Bounded-Lipschitz (Fortet–Mourier) distance

The bounded-Lipschitz (or Fortet–Mourier) distance is an IPM over functions that are both bounded and Lipschitz:

Restricting to bounded functions makes it always finite, even for heavy-tailed distributions where Wasserstein may be infinite. Restricting to Lipschitz functions means, like Wasserstein, it respects the geometry of the space — moving mass a little changes the distance a little.

It metrises weak convergence: $d_{BL}(P_n,P)\to 0$ exactly when $P_n$ converges in distribution to $P$. It is always at most both the TV-based and the Wasserstein distance.