Measure-preserving transformation

Summary

A measure-preserving transformation is a measurable map between measure spaces under which the preimage of every measurable set has the same measure as the set. Equivalently, the pushforward of the first measure is the second one. A measure-preserving map of a space to itself is the basic object of ergodic theory. It models a system that evolves in time without creating or destroying mass, as in Hamiltonian mechanics.