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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 $$\liminf_{n\to\infty}\int f\text{d}\mu_n\geq \int f\text{d}\mu.$$
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    $\begingroup$ These days I'd expect to find it in books instead of articles. Have you checked in Billingsley? $\endgroup$
    – Nate Eldredge
    Apr 20, 2018 at 20:10
  • $\begingroup$ You're welcome; was it there? If you find a satisfactory reference, you might like to post your own answer to this question. $\endgroup$
    – Nate Eldredge
    Apr 21, 2018 at 20:38
  • $\begingroup$ 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 $-f$. $\endgroup$
    – Nate Eldredge
    Apr 22, 2018 at 3:58
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    $\begingroup$ 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! $\endgroup$
    – Martha Łącka
    Apr 22, 2018 at 16:37

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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.

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