Definition (Bounded-Lipschitz Norm)

Let $\mathbb{X}$ be a Borel space and let $f:\mathbb{X}\to \mathbb{R}$ be a Borel function. We define the bounded-Lipschitz Norm as $$\lVert f \rVert _{BL}:=\lVert f \rVert _{\infty}+\sup_{x\not=y}\frac{|f(x)-f(y)|}{d(x,y)}$$

Definition (Bounded-Lipschitz Metric)

Let $\mu,\nu \in\mathcal{P}(\mathbb{X})$ we define the bounded Lipschitz metric as $$\rho_{BL}(\mu,\nu):=\sup_{\lVert f \rVert _{BL}\le 1}\left|\int\limits f \, d\mu-\int\limits f \, d\nu \right|$$