Suppose $X$ and $Y$ are Polish spaces and $\mu$ and $\nu$ are probability measures respectively on $X$ and $Y$. Define
$$ \Gamma(\mu, \nu) = \{ \pi \in \mathcal{P}(X \times Y) \mid \pi(A \times Y) = \mu(A) \, , \, \pi(X \times B) = \nu(B) \quad \forall \, (A,B) \in \mathcal{B}(X) \times \mathcal{B}(Y) \} $$
i.e. the space of probability measures on the product that have $\mu$ and $\nu$ as marginals.
I want to prove that this set is closed w.r.t. the weak topology of measures. In this question:
Proof that the set of transference plans is closed in the weak topology was proved that if $\{\pi_n\}_{n \in \mathbb{N}} \subset \Gamma(\mu, \nu)$ are s.t. $\pi_n \rightharpoonup \pi$ then $\pi$ has $\mu$ and $\nu$ as marginals.
Is it enough? Shouldn't I prove also that $\pi$ is a positive measure? How to do that?
Best Answer
Non-negative measures form a closed subset of the set of all signed/complex measures w.r.t. weak convergence. Hence any weak limit of Probability measures is automatically a positive mesure measure. Since $\int 1 d\pi_n \to \int 1d\pi$ it follows that $\pi$ is necessarily a probability measure.