A mixing transformation is a measure-preserving map 
on a probability
space 
for which
 |
as 
for every pair of measurable sets 
and 
. Thus observations separated by a long time become asymptotically
independent. A mixing transformation is an ergodic
transformation, since applying the condition to an invariant measurable
set 
gives 
.
For an invertible 
whose inverse is a measurable function, mixing
of order 
means
 |
whenever the integers 
satisfy 
. The order counts the number
of sets. Rokhlin's
multiple-mixing problem asks whether mixing of order 2 implies mixing of every
finite order for one transformation.
