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According to Billingsly, let $P$ and $Q$ be two probability measures. Then the Prohorov metric $\pi (P,Q)$ is the infimum of those positive $ \epsilon $ for which the two inequalities $PA \le QA^{\epsilon}+ \epsilon$ and $QA \le PA^{\epsilon}+ \epsilon$ for all Borel sets $A$.

Then the theorem 6.8 says that if the set on which probability measures is separable and complete, weak convergence is equivalent to $ \pi $ convergence.

Then,

Question: what is an intuitive explanation that the Prohorov metric metrizes weak convergence? How can I explain it using an example?

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