Birkhoff's Ergodic Theorem

Theorem
Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.

Let $f: X \to \overline \R$ be a $\mu$-integrable function.

Then:
 * $\ds \lim_{n\to\infty} \dfrac 1 n \sum_{n\mathop=0}^{n-1} \map f {T^n x} = \map {f^\ast} x$

for $\mu$-almost all $x \in X$, where:
 * $f^\ast$ is a $\mu$-integrable function such that $f^\ast \circ T = f^\ast$.