complete measure

A measure space $(X,𝒮,\mu )$ is said to be complete if every subset of a set of measure $0$ is measurable (and consequently, has measure $0$); i.e. if for all $E\in 𝒮$ such that $\mu (E)=0$ and for all $S\subset E$ we have $\mu (S)=0$.

If a measure space is not complete, there exists a completion (http://planetmath.org/CompletionOfAMeasureSpace) of it, which is a complete measure space $(X,\overline{𝒮},\overline{\mu })$ such that $𝒮\subset \overline{𝒮}$ and ${\overline{\mu }}_{|𝒮}=\mu$, where $\overline{𝒮}$ is the smallest $\sigma$-algebra containing both $𝒮$ and all subsets of elements of zero measure of $𝒮$.