Birkhoff's ergodic theorem states that if 
is an ergodic endomorphism
of a probability space 
and 
is 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 the first displayed equation 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 the first displayed equation with "everywhere."
