Ergodicity

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…)