Ergodic theory is concerned with the extent to which iterates of
a function mix subsets of a measure space.
I will use the following basic concepts of ergodic theory specialized
to the measure space with the measure .
A measure-preserving function is ergodic
if the only -measurable sets E for which
are and the empty set, i.e.,
such a function does such a good job of mixing points in the space
that it has no nontrivial -invariant sets.
It can be shown [, p. 36] that an equivalent condition for
ergodicity is that
for all and all integers .
This condition in turn is equivalent to the assertion that for
almost all the sequence of iterates
is uniformly distributed for all .
A function is
for all and all .
Strongly mixing functions are ergodic.