# existence of the Lebesgue measure

###### Theorem (Lebesgue).

Let $\mathrm{B}$ be the Borel $\sigma $-algebra (http://planetmath.org/BorelSigmaAlgebra) on the real number line. Then, there is a unique measure^{} $\mu $ on the measurable space^{} $\mathrm{(}\mathrm{R}\mathrm{,}\mathrm{B}\mathrm{)}$ satisfying

$$\mu \left((a,b)\right)=b-a$$ |

for all real numbers $$.

Title | existence of the Lebesgue measure^{} |
---|---|

Canonical name | ExistenceOfTheLebesgueMeasure |

Date of creation | 2013-03-22 18:33:12 |

Last modified on | 2013-03-22 18:33:12 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 4 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 28A12 |

Classification | msc 26A42 |

Related topic | LebesgueMeasure |

Related topic | Measure |

Related topic | CaratheodorysExtensionTheorem |