Let
be an ergodic Endomorphism of the Probability Space
and let
be a real-valued Measurable Function. Then for Almost Every
, we have
![\begin{displaymath}
{1\over n} \sum_{j=1}^{n} f \circ T^{j} (x) \rightarrow \int f\,dm
\end{displaymath}](b_1512.gif) |
(1) |
as
. To illustrate this, take
to be the characteristic function of some Subset
of
so that
![\begin{displaymath}
f(x)=\cases{
1 & if $x\in A$\cr
0 & if $x\notin A$.\cr}
\hrule width 0pt height 3.2pt
\end{displaymath}](b_1514.gif) |
(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.
© 1996-9 Eric W. Weisstein
1999-05-26