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- Sigma-ideal abstract "In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra (σ, read "sigma," means countable in this context) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is perhaps in probability theory.Let (X,Σ) be a measurable space (meaning Σ is a σ-algebra of subsets of X). A subset N of Σ is a σ-ideal if the following properties are satisfied:(i) Ø ∈ N;(ii) When A ∈ N and B ∈ Σ , B ⊆ A ⇒ B ∈ N;(iii) Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. The concept of σ-ideal is dual to that of a countably complete (σ-) filter.If a measure μ is given on (X,Σ), the set of μ-negligible sets (S ∈ Σ such that μ(S) = 0) is a σ-ideal.The notion can be generalized to preorders (P,≤,0) with a bottom element 0 as follows: I is a σ-ideal of P just when(i') 0 ∈ I,(ii') x ≤ y & y ∈ I ⇒ x ∈ I, and(iii') given a family xn ∈ I (n ∈ N), there is y ∈ I such that xn ≤ y for each nThus I contains the bottom element, is downward closed, and is closed under countable suprema (which must exist). It is natural in this context to ask that P itself have countable suprema.".
- Sigma-ideal wikiPageID "1567410".
- Sigma-ideal wikiPageRevisionID "354005516".
- Sigma-ideal hasPhotoCollection Sigma-ideal.
- Sigma-ideal subject Category:Measure_theory.
- Sigma-ideal subject Category:Set_families.
- Sigma-ideal type Abstraction100002137.
- Sigma-ideal type Family108078020.
- Sigma-ideal type Group100031264.
- Sigma-ideal type Organization108008335.
- Sigma-ideal type SetFamilies.
- Sigma-ideal type SocialGroup107950920.
- Sigma-ideal type Unit108189659.
- Sigma-ideal type YagoLegalActor.
- Sigma-ideal type YagoLegalActorGeo.
- Sigma-ideal type YagoPermanentlyLocatedEntity.
- Sigma-ideal comment "In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra (σ, read "sigma," means countable in this context) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is perhaps in probability theory.Let (X,Σ) be a measurable space (meaning Σ is a σ-algebra of subsets of X).".
- Sigma-ideal label "Sigma-ideal".
- Sigma-ideal sameAs m.05b_yp.
- Sigma-ideal sameAs Q7512218.
- Sigma-ideal sameAs Q7512218.
- Sigma-ideal sameAs Sigma-ideal.
- Sigma-ideal wasDerivedFrom Sigma-ideal?oldid=354005516.
- Sigma-ideal isPrimaryTopicOf Sigma-ideal.