# 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

Xbe a set and be an algebra of subsets of X. Let be a countably additive non-negative set function on . Then there exists a measure on the -algebra (i.e., the smallest sigma algebra containing ) such that for all . Furthermore, if then the extension is unique.

is also referred to as the *sigma algebra generated by* . 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 of subsets of *X* satisfies the following requirements:

- If then
- For any positive integer
*n*, if then

The last two properties imply that is also closed under the operation of taking finite intersections of elements of .

## References

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