 You can imagine the task as the need to move a pile of earth of shape formula_7 to the hole in the ground of shape formula_15 such that at the end, both the pile of earth and the hole in the ground completely vanish.In order for this plan to be meaningful, it must satisfy the following properties

That is, that the total mass moved "out of" an infinitesimal region around formula_43 must be equal to formula_54 and the total mass moved "into" a region around formula_44 must be formula_56.This is equivalent to the requirement that formula_57 be a joint probability distribution with marginals formula_7 and formula_15.