# $\sigma $-finite

A measure space^{} $(\mathrm{\Omega},\mathcal{B},\mu )$ is a finite measure space if $$; it is $\sigma $-finite if the total space is the union of a finite or countable^{} family of sets of finite measure, i.e. if there exists a countable set $\mathcal{F}\subset \mathcal{B}$ such that $$ for each $A\in \mathcal{F}$, and
$\mathrm{\Omega}={\bigcup}_{A\in \mathcal{F}}A.$
In this case we also say that $\mu $ is a $\sigma $-finite measure.
If $\mu $ is not $\sigma $-finite, we say that it is $\sigma $-infinite^{}.

Examples. Any finite measure space is $\sigma $-finite. A more interesting example is the Lebesgue measure^{} $\mu $ in ${\mathbb{R}}^{n}$: it is $\sigma $-finite but not finite. In fact

$${\mathbb{R}}^{n}=\bigcup _{k\in \mathbb{N}}{[-k,k]}^{n}$$ |

(${[-k,k]}^{n}$ is a cube with center at $0$ and side length $2k$, and its measure is ${(2k)}^{n}$), but $\mu ({\mathbb{R}}^{n})=\mathrm{\infty}$.

Title | $\sigma $-finite |
---|---|

Canonical name | sigmafinite |

Date of creation | 2013-03-22 12:29:48 |

Last modified on | 2013-03-22 12:29:48 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 13 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 28A10 |

Synonym | $\sigma $ finite |

Synonym | sigma-finite |

Synonym | sigma finite |

Related topic | Measure |

Related topic | MeasureSpace |

Related topic | AlternativeDefinitionOfSigmaFiniteMeasure |

Related topic | AnySigmaFiniteMeasureIsEquivalentToAProbabilityMeasure |

Defines | $\sigma $-infinite |

Defines | sigma-infinite |

Defines | finite measure space |