In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets R of a given set Ω can be extended to a measure on the σ-ring generated by R, and this extension is unique if the pre-measure is σ-finite. Consequently, any pre-measure on a ring containing all intervals of real numbers can be extended to the Borel algebra of the set of real numbers. This is an extremely powerful result of measure theory, and leads, for example, to the Lebesgue measure.

The theorem is also sometimes known as the Carathéodory–Fréchet extension theorem, the Carathéodory–Hopf extension theorem, the Hopf extension theorem and the Hahn–Kolmogorov extension theorem.^[1]

Introductory statement

Several very similar statements of the theorem can be given. A slightly more involved one, based on semi-rings of sets, is given further down below. A shorter, simpler statement is as follows. In this form, it is often called the Hahn–Kolmogorov theorem.

Let ${\displaystyle \Sigma _{0}}$ be an algebra of subsets of a set ${\displaystyle X.}$ Consider a set function

which is finitely additive, meaning that for any positive integer ${\displaystyle N}$ and ${\displaystyle A_{1},A_{2},\ldots ,A_{N}}$ disjoint sets in ${\displaystyle \Sigma _{0}.}$

Assume that this function satisfies the stronger sigma additivity assumption

for any disjoint family ${\displaystyle \{A_{n}:n\in \mathbb {N} \}}$ of elements of ${\displaystyle \Sigma _{0}}$ such that ${\displaystyle \cup _{n=1}^{\infty }A_{n}\in \Sigma _{0}.}$ (Functions ${\displaystyle \mu _{0}}$ obeying these two properties are known as pre-measures.) Then, ${\displaystyle \mu _{0}}$ extends to a measure defined on the ${\displaystyle \sigma }$-algebra ${\displaystyle \Sigma }$ generated by ${\displaystyle \Sigma _{0}}$; that is, there exists a measure such that its restriction to ${\displaystyle \Sigma _{0}}$ coincides with ${\displaystyle \mu _{0}.}$

If ${\displaystyle \mu _{0}}$ is ${\displaystyle \sigma }$-finite, then the extension is unique.

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 ${\displaystyle \mu _{0}}$ from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (if ${\displaystyle \mu _{0}}$ is ${\displaystyle \sigma }$-finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function.

Semi-ring and ring

Definitions

For a given set ${\displaystyle \Omega ,}$ we call a family ${\displaystyle {\mathcal {S}}}$ of subsets of ${\displaystyle \Omega }$ a semi-ring of sets if it has the following properties:

• ${\displaystyle \varnothing \in {\mathcal {S}}}$
• For all ${\displaystyle A,B\in {\mathcal {S}},}$ we have ${\displaystyle A\cap B\in {\mathcal {S}}}$ (closed under pairwise intersections)
• For all ${\displaystyle A,B\in {\mathcal {S}},}$ there exists a finite number of disjoint sets ${\displaystyle K_{i}\in {\mathcal {S}},i=1,2,\ldots ,n,}$ such that ${\displaystyle A\setminus B=\bigcup _{i=1}^{n}K_{i}}$ (relative complements can be written as finite disjoint unions).

The first property can be replaced with ${\displaystyle {\mathcal {S}}\neq \varnothing }$ since ${\displaystyle }$Zdroj:https://en.wikipedia.org?pojem=Carathéodory's_extension_theorem
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