Sigma-idealFrom Wikipedia, the free encyclopedia

In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra (σ, read “sigma,” means countable in thiscontext) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent applicationis 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 thefollowing properties are satisfied:(i) Ø ∈ N;(ii) When A ∈ N and B ∈ Σ , B ⊆ A⇒ B ∈ N;(iii) {An}n∈N ⊆ N ⇒

∪n∈N An ∈ N.

Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. Theconcept 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.

1 References• Bauer, Heinz (2001): Measure and Integration Theory. Walter de Gruyter GmbH & Co. KG, 10785 Berlin,Germany.

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