English Dictionary

μ-completion

名詞

(mathematical 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).

日本語の意味

（数学解析において）与えられたσ‐加法族（測度空間の可測集合の族）を補完する操作によって得られるσ‐加法族。つまり、もともとのσ‐加法族に含まれないが、ある内包集合と包含集合が同じ測度を持つ場合にその共通の測度を割り当てることで、補足的に定義される集合を加えた結果得られる完全なσ‐加法族。 / 測度の拡張という観点から、既存のσ‐加法族が持つ測度定義を補完し、測度空間内の例外的（零集合的な）部分集合も測度の一貫性を保って含める拡張されたσ‐加法族を表す。

When constructing the μ-completion of the measurable space, we adjoin all subsets that lie between two measurable sets of equal measure so that the measure becomes complete as well.

測度空間のミューの完備化を構成する際には、同じ測度を持つ二つの可測集合の間に挟まれたすべての部分集合を付け加え、測度も同時に完備化します。

意味（1）

(mathematical 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.

μ-completions （ plural ）