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Concept
Carathéodory's extension theorem
Summary
In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets R of a given set Ω can be extended to a measure on the σ-ring generated by R, and this extension is unique if the pre-measure is σ-finite. Consequently, any pre-measure on a ring containing all intervals of real numbers can be extended to the Borel algebra of the set of real numbers. This is an extremely powerful result of measure theory, and leads, for example, to the Lebesgue measure.
The theorem is also sometimes known as the Carathéodory–Fréchet extension theorem, the Carathéodory–Hopf extension theorem, the Hopf extension theorem and the Hahn–Kolmogorov extension theorem.
Several very similar statements of the theorem can be given. A slightly more involved one, based on semi-rings of sets, is given further down below. A shorter, simpler statement is as follows. In this form, it is often called the Hahn–Kolmogorov theorem.
Let be an algebra of subsets of a set Consider a set function
which is finitely additive, meaning that
for any positive integer and disjoint sets in
Assume that this function satisfies the stronger sigma additivity assumption
for any disjoint family of elements of such that (Functions obeying these two properties are known as pre-measures.) Then,
extends to a measure defined on the -algebra generated by ; that is, there exists a measure
such that its restriction to coincides with
If is -finite, then the extension is unique.
This theorem is remarkable for it allows one to construct a measure by first defining it on a small algebra of sets, where its sigma additivity could be easy to verify, and then this theorem guarantees its extension to a sigma-algebra. The proof of this theorem is not trivial, since it requires extending from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (if is -finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function.
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