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

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. $$W_p (\mu_n, \mu) \to 0$$.
1. $$\int f \mathrm d \mu_n \to \int f \mathrm d \mu$$ 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.

• You can combine this answer with the fact that one can test weak convergence with a countable set of functions in $C_b(E)$, which is, for example, Theorem 6.6 in "Probability measures on metric spaces" by Parthasarathy. Nov 28, 2022 at 22:29
• @MichaelGreinecker Thank you so much for your help! I got it. Could you post your comment as an answer? Nov 29, 2022 at 13:26

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