Let {(X, \mu)} be a probability space and {T: X \rightarrow X} a measure-preserving transformation. In many cases, it turns out that the averages of a function {f} given by

\displaystyle \frac{1}{N} \sum_{i=0}^{N-1} f \circ T^i

actually converge a.e. to a constant.

This is the case if {T} is ergodic, which we define as follows: {T} is ergodic if for all {E \subset X} with {T^{-1}E = E}, {m(E)=1} or {0}. This is a form of irreducibility; the system {X,T} has no smaller subsystem (disregarding measure zero sets). It is easy to see that this is equivalent to the statement: {f} measurable (one could assume measurable and bounded if one prefers) and {T}-invariant implies {f} constant a.e. (One first shows that if {T} is ergodic, then {\mu(T^{-1}E \Delta E )} implies {\mu(E)=0,1}, by constructing something close to {E} that is {T}-invariant.)

In this case, therefore, the ergodic theorem takes the following form. Let {f: X \rightarrow \mathbb{C}} be integrable. Then almost everywhere,

\displaystyle \boxed{ \frac{1}{N} \sum_{i=0}^{N-1} f ( T^i (x)) \rightarrow \int_X f d\mu .}

This is a very useful fact, and it has many applications. (more…)