# Maximal ergodic theorem

Let $(X,\mathcal{B},\mu )$ be a probability space^{} and $T:X\to X$ a measure preserving transformation. Let $f$ be a ${L}^{1}(\mu )$ function.
Define the averages

$${f}^{*}(x)=\underset{N\ge 1}{sup}\frac{1}{N}\sum _{i=0}^{N-1}f({T}^{i}(x))$$ |

Then, for any $\lambda \in \text{\mathbf{R}}$, we have:

$${\int}_{{f}^{*}>\lambda}fd\mu \ge \lambda \mu (\{{f}^{*}>\lambda \})$$ |

This theorem may be used in the proof of the ergodic theorem (also known as Birkhoff ergodic theorem, or pointwise or strong ergodic theorem)

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

Canonical name | MaximalErgodicTheorem |

Date of creation | 2014-03-19 22:15:48 |

Last modified on | 2014-03-19 22:15:48 |

Owner | Filipe (28191) |

Last modified by | Filipe (28191) |

Numerical id | 3 |

Author | Filipe (28191) |

Entry type | Theorem |

Related topic | birkhoff ergodic theorem |