# 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 $\lambda^*$ 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.