the bounded-Lipschitz metric
/ LIP-shits /
The Levy-Prokhorov metric metrizes weak convergence but is defined through an awkward infimum over sets; the Wasserstein-1 metric is a clean supremum over Lipschitz functions but blows up when mass runs to infinity (it sees tails). The bounded-Lipschitz (or Dudley) metric is the happy compromise: a clean dual-norm formula like Wasserstein-1, but using BOUNDED Lipschitz test functions so it stays finite and metrizes exactly weak convergence on all of P(S).
On a metric space (S, d), the bounded-Lipschitz norm of a function f is ||f||_BL = ||f||_infinity + Lip(f), where ||f||_infinity is the sup norm and Lip(f) is the smallest Lipschitz constant. The bounded-Lipschitz distance between probability measures is d_BL(mu, nu) = sup { integral f dmu - integral f dnu : ||f||_BL <= 1 }. Because the test functions are both bounded and Lipschitz, escaping mass at infinity is damped (boundedness caps the contribution) while local discrepancies are still felt (the Lipschitz part). The fundamental theorem: on a separable metric space, d_BL metrizes the topology of weak convergence, d_BL(mu_n, mu) -> 0 iff mu_n => mu. Moreover d_BL and the Levy-Prokhorov metric are topologically equivalent and even satisfy explicit two-sided bounds (each is controlled by a power of the other).
Its practical virtues are real. The dual formula is differentiable-friendly and easy to estimate, so d_BL is the metric of choice for many quantitative weak-convergence results, for Stein's method bounds on distances to the normal, and in empirical-process and machine-learning settings (the integral-probability-metric viewpoint generalizes it). Compared to W_1, it never diverges, since the unit ball of bounded-Lipschitz functions is contained in functions bounded by 1; compared to Levy-Prokhorov, its supremum-over-functions form is far easier to bound by hand. It is, in short, the most convenient metric that exactly metrizes weak convergence.
For the escaping example nu_n = (1-1/n) delta_0 + (1/n) delta_n, any bounded-Lipschitz f with ||f||_BL <= 1 has |f| <= 1, so |integral f dnu_n - f(0)| <= (1/n)|f(n) - f(0)| <= (1/n)*2 -> 0. Hence d_BL(nu_n, delta_0) -> 0, correctly registering weak convergence, whereas W_1 stayed at 1. Boundedness tames the runaway mass.
Boundedness of the test functions makes d_BL ignore far-away mass, so it metrizes weak convergence where W_1 does not.
Drop the boundedness and you get W_1, which no longer metrizes weak convergence on all of P(S) (it adds first-moment convergence). The two agree only on uniformly integrable families; the bounded part of bounded-Lipschitz is doing essential work.