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:

$f^\ast = \expect {f \mid \II_T}$

where:

$\expect {f \mid \II_T}$ denotes the conditional expectation of $f$ given $\II_T$

$\II_T := \set { A \in \BB : T^{-1} \sqbrk A = A }$

Proof

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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 John von Neumann

Source of Name

This entry was named for George David Birkhoff.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Birkhoff (or strong or pointwise) ergodic theorem
• 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $2.6$: Pointwise Ergodic Theorem