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 Pierre Lévy and the Soviet mathematician Yuri Vasilevich Prokhorov; Prokhorov introduced it in 1956 as a generalization of the earlier Lévy metric.
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Let be a metric space with its Borel sigma algebra . Let denote the collection of all probability measures on the measurable space .
For a subset , define the ε-neighborhood of by
where is the open ball of radius centered at .
The Lévy–Prokhorov metric is defined by setting the distance between two probability measures and to be
For probability measures clearly .
Some authors omit one of the two inequalities or choose only open or closed ; either inequality implies the other, but restricting to open/closed sets changes the metric so defined.