CarathéodoryExtension

CarathéodoryExtension refers to the Carathéodory extension theorem, a foundational result in measure theory named after Constantin Carathéodory. It provides a method to extend a pre-measure defined on a practical collection of sets to a full measure on a larger, generated sigma-algebra.

The theorem states that if μ0 is a nonnegative pre-measure defined on an algebra (or semiring) A of subsets of a set X, and μ0 is countably additive on disjoint unions, then there exists a measure μ defined on the sigma-algebra σ(A) generated by A that extends μ0. The standard construction uses an outer measure μ*(E) = inf{∑ μ0(Ai) : E ⊆ ⋃ Ai, Ai ∈ A}. A set E is deemed Carathéodory-measurable if μ*(E) = μ*(E ∩ A) + μ*(E ∩ A^c) for every A ⊆ X. The measure μ is then obtained by restricting μ* to the collection of Carathéodory-measurable sets, yielding a complete measure on σ(A) whose restriction to A agrees with μ0.

Uniqueness of the extension holds when μ0 is σ-finite on A; without σ-finiteness, the same statement may fail in general. The theorem underpins the construction of many familiar measures.

Applications include defining the Lebesgue measure on the real line by extending from intervals, and more broadly enabling the extension of measures from simple set systems to the sigma-algebras they generate. It is a core tool in probability, integration, and real analysis, providing a bridge from finite-additivity on simple collections to full measure on a sigma-algebra.