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 transformations
include the map 
modulo 1 on 
with Lebesgue measure,
certain automorphisms of the torus,
Bernoulli schemes, and finite-state irreducible
Markov shifts with their stationary probability
measures (Walkden 2018, pp. 43-44, 59-60).
A map 
on a measurable space
is called uniquely ergodic if it has exactly one invariant probability
measure. An example is the map 
modulo 1 on 
when 
is irrational. Here the
unique invariant probability measure is Lebesgue measure (Walkden 2018, pp. 78-80).
