PlanetMath: Hahn-Kolmogorov theorem

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (210)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (35)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Hahn-Kolmogorov theorem

(Theorem)

Let $\mathscr{A}_0$ be an algebra of subsets of a set . If a finitely additive measure $\mu_0\colon \mathscr{A}\to\mathbb{R}\cup \{\infty\}$ satisfies

$\displaystyle \mu_0(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu_0(A_n)$

for any disjoint family $\{A_n:n\in \mathbb{N}\}$ of elements of $\mathscr{A}_0$ such that $\cup_{n=0}^\infty A_n\in \mathscr{A}_0$ , then $\mu_0$ extends uniquely to a measure defined on the $\sigma$ -algebra $\mathscr{A}$ generated by $\mathscr{A}_0$ ; i.e. there exists a unique measure $\mu\colon\mathscr{A}\to \mathbb{R}\cup\{\infty\}$ such that its restriction to $\mathscr{A}_0$ coincides with $\mu_0$

"Hahn-Kolmogorov theorem" is owned by Koro.

(view preamble)

Other names:

Hahn-Kolmogorov extension theorem, Kolmogorov extension theorem

Log in to rate this entry.
(view current ratings)

Cross-references: generated by, disjoint, measure, finitely additive, subsets, algebra

This is version 3 of Hahn-Kolmogorov theorem, born on 2003-10-24, modified 2003-10-24.
Object id is 5409, canonical name is HahnKolmogorovTheorem.
Accessed 4899 times total.

Classification:

AMS MSC:

28A10 (Measure and integration :: Classical measure theory :: Real- or complex-valued set functions)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact