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
$$\liminf\_{n\to\infty}\int f\text{d}\mu\_n\geq \int f\text{d}\mu.$$



A version of the Portmanteau theorem - reference request
The Portmanteau theorem does not seem to be stated in this form in Billingsley or other classical references that I checked. A possible reference for the direct implication is Theorem A.3.12. p.378 of



> 
> Dupuis, P., Ellis, R.S., A weak convergence approach to the theory of
>  large deviations. Wiley Series in Probability and Statistics,
>  Wiley-Interscience, New York, 1997
> 
> 
> 


The second statement implies that for an open set $O$ of $M$,
$$\liminf\_{n\to\infty}\mu\_n(O)\geq\mu(O),$$
so the reverse implication just follows from the above inequality since it is one of the equivalent properties to weak-\* convergence, as stated in the usual version of the Portmanteau theorem.


