Hahn-Kolmogorov theorem

In mathematics, the Hahn-Kolmogorov theorem characterizes when a finitely additive function with non-negative (possibly infinite) values can be extended to a bona fide measure. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, additivity and sigma additivity of a function defined on Subsets of a given set are abstractions of the intuitive properties The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function A negative number is a Number that is less than zero, such as −2 In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with It is named after the Austrian mathematician Hans Hahn and the Russian/Soviet mathematician Andrey Kolmogorov. Austria (Österreich ( officially the Republic of Austria (Republik Österreich A mathematician is a person whose primary area of study and research is the field of Mathematics. Hans Hahn ( September 27, 1879 - July 24, 1934) was an Austrian mathematician who made contributions to Functional analysis Russia (Россия Rossiya) or the Russian Federation ( Rossiyskaya Federatsiya) is a transcontinental Country extending A soviet (сове́т, "council" originally was a workers' local council in late Imperial Russia. Andrey Nikolaevich Kolmogorov (Андрей Николаевич Колмогоров ( April 25, 1903 - October 20, 1987) was a Soviet

Statement of the theorem

Let $Σ 0$ be an algebra of subsets of a set $X. In Mathematics a field of sets is a pair \langle X \mathcal{F} \rangle where X is a set and \mathcal{F} is an algebra$ Consider a function

$\mu_0\colon \Sigma_0 \to\mathbb{R}\cup \{\infty\}$

which is finitely additive, meaning that

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

for any positive integer N and $A 1, A 2,. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French . . , A N$ disjoint sets in $Σ 0$ . In Mathematics, two sets are said to be disjoint if they have no element in common

Assume that this function satisfies the stronger sigma additivity assumption

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

for any disjoint family $\{A_n:n\in \mathbb{N}\}$ of elements of $Σ 0$ such that $\cup_{n=1}^\infty A_n\in \Sigma_0$ . Then, $μ 0$ extends uniquely to a measure defined on the sigma-algebra $Σ$ generated by $Σ 0$ ; i. In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty e. , there exists a unique measure

$\mu\colon\Sigma\to \mathbb{R}\cup\{\infty\}$

such that its restriction to $Σ 0$ coincides with $μ 0 . In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined$

Comments

This theorem is remarkable for it allows one to construct a measure by first defining it on a small algebra of sets, where its sigma additivity could be easy to verify, and then this theorem guarantees its extension to a sigma-algebra. The proof of this theorem is not trivial, since it requires extending $μ 0$ from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique, and moreover that it does not fail to satisfy the sigma-additivity of the original function.

This article incorporates material from Hahn-Kolmogorov theorem on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative online Mathematics Encyclopedia.