# dominated convergence theorem

Let $X$ be a measure space^{}, and let $\mathrm{\Phi},{f}_{1},{f}_{2},\mathrm{\dots}$ be measurable functions^{} such that $$ and $|{f}_{n}|\le \mathrm{\Phi}$ for each $n$.
If ${f}_{n}\to f$ almost everywhere, then $f$ is integrable and

$$\underset{n\to \mathrm{\infty}}{lim}{\int}_{X}{f}_{n}={\int}_{X}f.$$ |

This theorem is a corollary of the Fatou-Lebesgue theorem.

Title | dominated convergence theorem |
---|---|

Canonical name | DominatedConvergenceTheorem |

Date of creation | 2013-03-22 13:12:47 |

Last modified on | 2013-03-22 13:12:47 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 13 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 28A20 |

Synonym | Lebesgue’s dominated convergence theorem |

Related topic | MonotoneConvergenceTheorem |

Related topic | FatousLemma |

Related topic | VitaliConvergenceTheorem |