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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 <math>\lambda^*</math> denote the [[Lebesgue measure|Lebesgue]] [[outer measure]] on <math>\mathbb{R}^n</math>, and let <math>E\subseteq\mathbb{R}^n</math>. Then <math>E</math> is [[Lebesgue measurable]] if and only if <math>\lambda^* (A) = \lambda^* (A \cap E) + \lambda^* (A \cap E^c)</math> for every <math>A\subseteq \mathbb{R}^n</math>. |