This might be a trivial question, but I am trying to prove equi-coerciveness of some family of functions on the space of Probability measures on some space. I could reduce the problem to showing that $$\{\nu:\mathcal{W}\_2^2(\mu, \nu)\le t\}$$ is compact (or is at least contained in some compact subset of $\mathcal{P}(\mathbb{R})$.)


The space of probability measures on $\mathbb{R}$ is equipped with weak topology. Now, I know from Banach Alaglou theorem unit ball in weak topology is compact. But here I am taking the ball in $\mathcal{W}\_2$ metric. Can anyone tell me if this is correct or I am just wasting time?



Is unit ball in 2-Wassestein metric weakly compact?
Yes, it is true. It follows from Prokhorov's theorem that in order to prove (pre-)compactness, it suffices to prove tightness. However, if we define $K$ to be the compact set such that $\mu(\mathbb{R}\setminus K)<\varepsilon$, and $K\_T:=\{x\in \mathbb{R}:\mathrm{dist}(K,x)\leq T\}$, then $\nu(\mathbb{R}\setminus K\_T)>2\varepsilon$ implies that $\mathcal{W}\_2^2(\mu,\nu)>\varepsilon T^2$ (since you have to move a mass $\geq\varepsilon$ over a distance $\geq T$), and so it is impossible for $\nu$ in your set provided that $\varepsilon T^2>t$. This proves tightness.


