# complete measure

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 (http://planetmath.org/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 $\mathscr{S}$.

Title complete measure CompleteMeasure 2013-03-22 14:06:56 2013-03-22 14:06:56 Koro (127) Koro (127) 5 Koro (127) Definition msc 28A12 UniversallyMeasurable completion complete