A measure-preserving transformation of a measure space 
is a measurable function 
such that
 |
for every measurable set 
, where 
is its pre-image. Equivalently,
is an invariant measure for 
, or the pushforward measure 
equals 
(Walkden 2018, p. 27). The map 
need not be surjective. In the narrower convention
for an endomorphism of a probability
space, 
is additionally required to be surjective.
