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# 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."

Birkhoff's Theorem, Ergodic Theory

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## References

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

## Referenced on Wolfram|Alpha

Birkhoff's Ergodic Theorem

## Cite this as:

Weisstein, Eric W. "Birkhoff's Ergodic Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BirkhoffsErgodicTheorem.html