The Gromov-Hausdorff-Prokhorov distance between compact metric spaces carrying probability measures compares both their geometry and their measures after isometric embedding into a common metric space. One convention is
 |
where 
and 
are isometric embeddings into
,
is Hausdorff distance, and 
is the Prokhorov metric.
The distance is zero precisely when there is a measure-preserving
isometry between the spaces. Random
planar map scaling limits use this distance to retain
the corner probability measure along with
graph distance.
