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J.R.
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Checking whether a Is this set is $\sigma$-compact in the Wasserstein space?

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J.R.
  • 291
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  • 6

Checking whether a set is $\sigma$-compact in the Wasserstein space

This is a follow-up to this question.

Fix a finite first moment probability measure $q\in\mathcal{P}_1(\mathbb R ^d)$, and real numbers $K,M,R$. Consider the following set:

$$A:=\left\{p\in\mathcal{P}_1(\mathbb R ^d): \int |x|dp\leq K, \int x dp=M, \mathcal{W}_1(p,q)\leq R \right\}$$

Is it a $\sigma$-compact set in $(\mathcal{P}_1(\mathbb R ^d),\mathcal W _1)$? It is not hard to check that this set is closed and that it is tight (weakly). As was already seen in the original question, this set is not compact, but I am hoping that it is still $\sigma$-compact.

Many thanks in advance!