= 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> is
$$
\overline{\mathcal F}=\{A\cup N:A\in\mathcal F,\ N\subseteq Z\in\mathcal F,\ \mu(Z)=0\},
$$
and 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>.<Measure Theory by John K. Hunter>[Chapter 1, Theorem 1.15]{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.
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