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

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

as $n \rightarrow \infty$. To illustrate this, take $f$ to be the characteristic function of some Subset $A$ of $X$ so that

$$f(x)=\cases{ 1 & if $x\in A$\cr 0 & if $x\notin A$.\cr} \hrule width 0pt height 3.2pt$$ (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.