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.
