Carathéodory's criterion

Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory that characterizes when a set is Lebesgue measurable.

Statement

Carathéodory's criterion:
Let denote the Lebesgue outer measure on where denotes the power set of and let Then is Lebesgue measurable if and only if for every where denotes the complement of
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:
If is an outer measure on a set where denotes the power set of then a subset is called ' or ' if for every the equalityholds where is the complement of
The family of all –measurable subsets is a σ-algebra and the restriction of the outer measure to this family is a measure.