**Ergodicity **

Let be a **probability** space and a measure-preserving transformation. In many cases, it turns out that the averages of a function given by

actually converge a.e. to a constant.

This is the case if is **ergodic**, which we define as follows: is ergodic if for all with , or . This is a form of irreducibility; the system has no smaller subsystem (disregarding measure zero sets). It is easy to see that this is equivalent to the statement: measurable (one could assume measurable and bounded if one prefers) and -invariant implies constant a.e. (One first shows that if is ergodic, then implies , by constructing something close to that is -invariant.)

In this case, therefore, the ergodic theorem takes the following form. Let be integrable. Then almost everywhere,

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