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.

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) ${\displaystyle \left\{A_{n}\right\}_{n\in \mathbb {N} }\subseteq N\Rightarrow \bigcup _{n\in \mathbb {N} }A_{n}\in N.}$

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.

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