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 $W_p$ 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 6.9Let $\mu_n, \mu \in \mathcal P_p (E)$ with $n\in \mathbb N$. The following statements are equivalent.

- $W_p (\mu_n, \mu) \to 0$.

- $\int f \mathrm d \mu_n \to \int f \mathrm d \mu$ for all $f \in \mathcal C_b(E)$ and $\int (d(x, a))^p \mathrm d \mu_n \to \int (d(x, a))^p \mathrm d \mu$.

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, $[(1) \implies (2)]$ holds for nets.

Could you elaborate on if the direction $[(2) \implies (1)]$ of

Theorem 6.9holds if we replace a sequence $(\mu_n)_{n \in \mathbb N}$ with a net $(\mu_d)_{d\in D}$?

Many thanks you so much for your explanation.