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
Uniqueness of the extension holds when μ0 is σ-finite on A; without σ-finiteness, the same statement may
Applications include defining the Lebesgue measure on the real line by extending from intervals, and more broadly