Let $(E, d)$ be a Polish space and $\mathcal C\_b(E)$ the space of all real-valued bounded continuous functions on $E$. Let $p \in [1, \infty)$ and $\mathcal P\_p (E)$ the space of all Borel probability measures on $E$ with finite $p$-th moments. We define the [Wasserstein metric](https://en.wikipedia.org/wiki/Wasserstein_metric#Definition) $W\_p$ on $\mathcal P\_p (E)$ by
$$
W\_p (\mu, \nu) := \inf\_{\gamma \in \Pi(\mu, \nu)} \left [ \int\_{E \times E} (d(x,y))^p \mathrm d \gamma (x, y) \right ]^{1/p} \quad \forall \mu, \nu \in \mathcal P\_p (E).
$$


Here $\Pi(\mu, \nu)$ is the set of all Borel probability measures on $E\times E$ whose marginals are $\mu, \nu$ respectively. Fix some $a \in E$. Then we have from Villani's *Optimal Transport: Old and New*,



> 
> **Theorem 6.9** Let $\mu\_n, \mu \in \mathcal P\_p (E)$ with $n\in \mathbb N$. The following statements are equivalent.
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> * 1. $W\_p (\mu\_n, \mu) \to 0$.
> * 2. $\int f \mathrm d \mu\_n \to \int f \mathrm d \mu$ for all $f \in \mathcal C\_b(E)$ and $\int (d(x, a))^p \mathrm d \mu\_n \to \int (d(x, a))^p \mathrm d \mu$.
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We define $g\_p:E \to \mathbb R, x \mapsto (d(x, a))^p$ and $\mathcal F\_p := \mathcal C\_b(E) \cup \{g\_p\}$. **Theorem 6.9** gives me a feeling that the metric topology of $\mathcal P\_p (E)$ is the initial topology induced by $\mathcal F\_p$. This would be true if **Theorem 6.9** holds not only for sequences but also for **nets**.


Because a metric space is a [sequential space](https://en.wikipedia.org/wiki/Sequential_space), $[(1) \implies (2)]$ holds for nets.



> 
> Could you elaborate on if the direction $[(2) \implies (1)]$ of **Theorem 6.9** holds if we replace a sequence $(\mu\_n)\_{n \in \mathbb N}$ with a net $(\mu\_d)\_{d\in D}$?
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Many thanks you so much for your explanation.



Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?
A topology generated by countably many point-separating real functions [is metrizable](https://mathoverflow.net/a/435477/35357). To apply this here, it suffices to show that there is a countable family $\mathcal{G}$ of bounded real functions on $E$ such that the topology of weak convergence of measures is generated by the functions $\mu\mapsto g~\mathrm d\mu$ with $g$ in $\mathcal{G}$. That this is possible is
Theorem 6.6 in "Probability measures on metric spaces" by Parthasarathy. The possibly first proof of this fact is in



> 
> Varadarajan, Veeravalli S. "Weak convergence of measures on separable
> metric spaces." *Sankhyā: The Indian Journal of Statistics*
> (1933-1960) 19.1/2 (1958): 15-22.
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