**Question:** What is known about algorithms for numerically computing/approximating the Prokhorov distance between two measures?

Recall that the Prokhorov distance metrizes the topology of weak(-*) convergence of measures on separable metric spaces, and is defined as follows.

Let $\mu_1$, $\mu_2$ be finite measures on a metric space $(X,d)$. The Prokhorov distance $\rho$ between them is, $$\rho(\mu_1,\mu_2):=\inf \left\{ \epsilon > 0 : \mu_1(A) \le \mu_2(A^\epsilon)+\epsilon~ \text{ for all } A \in \mathcal{B} \right\},$$ where $\mathcal{B}$ is the Borel $\sigma$-algebra on $X$ and $A^\epsilon$ is the $\epsilon$-neighborhood of $A$.

Has a constructive/algorithmic approach to the Prokhorov metric been studied in any contexts? How could one go about constructing numerical algorithms to compute it?

**Note:** *Per asked this nearly identical question at math.stackexchange, where it got no answers even after having a bounty for a week. I'm reposting here with his/her permission.*