μ-completions

Noun

μ-completion (plural μ-completions)

1. 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.