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 formula_1 denote the Lebesgue outer measure on formula_2, and let formula_3. Then formula_4 is Lebesgue measurable if and only if formula_5 for every formula_6, formula_7 denoting the complement of formula_4. Notice that formula_9 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 formula_10, 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 formula_11 be an outer measure on a set formula_12. Then formula_13 is called formula_11–"measurable" if for every formula_15, the equality formula_16 holds.