μ-completion

noun

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.”

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Meanings

nounμ-completion (plural μ-completions)

plural μ-completions

  1. mathematical analysisA 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.

Cite this entry
APA

μ-completion. (n.d.). In WatkinsX Dictionary. Retrieved November 3, 2026, from https://dictionary.watkinsx.com/word/%CE%BC-completion

MLA

“μ-completion.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/%CE%BC-completion. Accessed November 3, 2026.

Chicago

WatkinsX Dictionary, s.v. “μ-completion,” accessed November 3, 2026, https://dictionary.watkinsx.com/word/%CE%BC-completion.

Source: μ-completion on Wiktionary, CC BY-SA 4.0.