# Birkhoff's Ergodic Theorem

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## 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