Lebesgue measurable set

A Lebesgue measurable set is a member of the completion of the Borel sigma-algebra with respect to Lebesgue measure. In particular, it differs from a Borel set by a null set.

Every Lebesgue measurable set of finite measure can be approximated arbitrarily closely in measure from outside by an open set and from inside by a compact set. On a bounded interval it can consequently be approximated in symmetric-difference measure by a finite union of intervals.