 Completion of a measure

The completion of a measure extends a measure space $(X,\mathcal{F},\mu )$ so that every subset of a measurable null set is measurable. Its completed sigma-algebra isand the extension is $\overline{\mu }(A\cup N)=\mu (A)$. This is well-defined: two such representations differ only inside a measurable null set, so their measurable parts have the same measure. The extension agrees with $\mu$ on $\mathcal{F}$ and is a complete measure.^[ref]

For example, completing the Borel measure given by length on the real line gives Lebesgue measure on the Lebesgue measurable sets, including every subset of every Borel null set. Completion of a measure changes which sets are measurable; it does not identify distinct sets. If $A$ and $B$ differ only on a null set, they can still be different sets in the completed sigma-algebra. Equality modulo null sets is an additional quotient construction.