How to show, that the set of density functions with compact support is compact wrt. Wasserstein Metric?

Question

In an Optimal Transport Math project we are currently working on we want to show that the Infimum of a Functional $\inf_{~f\in\overline{\mathcal{P}_{ac,2}(\Omega)}}H[f]$ is reached, where $\mathcal{P}_{ac,2}(\Omega)$ denotes the set of density functions on a compact set $\Omega \subset \mathbb{R}$ with finite second moments:

$\int_{\Omega}f(x)dx = 1,~\int_{\Omega}|x|^2f(x)dx < \infty$

We introduced the Wasserstein Distance as a Metric on $\mathcal{P}_{ac,2}(\Omega)$ before:

$W(\mu, \nu) = \left(\inf_{\eta \in \text{ Adm}(\mu, \nu)} \int_{\mathbb{R} \times \mathbb{R}}|x-y|^2d \eta (x,y) \right)^{\frac{1}{2}} $

$\text{Adm}(\mu, \nu) = \{\eta \in \mathcal{P}(\mathbb{R} \times \mathbb{R}): \text{The Marginals of } \eta \text{ are } \mu \text{ and } \nu \} $

$\mu$ and $\nu$ being Probability Measures on $\mathbb{R}$.

As simplification we are only dealing with absolutely continuous probability measures on $\mathbb{R}$ wrt. Lebesgue Measure.

Since we already showed $H$ to be lower semicontinuous (and strictly convex) before, it seems obvious to show that $\overline{\mathcal{P}_{ac,2}(\Omega)}$ is compact to conclude that the Infimum is reached.

We can't seem to find a way to proof it though. Using the usual ways to show compactness seem to be very unhandy with the Wasserstein Metric. Is there an obvious point that we are just overlooking, or a theorem that could help? We didn't take any courses in Functional Analysis, so we are not very familiar with compactness in function spaces, and the Theorems we found (eg. Arzelà-Ascoli) don't seem to help either.

Answer 1

This is what I would suggest, but I would be open to any comments/suggestions to tighten up the analysis. Since you have already shown your functional $H$ is lower semicontinuous then you are already nearly there. A standard recipe for a problem of this sort is

1. Pick a minimising sequence $\mu_n \in \mathcal{P}_{ac,2}(\Omega)$ of your functional $H$.
2. Show that $\{ \mu_n \}$ has a convergent subsequence, $W(\mu_{n_k}, \mu_0) \rightarrow 0$ for some $\mu_0$ in $\overline{\mathcal{P}_{ac,2}(\Omega)}$
3. Since $H$ is lower-semicontinuous then $$ H(\mu_0 ) = \inf\left\{ H(\mu) \; : \mu \in \overline{\mathcal{P}_{ac,2}(\Omega)} \right\}. $$

Since you have already done $3$ and you have $H$ is strictly convex giving the uniqueness of the minima, the only challenge is $2$.

Now since $\Omega$ is compact it can be shown (see here) that the space of all probability measures on $\Omega$ is compact with respect to the weak topology which guarantees a weak convergent subsequence, but it is also true that convergence in the second Wasserstein distance metric is equivalent to weak convergence plus convergence of the first two moments of elements in the sequence. So that $\mu_{n_k} \rightarrow \mu_0$ weakly with $\mu_{n_k} \in \mathcal{P}_{ac,2}$ implies that $W(\mu_n,\mu_0) \rightarrow 0$.