I am trying to find peer-reviewed references to the following version of the Portmanteau theorem: Let $M$ be a metric space and let $(\mu_n)_{n\in\mathbb N}$ be a sequence of Borel probability measures. Then the following conditions are equivalent:

- $(\mu_n)_{n\in\mathbb N}$ converges to $\mu$ with respect to the weak$^*$ topology,
- for every lower semicontinuous function $f\colon M\to\mathbb R$ bounded from below one has $$\liminf_{n\to\infty}\int f\text{d}\mu_n\geq \int f\text{d}\mu.$$