## 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 Probability Measure and is continuous, then we can replace the Almost Everywhere convergence in (1) to everywhere.