Carathéodory's extension theorem

In measure theory, Carathéodory's extension theorem (named after the Greek mathematician Constantin Carathéodory) states that any measure defined on a given ring R of subsets of a given set Ω can be extended to the σ-algebra generated by R, and this extension is unique if the measure is σ-finite. Consequently, any measure on a space 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 proves, for example, the existence of the Lebesgue measure.

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

(The first property can be replaced with "S is not empty" since A \ A = ∅ must be in S if A is in S).

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

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