PlanetMath: complete measure

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complete measure

(Definition)

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 $\overline {\mu}_{\vert\mathscr{S}} = \mu$ , where $\overline{\mathscr{S}}$ is the smallest $\sigma$ -algebra containing both $\mathscr{S}$ and all subsets of elements of zero measure of $\mathscr{S}$ .

"complete measure" is owned by Koro.

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Also defines:

completion, complete

Attachments:: completion of a measure space (Derivation) by Koro

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Cross-references: measurable, measure, subset, measure space
There are 31 references to this entry.

This is version 2 of complete measure, born on 2004-01-18, modified 2004-01-18.
Object id is 5520, canonical name is CompleteMeasure.
Accessed 5676 times total.

Classification:

AMS MSC:

28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)

Pending Errata and Addenda

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