the Levy-Prokhorov metric
/ lev-EE pro-KHOR-ov /
Weak convergence was given as a topology, but for estimates and quantitative work you want an actual distance between measures that metrizes it: a number d(mu, nu) that is small exactly when the measures are weakly close. The Levy-Prokhorov metric is the canonical such distance on P(S) for a separable metric space S. It generalizes the classical Levy metric (which metrized weak convergence of CDFs on R via the largest gap of their graphs) to arbitrary metric spaces.
For a set A in S and r > 0 write A^r for its r-neighbourhood, the set of points within distance r of A. The Levy-Prokhorov distance between probability measures mu, nu is pi(mu, nu) = inf { r > 0 : mu(A) <= nu(A^r) + r and nu(A) <= mu(A^r) + r for all Borel sets A }. It says the two measures agree up to spreading mass by a little (the neighbourhood A^r) and losing a little (the additive r). The key theorem: on a separable metric space S, pi is a genuine metric on P(S) and it metrizes the topology of weak convergence, pi(mu_n, mu) -> 0 iff mu_n => mu. If S is also complete (Polish), then (P(S), pi) is itself a complete separable metric space, i.e. Polish.
Beyond metrizing weak convergence, the Levy-Prokhorov metric has a beautiful coupling interpretation through Strassen's theorem: pi(mu, nu) <= r is essentially equivalent to the existence of a coupling (X, Y) with X ~ mu, Y ~ nu and P(d(X, Y) > r) <= r. So small Levy-Prokhorov distance means the two measures can be coupled to be close with high probability. This makes it a workhorse in the theory of couplings and in quantitative weak-convergence bounds, though for convergence of moments one usually turns to Wasserstein distances instead, since Levy-Prokhorov controls only convergence in distribution, not in any moment.
Take mu = delta_0 and nu = delta_a on R with small a > 0. For r >= a, every set A has delta_0(A) <= delta_a(A^r) + r and vice versa (the point a sits in {0}^r once r >= a), while for r < a it can fail. So pi(delta_0, delta_a) = min(a, 1) up to the additive-r cap; as a -> 0 we get pi -> 0, matching delta_a => delta_0.
Two point masses: the Levy-Prokhorov distance shrinks with their separation, metrizing weak convergence.
Small Levy-Prokhorov distance controls only convergence in distribution, never moments. To bound E[X] differences you need a Wasserstein metric (which integrates the distance) plus a uniform-integrability or moment hypothesis.