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.
