Caratheodory extension theorem

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In the branch of mathematics known as measure theory, the Caratheodory extension theorem states that a countably additive non-negative set function on an algebra of subsets of a set can be extended to be a measure on the sigma algebra generated by that algebra. Measure in this context specifically refers to a non-negative measure.

Statement of the theorem

(Caratheodory extension theorem) Let X be a set and $\mathcal{F}_0$ be an algebra of subsets of X. Let $\mu_0$ be a countably additive non-negative set function on $\mathcal{F}_0$ . Then there exists a measure $\mu$ on the $\sigma$ -algebra $\mathcal{F}=\sigma(\mathcal{F}_0)$ (i.e., the smallest sigma algebra containing $\mathcal{F}_0$ ) such that $\mu(A)=\mu_0(A)$ for all $A \in \mathcal{F}_0$ . Furthermore, if $\mu(X)=\mu_0(X)<\infty$ then the extension is unique.

$\mathcal{F}$ is also referred to as the sigma algebra generated by $\mathcal{F}_0$ . The term "algebra of subsets" in the theorem refers to a collection of subsets of a set X which contains X itself and is closed under the operation of taking complements, finite unions and finite intersections in X. That is, any algebra $\mathcal{A}$ of subsets of X satisfies the following requirements:

$X \in \mathcal{A}$
If $A \in \mathcal{A}$ then $X-A \in \mathcal{A}$
For any positive integer n, if $A_1,A_2,\ldots,A_n \in \mathcal{A}$ then $A_1 \cup A_2 \cup \ldots \cup A_n \in \mathcal{A}$