An invariant measure for a measurable function 
on a measurable
space 
is a measure 
satisfying
 |
where 
is the pre-image of any 
. Equivalently, the pushforward
measure 
equals 
. An invariant probability
measure makes 
a measure-preserving transformation
of the corresponding probability space. For
example, a rotation 
of 
preserves normalized Lebesgue
measure. An invariant probability measure
is an ergodic measure when every measurable
set 
satisfying 
has measure 0 or 1 (Walters 1982; Walkden 2018, pp. 27,
29, and 43-44).
