# Hahn-Kolmogorov theorem

Let ${\mathcal{A}}_{0}$ be an algebra^{} of subsets of a set $X$. If a finitely additive measure ${\mu}_{0}:\mathcal{A}\to \mathbb{R}\cup \{\mathrm{\infty}\}$ satisfies

$${\mu}_{0}(\bigcup _{n=1}^{\mathrm{\infty}}{A}_{n})=\sum _{n=1}^{\mathrm{\infty}}{\mu}_{0}({A}_{n})$$ |

for any disjoint family $\{{A}_{n}:n\in \mathbb{N}\}$ of elements of ${\mathcal{A}}_{0}$ such that ${\cup}_{n=0}^{\mathrm{\infty}}{A}_{n}\in {\mathcal{A}}_{0}$, then ${\mu}_{0}$
extends to a measure defined on the $\sigma $-algebra $\mathcal{A}$ generated by ${\mathcal{A}}_{0}$; i.e. there exists a measure $\mu :\mathcal{A}\to \mathbb{R}\cup \{\mathrm{\infty}\}$ such that its restriction^{} (http://planetmath.org/RestrictionOfAFunction) to ${\mathcal{A}}_{0}$ coincides with ${\mu}_{0}$.

If ${\mu}_{0}$ is $\sigma $-finite (http://planetmath.org/SigmaFinite), then the extension^{} is unique.

Title | Hahn-Kolmogorov theorem |
---|---|

Canonical name | HahnKolmogorovTheorem |

Date of creation | 2013-03-22 14:03:10 |

Last modified on | 2013-03-22 14:03:10 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 7 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 28A10 |

Synonym | Hahn-Kolmogorov extension theorem |

Synonym | Kolmogorov extension theorem |