The Wasserstein distance of order 
between probability
measures 
on a metric space 
is the least transport cost
 |
where 
is the set of probability
measures with marginals 
and 
. The distance
is finite when both measures have finite 
th distance moments
about a fixed point. For two point
masses it equals 
. On 
, convergence in 
is equivalent to weak
convergence together with convergence of the
first absolute moments.
For quadratic Euclidean cost and an absolutely continuous source, an optimal transport map is a Brenier map.
