Complete measure

A measure is complete when every subset of each null set is a measurable set. Such a subset also has measure zero by monotonicity. The completion of a measure extends it to a complete measure by adjoining all subsets of measurable null sets to its sigma-algebra.

The completion of a measure space adjoins every subset of every measurable null set to its sigma-algebra. A statement that two events or σ-algebras agree modulo null sets becomes literal after passing to the corresponding measure-algebra completion.