*μ*-completion

Definition from Wiktionary, the free dictionary

## English[edit]

### Noun[edit]

** μ-completion** (

*plural*

**μ-completions**)

- (analysis) A
*σ*-algebra which is obtained as a "completion" of a given*σ*-algebra, which includes all subsets of the given measure space which simultaneously contain a member of the given*σ*-algebra and are contained by a member of the given*σ*-algebra, as long as the contained and containing measurable sets have the same measure, in which case the subset in question is assigned a measure equal to the common measure of its contained and containing measurable sets (so the measure is also being completed, in parallel with the*σ*-algebra).*Every σ-algebra has a***μ-completion**: if a σ-algebra is complete, then it is equal to its μ-completion, otherwise it is contained by its μ-completion.