Definitions

# Carathéodory's extension theorem

In measure theory, Carathéodory's extension theorem proves that for a given set Ω, you can always extend a σ-finite measure defined on R to the σ-algebra generated by R, where R is a ring included in the power set of Ω; moreover, the extension is unique. For example, it proves that if you construct a measure on a space which contains all intervals of the set of real numbers, you can extend it to the Borel algebra of the set of real numbers. This is an extremely powerful result of measure theory, and proves for example the existence of the Lebesgue measure.

## Semi-ring and ring

### Definitions

For a given set Ω, we may define a semi-ring as a subset S of $mathcal\left\{P\right\}\left(Omega\right)$, the power set of Ω, which has the following properties:

• $varnothing in S$
• For all $A, B in S$, we have $A cap B in S$ (closed under pairwise intersections)
• For all $A, B in S$, there exist disjoint sets $K_i in S$, with $i = 1,2,ldots,n$, such that $Asetminus B = biguplus K_i$ (relative complements can be written as finite disjoint unions).

With the same notation, we define a ring R as a subset of the power set of Ω which has the following properties:

• $varnothing in R$
• For all $A, B in R$, we have $A cup B in R$ (closed under pairwise unions)
• For all $A, B in R$, we have $A setminus B in R$ (closed under relative complements).

Thus any ring on Ω is also a semi-ring.

Sometimes, the following constraint is added in the measure theory context:

• Ω is the disjoint union of a countable family of sets in S.

### Properties

• Arbitrary intersections of rings on Ω are still rings on Ω (the intersection needs not to be countable)
• If A is a non-empty subset of $mathcal\left\{P\right\}\left(Omega\right)$, then we define the ring generated by A (noted R(A)) as the smallest ring containing A. It is straightfoward to see that the ring generated by A is equivalent to the intersection of all rings containing A.
• For a semi-ring S, the set containing all finite disjoint union of sets of S is the ring generated by S:

$R\left(S\right) = \left\{ A: A = bigcup_\left\{i=1\right\}^\left\{n\right\}\left\{A_i\right\}, A_i in S \right\}$
(R(S) is simply the set containing all finite unions of sets in S)

• A measure μ defined on a semi-ring S can be extended on the ring generated by S; such an extension is unique. The extended measure can be written:

$mu\left(A\right) = sum_\left\{p=1\right\}^\left\{n\right\}\left\{mu\left(A_p\right)\right\}$ for $A = biguplus_\left\{p=1\right\}^\left\{n\right\}\left\{A_p\right\}$, with the Ap in S.
It can be proven that such a definition indeed defines a countably additive measure, and that any measure on R(S) which extends the measure on S is necessary of this form.

### Motivation

In the theory of measure, we are not interested in semi-rings and rings themselves, but rather in σ-algebra generated by them. The idea is that it is possible to build measures on semi-rings S (for example Stieltjes measures), which can then be extended on rings generated by S , and then extended to σ-algebra through Caratheodory's extension theorem. As σ-algebra generated by semi ring and rings are the same, the difference does not really matter (in the theory of measure's context at least). Actually, the Carathéodory's extension theorem can be slightly generalized by replacing ring by semi ring.

The definition of semi-ring may seem a bit convoluted, but the following example will show you why it is useful.

### Example

Think about the subset of $mathcal\left\{P\right\}\left(Bbb\left\{R\right\}\right)$ defined by the set of all intervals ]a, b] for a and b reals. This is a semi-ring, but not a ring. Stieltjes measures are defined on intervals; the countable additivity on the semi ring is not too difficult to prove because we only consider countable unions of intervals which are intervals themselves. Proving it for arbitrary countably union of intervals is proved using Caratheodory's theorem.