In Prokhorov's 1956 paper "Convergence of Random Processes…" it states the following.
Where $\mathcal{R}$ is a complete, separable, metric space.
Additionally, it says that any weakly convergent sequence of measures satisfies conditions 1,2. This would give that any sequence is compact.
Isn't a counterexample to this the set of Dirac probability measures $\delta_{(1/n)}$ for $n\geq1$ on $\mathbb{R}$? Every element in the sequence has measure 1, and the set $[-2,2]$ is a compact set giving the tightness bound, but the limit $\delta_{(0)}$ is not in the sequence.
This seems to be in contradiction to results stated elsewhere, e.g. (https://mathoverflow.net/questions/140760/prokhorovs-theorem-for-finite-signed-measures).
Should it be the case that it should say relatively compact instead?
Thanks.
Best Answer
Yes, he means the set of measures $\mathfrak N$ is relatively compact if the two conditions are met. I don't have access to the Russian original, so cannot tell if the mistake is Prohorov's or his translator's.