# monotone convergence theorem

Let $X$ be a measure space^{}, and let $0\le {f}_{1}\le {f}_{2}\le \mathrm{\cdots}$ be a monotone increasing sequence^{} of nonnegative measurable functions^{}. Let $f:X\to \mathbb{R}\cup \{\mathrm{\infty}\}$ be the
function defined by $f(x)={lim}_{n\to \mathrm{\infty}}{f}_{n}(x)$.
Then $f$ is measurable, and

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

Remark. This theorem is the first of several theorems which allow us to “exchange integration and limits”. It requires the use of the Lebesgue integral^{}: with the Riemann integral, we cannot even formulate the theorem, lacking, as we do, the concept^{} of “almost everywhere”. For instance, the characteristic function^{} of the rational numbers in $[0,1]$ is not Riemann integrable, despite being the limit of an increasing sequence of Riemann integrable functions.

Title | monotone convergence theorem^{} |
---|---|

Canonical name | MonotoneConvergenceTheorem |

Date of creation | 2013-03-22 12:47:27 |

Last modified on | 2013-03-22 12:47:27 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 9 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 26A42 |

Classification | msc 28A20 |

Synonym | Lebesgue’s monotone convergence theorem |

Synonym | Beppo Levi’s theorem |

Related topic | DominatedConvergenceTheorem |

Related topic | FatousLemma |