Let $(E, d)$ be a metric space, $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$, and $\mathcal P(E)$ the space of all Borel probability measures on $E$. For $f \in \mathcal C_b(E)$, we define $$ L_f:\mathcal P(E) \to \mathbb R, \mu \mapsto \int_E f\mathrm d \mu. $$
Let $\tau$ be the initial topology on $\mathcal P(E)$ that is induced by the collection $\{L_f : f \in \mathcal C_b(E)\}$. If $(\mu_d)_{d\in D}$ is a net in $\mathcal P(E)$ and $\mu \in \mathcal P(E)$ then $\mu_d \to \mu$ in $\tau$ IFF $L_f(\mu_d) \to L_f(\mu)$ for all $f \in \mathcal C_b(E)$. We have the famous result, i.e.,
Portmanteau theorem Let $(\mu_n)_{n\in \mathbb N}$ be a sequence in $\mathcal P(E)$ and $\mu \in \mathcal P(E)$. The following statements are equivalent:
- $\mu_n \to \mu$ in $\tau$;
- $\int f \mathrm d \mu_n \to \int f \mathrm d \mu$ for every real-valued bounded Lipschitz function $f$ on $E$;
- $\lim \inf \int f \mathrm d \mu_n \ge \int f \mathrm d \mu$ for every real-valued lower semi-continuous function $f$ on $E$ that is bounded from below;
- $\liminf \mu_{n}(U) \ge \mu(U)$ for every open subset $U$ of $E$;
- $\lim \mu_{n}(A) = \mu(A)$ for every Borel subset $A$ of $E$ such that $\mu(\partial A) = 0$.
Of course, $\liminf$ and $\limsup$ are well-defined for nets.
Are there references that contain a version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$?
Thank you so much for your help!
