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Birkhoff's Ergodic Theorem

Let $T$ be an ergodic Endomorphism of the Probability Space $X$ and let $f:X \rightarrow \Bbb{R}$ be a real-valued Measurable Function. Then for Almost Every $x\in X$, we have

{1\over n} \sum_{j=1}^{n} f \circ T^{j} (x) \rightarrow \int f\,dm
\end{displaymath} (1)

as $n \rightarrow \infty$. To illustrate this, take $f$ to be the characteristic function of some Subset $A$ of $X$ so that
1 & if $x\in A$\cr
0 & if $x\notin A$.\cr}
\hrule width 0pt height 3.2pt
\end{displaymath} (2)

The left-hand side of (1) just says how often the orbit of $x$ (that is, the points $x$, $Tx$, $T^2x$, ...) lies in $A$, and the right-hand side is just the Measure of $A$. Thus, for an ergodic Endomorphism, ``space-averages = time-averages almost everywhere.'' Moreover, if $T$ is continuous and uniquely ergodic with Borel Probability Measure $m$ and $f$ is continuous, then we can replace the Almost Everywhere convergence in (1) to everywhere.

© 1996-9 Eric W. Weisstein