Hahn-Kolmogorov theorem - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Hahn–Kolmogorov theorem characterizes when a finitely additive function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

with non-negative (possibly infinite) values can be extended to a bona fide measure

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

. It is named after the Austria

Austria

Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...

n mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

Hans Hahn

Hans Hahn was an Austrian mathematician who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory.-Biography:...

and the Russia

Russia

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n/Soviet mathematician Andrey Kolmogorov

Andrey Kolmogorov

Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...

Statement of the theorem

Let

be an algebra of subsets of a set

Consider a function

which is finitely additive, meaning that

for any positive integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

N and

disjoint sets in

.

Assume that this function satisfies the stronger sigma additivity assumption

for any disjoint family

of elements of

such that

. (Functions

obeying these two properties are known as pre-measures.) Then,

extends to a measure defined on the sigma-algebra

Sigma-algebra

In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...

generated by

; i.e., there exists a measure

such that its restriction to

coincides with

-finite, then the extension is unique.

Non-uniqueness of the extension

is not

-finite then the extension need not be unique, even if the extension itself is

-finite.

Here is an example:

We call rational closed-open interval, any subset of

of the form

, where

.

Let

and let

be the algebra of all finite union of rational closed-open intervals contained in

. It is easy to prove that

is, in fact, an algebra. It is also easy to see that every non-empty set in

is infinite.

Let

be the counting set function (

) defined in

.
It is clear that

is finitely additive and

-additive in

. Since every non-empty set in

is infinite, we have, for every non-empty set

Now, let

be the

-algebra generated by

. It is easy to see that

is the Borel

-algebra of subsets of

, and both

and

are measures defined on

and both are extensions of

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

from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (if

-finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function.

Statement of the theorem

Non-uniqueness of the extension

Comments

See also