A mixing transformation is a measure-preserving transformation 
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 a measurable
set 
invariant modulo a set of measure
zero gives 
.
For an invertible 
whose inverse is a measurable function, mixing of order 
, where 
is an integer, means that
for every choice of measurable
sets 
,
 |
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.
