An Endomorphism is called ergodic if it is true that Implies or 1, where . Examples of ergodic endomorphisms include the Map mod 1 on the unit interval with Lebesgue Measure, certain Automorphisms of the Torus, and ``Bernoulli shifts'' (and more generally ``Markov shifts'').
Given a Map and a Sigma Algebra, there may be many ergodic measures. If there is only one ergodic measure, then is called uniquely ergodic. An example of a uniquely ergodic transformation is the Map mod 1 on the unit interval when is irrational. Here, the unique ergodic measure is Lebesgue Measure.