Bounded-Lipschitz metric ($\beta (\mu ,\nu )={\sup }_{\Vert h{\Vert }_{\infty }+\mathrm{Lip}(h)\le 1}|\mu (h)-\nu (h)|$)

The bounded-Lipschitz metric tests probability measures against a uniformly bounded Lipschitz class. It metrizes weak convergence of probability measures on separable metric spaces. With a complete compatible base metric, it is complete on the probability measures of a Polish space. Replacing the sum norm by the maximum norm changes the metric only by uniform constant factors.