Carathéodory's criterion

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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.

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