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."
