complete measureCompleteMeasure2004-01-18 16:24:022004-01-18 16:57:03Definitioncompletioncomplete% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Per}{\operatorname{Per}}A measure space $(X,\mathscr{S},\mu)$ is said to be \emph{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 \PMlinkname{completion}{CompletionOfAMeasureSpace} 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 $\mathscr{S}$.