Lévy–Prokhorov metric

In mathematics, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a metric (i.e., a definition of distance) on the collection of probability measures on a given metric space.

It is named after the French mathematician Paul Lévy and the Soviet mathematician Yuri Vasilyevich Prokhorov; Prokhorov introduced it in 1956 as a generalization of the earlier Lévy metric.

be a metric space with its Borel sigma algebra

denote the collection of all probability measures on the measurable space

, define the ε-neighborhood of

ε

is the open ball of radius

ε

is defined by setting the distance between two probability measures

to be For probability measures clearly

π ( μ , ν ) ≤ 1

Some authors omit one of the two inequalities or choose only open or closed

; either inequality implies the other, and

, but restricting to open sets may change the metric so defined (if