Portmanteau Theorem on a Metric Space

Let $(X,d)$ be a metric space with $X$ nonempty, let $\mathcal{T}_d$ be its topology of open subsets, and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra. Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of probability measures on $(X,\mathcal{B}(X))$ and let $\mu$ be a probability measure on $(X,\mathcal{B}(X))$.

Regard $\mathbb{R}$ as a metric space with the absolute-value metric, and let bounded have the meaning fixed there for a real-valued function on a set. All the integrals below are those of Integrable Function and the Lebesgue Integral and exist by claim 6 of Borel Measurability and Bounded Integration on a Metric Space. The four claims below are implications; they are not offered as a complete list of characterizations equivalent to weak convergence.

1. (Lipschitz test functions suffice) Suppose that for every bounded Lipschitz function $f:X\to\mathbb{R}$ the sequence $\bigl(\int_X f\,d\mu_n\bigr)_{n\in\mathbb{N}}$ converges to $\int_X f\,d\mu$. Then $(\mu_n)_{n\in\mathbb{N}}$ converges weakly to $\mu$.

2. (Lower semicontinuous functions) Suppose $(\mu_n)_{n\in\mathbb{N}}$ converges weakly to $\mu$, and let $f:X\to\mathbb{R}$ be bounded and lower semicontinuous on $X$. Then the sequence $\bigl(\int_X f\,d\mu_n\bigr)_{n\in\mathbb{N}}$ is bounded and

$$\int_X f\,d\mu\;\le\;\liminf_{n}\int_X f\,d\mu_n ,$$

the limit inferior being that of a bounded real sequence.

3. (Open sets) Suppose $(\mu_n)_{n\in\mathbb{N}}$ converges weakly to $\mu$ and let $U\in\mathcal{T}_d$. Then $\mu(U)\le\liminf_{n}\mu_n(U)$.

4. (Closed sets) Suppose $(\mu_n)_{n\in\mathbb{N}}$ converges weakly to $\mu$ and let $F\subseteq X$ be closed in $(X,\mathcal{T}_d)$. Then

$$\limsup_{n}\mu_n(F)\;\le\;\mu(F),$$

the limit superior being that of a bounded real sequence.