A set $E$ is measurable exactly when, for every set $A$, the outer measure of $A$ is additive over $A\cap E$ and $A\setminus E$.Suggest a slogan
Let \(m_*\) be the outer measure on \(\RR^n\). A set \(E\subseteq \RR^n\) is Lebesgue measurable if and only if for every \(A\subseteq \RR^n\), \[ m_*(A) = m_*(A\cap E) + m_*(A\setminus E). \]
What is the Carathéodory criterion for Lebesgue measurability?