $$
I think $K$ could be $ \text{supp}(\mu) \times \text{supp}(\nu)$. Do you think it is true ? How to prove it ?
Thanks for your help and regards.
Well, I think I found a proof using this definition of the support
Let $x \in \text{supp}(\gamma)$ which we can write $(x_{1},x_{2})$ now we would like to show that $x_{1} \in \text{supp}(\mu)$ and $x_{2} \in \text{supp}(\nu)$ .
Let $B_{i}$ a open set containing $x_{i}$. As $B_{1} \times B_{2}$ is a open set containing $x$ we can deduce $\gamma(B_{1} \times B_{2}) >0$
As $B_{1} \times B_{2} \subseteq R^{d} \times B_{2}$ we have $\nu(B_{2})>0$ and for the same reason $\mu(B_{1})>0$.
We proved that $\text{supp}(\gamma) \subseteq \text{supp}(\mu) \times \text{supp}(\nu)$
Do you agree ?
I used very often the notion of support of a meausre during my project (for the university) but I didn't find any book containing general result. So if you have any general comment about my question or the support of a measure, I would feel glad to hear it.