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 –measurable or Carathéodory-measurable if for every the equality
The family of all –measurable subsets is a σ-algebra (so for instance, the complement of a –measurable set is –measurable, and the same is true of countable intersections and unions of –measurable sets) and the restriction of the outer measure to this family is a measure.