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 \lambda^* denote the Lebesgue outer measure on \mathbb{R}^n, and let E\subseteq\mathbb{R}^n. Then E is Lebesgue measurable if and only if \lambda^* (A) = \lambda^* (A \cap E) + \lambda^* (A \cap E^c) for every A\subseteq \mathbb{R}^n. Notice that A is not required to be a measurable set.

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