Sigma-ideal

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. 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) { A n } n ∈ N ⊆ N ⇒ ⋃ n ∈ N A n ∈ N . {displaystyle left{A_{n} ight}_{nin mathbb {N} }subseteq NRightarrow igcup _{nin mathbb {N} }A_{n}in N.} 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,