A pushforward measure is the measure 
on a measurable space 
induced by a measure 
on a measurable
space 
and a measurable function 
. It is defined by
 |
for every 
, where 
is the pre-image of
(Tao 2009).
It satisfies 
, so the pushforward of a probability
measure is a probability measure. For
a measurable function 
, 
iff 
is an invariant measure
for 
, or equivalently 
is a measure-preserving
transformation (Walkden 2018, p. 27).
