 Lebesgue measurable set

A subset $E\subseteq \mathbb{R}$ is a Lebesgue measurable set when it satisfies Carathéodory's criterion for Lebesgue outer measure: for every subset $A\subseteq \mathbb{R}$,These sets form the Lebesgue sigma-algebra, the completion of the Borel sigma-algebra with respect to Lebesgue measure. Equivalently, $E$ differs from a Borel set by a subset of a Borel null set.