Caratheodory extension theorem

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 ${\displaystyle {\mathcal {F}}_{0}}$ be an algebra of subsets of X. Let ${\displaystyle \mu _{0}}$ be a countably additive non-negative set function on ${\displaystyle {\mathcal {F}}_{0}}$. Then there exists a measure ${\displaystyle \mu }$ on the ${\displaystyle \sigma }$-algebra ${\displaystyle {\mathcal {F}}=\sigma ({\mathcal {F}}_{0})}$ (i.e., the smallest sigma algebra containing ${\displaystyle {\mathcal {F}}_{0}}$) such that ${\displaystyle \mu (A)=\mu _{0}(A)}$ for all ${\displaystyle A\in {\mathcal {F}}_{0}}$. Furthermore, if ${\displaystyle \mu (X)=\mu _{0}(X)<\infty }$ then the extension is unique.

${\displaystyle {\mathcal {F}}}$ is also referred to as the sigma algebra generated by ${\displaystyle {\mathcal {F}}_{0}}$. The term "algebra of subsets" in the theorem refers to a collection of subsets of a set X which contains X itself, is closed under the operation of taking complements, finite unions and finite intersections in X. That is, any algebra ${\displaystyle {\mathcal {A}}}$ of subsets of X satisfies the following requirements:

1. ${\displaystyle X\in {\mathcal {A}}}$
2. If ${\displaystyle A\in {\mathcal {A}}}$ then ${\displaystyle X-A\in {\mathcal {A}}}$
3. For any positive integer n, if ${\displaystyle A_{1},A_{2},\ldots ,A_{n}\in {\mathcal {A}}}$ then ${\displaystyle A_{1}\cup A_{2}\cup \ldots \cup A_{n}\in {\mathcal {A}}}$

The last two properties imply that ${\displaystyle {\mathcal {A}}}$ is also closed under the operation of taking finite intersections of elements of ${\displaystyle {\mathcal {A}}}$.

References

1. D. Williams, Probability with Martingales, Cambridge : Cambridge University Press, 1991.