# ergodic theorem

Let $(X,\U0001d505,\mu )$ be a probability space^{}, $f\in {L}^{1}(\mu )$, and $T:X\to X$ a measure preserving transformation. Birkhoff’s *ergodic theorem* (often called the *pointwise* or *strong* ergodic theorem) states that there exists ${f}^{*}\in {L}^{1}(\mu )$ such that

$$\underset{n\to \mathrm{\infty}}{lim}\frac{1}{n}\sum _{k=0}^{n-1}f({T}^{k}x)={f}^{*}(x)$$ |

for almost all $x\in X$. Moreover, ${f}^{*}$ is $T$-invariant (i.e., ${f}^{*}\circ T={f}^{*}$) almost everywhere and

$$\int {f}^{*}\mathit{d}\mu =\int f\mathit{d}\mu .$$ |

In particular, if $T$ is ergodic then the $T$-invariance of ${f}^{*}$ implies that it is constant almost everywhere, and so this constant must be the integral of ${f}^{*}$; that is, if $T$ is ergodic, then

$$\underset{n\to \mathrm{\infty}}{lim}\frac{1}{n}\sum _{k=0}^{n-1}f({T}^{k}x)=\int f\mathit{d}\mu $$ |

for almost every $x$. This is often interpreted in the following way: for an ergodic transformation, the time average equals the space average almost surely.

Title | ergodic theorem |
---|---|

Canonical name | ErgodicTheorem |

Date of creation | 2013-03-22 12:20:52 |

Last modified on | 2013-03-22 12:20:52 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 11 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 37A30 |

Classification | msc 47A35 |

Synonym | strong ergodic theorem |

Synonym | Birkhoff ergodic theorem |

Synonym | Birkhoff-Khinchin ergodic theorem |

Related topic | ErgodicTransformation |