A formal difficulty arises when one tries to reconcile the volume of sets with the need to preserve their size under a map. The problem arises because, in general, several different points in the domain of a function can map to the same point in its range; that is, there may be formula_17 with formula_18. Worse, a single point formula_19 has no size. These difficulties can be avoided by working with the inverse map formula_20; it will map any given subset formula_6 to the parts that were assembled to make it: these parts are formula_22. It has the important property of not losing track of where things came from. More strongly, it has the important property that "any" (measure-preserving) map formula_23 is the inverse of some map formula_24. The proper definition of a volume-preserving map is one for which formula_25 because formula_26 describes all the pieces-parts that formula_14 came from.