Let be an ergodic Endomorphism of the Probability Space and let
be a realvalued 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 lefthand side of (1) just says how often the orbit of (that is, the points , , , ...)
lies in , and the righthand side is just the Measure of . Thus, for an ergodic Endomorphism,
``spaceaverages = timeaverages 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.
© 19969 Eric W. Weisstein
19990526