  A measurable map between measure spaces is shown to have bounded compression
if and only if its image via the measure-algebra functor is
Lipschitz-continuous w.r.t. the measure-algebra distances. This provides a
natural interpretation of maps of bounded compression/deformation by means of
the measure-algebra functor and corroborates the assertion that maps of bounded
deformation are the natural class of morphisms for the category of complete and
separable metric measure spaces.

Metric Geometry;Category Theory