# Fatou-Lebesgue theorem

Let $(X,\mu )$ be a measure space^{}. If $\mathrm{\Phi}:X\to \mathbb{R}$ is a nonnegative function with $$, and if ${f}_{1},{f}_{2},\mathrm{\dots}$ is a sequence of measurable functions^{} such that $|{f}_{n}|\le \mathrm{\Phi}$ for each $n$, then

$$g=\underset{n\to \mathrm{\infty}}{lim\; inf}{f}_{n}\text{and}h=\underset{n\to \mathrm{\infty}}{lim\; sup}{f}_{n}$$ |

are both integrable, and

$$ |

Title | Fatou-Lebesgue theorem |
---|---|

Canonical name | FatouLebesgueTheorem |

Date of creation | 2013-03-22 13:12:53 |

Last modified on | 2013-03-22 13:12:53 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 7 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 28A20 |

Related topic | FatousLemma |