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 a $\mu$-integrable function $f^\ast$ exists such that:
 * $\forall x \in X : \map {f^\ast} {\map T x} = \map {f^\ast} x$

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

converges $\mu$-almost everywhere and in $L^1$-norm.

Furthermore, we have:
 * $f^\ast = \expect {f \mid \II_T}$ almost surely

where:
 * $\expect {f \mid \II_T}$ is a version of conditional expectation of $f$ given $\II_T$
 * $\II_T := \set { A \in \BB : T^{-1} \sqbrk A = A }$

Also known as
'''Birkhoff's Ergodic Theorem is also known as
 * The Birkhoff ergodic theorem
 * The strong ergodic theorem
 * The pointwise ergodic theorem.

Also see

 * Mean Ergodic Theorem (also known as the Weak Ergodic Theorem) of