A measure space $(X,\mathscr{S},\mu)$ 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 $E\in\mathscr{S}$ 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 of it, which is a complete measure space $(X,\overline{\mathscr{S}},\overline{\mu})$ such that $\mathscr{S}\subset\overline{\mathscr{S}}$ and $\overline {\mu}_{|\mathscr{S}} = \mu$ where $\overline{\mathscr{S}}$ is the smallest $\sigma$ algebra containing both $\mathscr{S}$ and all subsets of elements of zero measure of $\mathscr{S}$