Let $\mathbf{M}_1(X)$ denote the set of Borel probability measures on a compact metric space $X$. Given $\mu_1,\mu_2 \in \mathbf{M}_1(X)$, let $J(\mu_1,\mu_2)$ be the set of Borel probability measures $\nu$ on $X \times X$ with the property that $$\nu(A \times X)=\mu_1(A), \nu(X \times B)=\mu_2(B); \space \space \text{for all $A,B \subset X$ Borel.}$$ If $d$ is the metric on $X$, then define $$\rho(\mu_1,\mu_2):=\inf_{\nu \in J(\mu_1,\mu_2)} \int_{X \times X} d(x,y) d \nu(x,y).$$ Prove $\rho$ is a metric on $\mathbf{M}_1(X)$ and given $\mu,\mu_n \in \mathbf{M}_1(X)$, $\mu_n \to \mu$ in the weak$^*$ topology if and only if $\rho(\mu_n,\mu) \to 0$.
Attempt:
Non-negativity is immediate as $\nu$ is non-negative and $d$ is non-negative thus taking an integral we obtain a non-negative value and taking an infimum keep things non-negative. For symmetry and the triangle inequality, I wanna say I need to express the integral as a double integral and invoke Fubini theorem? Any help or hints greatly appreciated!! I am having trouble interpreting $d \nu(x,y)$.. Also as $\nu$ is a probability measure, does this mean $\nu(X \times X)=1$? Furthermore, does $\mu_i(X)=1$ for $i=1,2$?