Carathéodory's criterion
Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory. Its statement is as follows: Let denote the Lebesgue outer measure on, and let. Then is Lebesgue measurable if and only if for every.
Notice that is not required to be a measurable set.
The Carathéodory criterion is of considerable importance because, in contrast to Lebesgue's original formulation of measurability, which relies on certain topological properties of, this criterion readily generalizes to a characterization of measurability in abstract spaces. Indeed, in the generalization to abstract measures, this theorem is sometimes extended to a definition of measurability. Thus, we have the following definition: Let be an outer measure on a set. Then is called –measurable if for every, the equality holds.
