The difference between Μ-completion and Σ-algebra
When used as nouns, μ-completion means 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), whereas σ-algebra means a collection of subsets of a given set, such that the empty set is part of this collection, the collection is closed under complements (with respect to the given set) and the collection is closed under countable unions.
check bellow for the other definitions of Μ-completion and Σ-algebra
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Μ-completion as a noun (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).
Examples:
"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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Σ-algebra as a noun (analysis):
A collection of subsets of a given set, such that the empty set is part of this collection, the collection is closed under complements (with respect to the given set) and the collection is closed under countable unions.