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 ${\mathcal {F}}_{0}$ 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, 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:

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

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

References

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