# A version of the Portmanteau theorem - reference request

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 $$\liminf_{n\to\infty}\int f\text{d}\mu_n\geq \int f\text{d}\mu.$$
• These days I'd expect to find it in books instead of articles. Have you checked in Billingsley? Apr 20, 2018 at 20:10
• You're welcome; was it there? If you find a satisfactory reference, you might like to post your own answer to this question. Apr 21, 2018 at 20:38
• The proof is about two lines, so if I were using this in a paper, I probably wouldn't bother with a reference - maybe just include a brief hint at the proof. Forward: consider a sequence of bounded continuous functions increasing to $f$. Reverse: what user111 said, or: if $f$ is bounded continuous, apply the second statement to $f$ and $-f$. Apr 22, 2018 at 3:58
• I know the proof, but I would prefer to cite it. I think the reference posted below by user111 is very good. Anyway, thank you for your help! Apr 22, 2018 at 16:37

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.