An ergodic measure for a measurable function 
on a measurable
space is an invariant measure 
that is a probability measure
such that every measurable set 
invariant modulo a set of measure
zero has 
or 1. 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.
