Birkhoff's Ergodic Theorem

Let  be an ergodic endomorphism of the probability space  and let  be a real-valued measurable function. Then for almost every , we have

(1)

as . To illustrate this, take  to be the characteristic function of some subset  of  so that

(2)

The left-hand side of (1) just says how often the orbit of  (that is, the points , , , ...) lies in , and the right-hand side is just the measure of . Thus, for an ergodic endomorphism, "space-averages = time-averages almost everywhere." Moreover, if  is continuous and uniquely ergodic with Borel measure  and  is continuous, then we can replace the almost everywhere convergence in (1) with "everywhere."

REFERENCES:

Cornfeld, I.; Fomin, S.; and Sinai, Ya. G. Appendix 3 in Ergodic Theory. New York: Springer-Verlag, 1982.