# outer regular

Let $X$ be a locally compact Hausdorff topological space with Borel $\sigma $–algebra $\mathcal{B}$, and suppose $\mu $ is a measure^{} on $(X,\mathcal{B})$. For any Borel set $B\in \mathcal{B}$, the measure $\mu $ is said to be outer regular on $B$ if

$$\mu (B)=inf\{\mu (U)\mid U\supset B,U\mathrm{open}\}.$$ |

We say $\mu $ is inner regular on $B$ if

$$\mu (B)=sup\{\mu (K)\mid K\subset B,K\mathrm{compact}\}.$$ |

Title | outer regular |
---|---|

Canonical name | OuterRegular |

Date of creation | 2013-03-22 12:39:57 |

Last modified on | 2013-03-22 12:39:57 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 4 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 28A12 |

Related topic | BorelSigmaAlgebra |

Defines | inner regular |