Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?

Question

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

• 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 $\int (d(x, a))^p \mathrm d \mu_n \to \int (d(x, a))^p \mathrm d \mu$.

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 6.9 holds 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.

Answer 1

A topology generated by countably many point-separating real functions is metrizable. 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.