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.
