http://en.wikipedia.org/wiki/Wasserstein_metric
I would like to prove that $$W^1(μ,δx_0)=∫d(x_0,y) μ(dy)$$ let $$γ∈Γ(μ,δx_0)$$ Can we say that it is the product of its marginal distributions $$γ=μ×δx_0 $$ and then apply Fubini's theorem? But being the product of its marginal distributions would mean $$γ({(x,y)/x∈A ,y∈B})=γ({(x,y)/x∈A})*γ({(x,y)/y∈B}) $$ which doesn't hold in general. So how can we use marginal distributions to compute $$∫d(x,y) γ(dx,dy)$$
Indeed, in general, if $\mu_1$ and $\mu_2$ are two probability measures and $\nu$ has marginals $\mu_1$ and $\mu_2$ respectively, it is not true that $\nu=\mu_1\times\mu_2$. However, in the case $\mu_2=\delta_{x_0}$, this becomes true.