# complete measure

A measure space^{} $(X,\mathcal{S},\mu )$ is said to be *complete ^{}* if every subset of a set of measure $0$ is measurable (and consequently, has measure $0$); i.e. if for all $E\in \mathcal{S}$ such that $\mu (E)=0$ and for all $S\subset E$ we have $\mu (S)=0$.

If a measure space is not complete, there exists a completion (http://planetmath.org/CompletionOfAMeasureSpace) of it, which is a complete measure space $(X,\overline{\mathcal{S}},\overline{\mu})$ such that $\mathcal{S}\subset \overline{\mathcal{S}}$ and ${\overline{\mu}}_{|\mathcal{S}}=\mu $, where $\overline{\mathcal{S}}$ is the smallest $\sigma $-algebra containing both $\mathcal{S}$ and all subsets of elements of zero measure of $\mathcal{S}$.

Title | complete measure |
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Canonical name | CompleteMeasure |

Date of creation | 2013-03-22 14:06:56 |

Last modified on | 2013-03-22 14:06:56 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 5 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 28A12 |

Related topic | UniversallyMeasurable |

Defines | completion |

Defines | complete |