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 
equals 
. An invariant probability
measure makes 
measure preserving on the corresponding probability
space. For example, a rotation 
of 
preserves normalized Lebesgue
measure. An invariant probability measure
is an ergodic measure when every invariant measurable set has measure
0 or 1 (Walters 1982).
