Birkhoff's ergodic theorem states that if 
is an ergodic measure-preserving endomorphism
of a probability space 
and 
is an integrable real-valued measurable
function, then for almost every 
, we have
 |
(1)
|
as 
.
To illustrate this, take 
to be the characteristic
function of some measurable subset 
of 
so that
 |
(2)
|
The left-hand side of the first displayed equation just says how often the map 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 a continuous map on a compact
space and is uniquely ergodic with Borel measure 
and 
is a continuous function, then we can replace
the almost everywhere convergence in the first
displayed equation with "everywhere."
