Why is the weak convergence of probability measures on $\mathbb{R}$ with respect to bounded continuous test functions $C^0_b(\mathbb{R})$ metrizable by the bounded Lipschitz metric $$d(\mu, \nu) = \sup_{f \in \text{Lip}(\mathbb{R})} \Big | \int_{\mathbb{R}} f d \nu - \int_{\mathbb{R}} f d \mu \Big |$$ where $$\text{Lip}(\mathbb{R}) = \Big \{ f \in C_b(\mathbb{R}) : \sup_x |f(x) | \leq 1, \sup_{x \neq y} \frac{| f(x) - f(y) |}{|x-y|} \leq 1 \Big \}?$$ For those who would like a reference, this is invoked in the proof of the truncated version of Wigner's semicircle law in Anderson-Guionnet-Zeitouni's $\textit{Introduction to Random Matrices}$ and is cited in the appendix as part of Theorem C.8, though no proof is given there. If anyone could help me with this fact, I'd greatly appreciate it!
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$\begingroup$ Are $\mu$ and $\nu$ probability measures? I think that a uniform bound of some kind must be given in order to metrize a weak convergence. $\endgroup$– Giuseppe NegroJan 7, 2013 at 2:10
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$\begingroup$ Thanks for pointing this out - $\mu$ and $\nu$ are definitely probability measures! I'll fix this. $\endgroup$– StackQsJan 7, 2013 at 17:08
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There is a proof in Section 8.3 of Bogachev's Measure Theory.
answered Jan 7, 2013 at 2:57
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1$\begingroup$ And another one in the book by Ambrosio, Gigli, and Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures. It is Proposition 7.1.5 on pages 154-155. They prove a more general result for $p$-Wasserstein distance, for which convergence is equivalent to weak ("narrow") plus uniform integrability of $p$th moments. $\endgroup$– user53153Jan 7, 2013 at 3:03