CarathéodoryDefinition

Carathéodory definition refers to a criterion for determining measurability of sets with respect to an outer measure, named after Constantin Carathéodory. Given a set X and an outer measure m* on X (a function defined on all subsets of X that is monotone, null-empty on the empty set, and countably subadditive), a subset E ⊆ X is called Carathéodory-measurable if for every A ⊆ X, the equality m*(A) = m*(A ∩ E) + m*(A \ E) holds. This condition expresses that, with respect to m*, there is no loss or redundancy when splitting A along E.

The collection of all Carathéodory-measurable sets forms a sigma-algebra, denoted here as M. The outer measure m* restricted to M becomes a complete measure on M. This construction allows one to define a genuine measure from an outer measure, extending existing measures on smaller algebras. A central example is the Lebesgue outer measure on the real line: the Carathéodory criterion identifies the Lebesgue measurable sets, and m* restricted to these sets yields the Lebesgue measure.

The definition is widely used beyond Euclidean spaces, serving as a standard tool to define measurable sets in spaces equipped with an outer measure, such as in probability, geometric measure theory, and various generalizations. It is a foundational concept in measure theory, enabling the rigorous development of integration and probability with respect to generalized notions of size.