σ-algebraとは 意味・読み方・使い方

別の表記

• sigma-algebra
• Σ-algebra

発音

• IPA: /ˈsɪɡ.mə ˈæl.dʒɪ.bɹə/

名詞

σ-algebra (plural σ-algebras)

1. (mathematical 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.
   • 2001, Elliott H. Lieb, Michael Loss, Analysis, American Mathematical Society, page 4:

     Consider all the sigma-algebras that contain ${\displaystyle {\mathcal {F}}}$ and take their intersection, which we call ${\displaystyle \Sigma }$, i.e., a subset ${\displaystyle A\subset \Omega }$ is in ${\displaystyle \Sigma }$ if and only if ${\displaystyle A}$ is in every sigma-algebra containing ${\displaystyle {\mathcal {F}}}$. It is easy to check that ${\displaystyle \Sigma }$ is indeed a sigma-algebra.

   • 2013, Alexandr A. Borovkov, Probability Theory, Springer, page 15:

     Consider all the σ-algebras on [0,1] containing all intervals from that segment (there is at least one such σ-algebra, for the collection of all the subsets of a given set clearly forms a σ-algebra).

   • 2017 February 4, Marco Taboga, Lectures on Probability Theory and Mathematical Statistics‎, 2nd edition, San Bernardino, CA, USA, →ISBN, §10.5.1, page 75:

         Denote by ${\displaystyle {\mathcal {F}}}$ the set of subsets of [the sample space] Ω which are considered events. ${\displaystyle {\mathcal {F}}}$ is called the space of events. In rigorous probability theory, ${\displaystyle {\mathcal {F}}}$ is required to be a sigma-algebra on Ω.

同意語

• (collection of subsets that obeys certain conditions): σ-field

上位語

• (collection of subsets that obeys certain conditions): field of sets, set algebra

下位語

• (collection of subsets that obeys certain conditions): Borel σ-algebra, μ-completion

Meronyms

• measurable set

Holonyms

• measurable space

参考

• σ-ring

Further reading

• Algebra of sets on Wikipedia.Wikipedia
• Sigma-ring on Wikipedia.Wikipedia
• sigma-algebra on nLab
• Algebra of sets on Encyclopedia of Mathematics
• Sigma-algebra (Computer Science) on Encyclopedia of Mathematics

アナグラム

• Labarge