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 (Ω,F) be a measurable space (meaning F is a σ-algebra of subsets of Ω). A subset N of F is a σ-ideal if the following properties are satisfied:

(i) Ø ∈ N;

(ii) When A ∈ N and B ∈ F , 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 m is given on (Ω,F), the set of m-negligible sets (S ∈ F such that m(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 x_n ∈ I (n ∈ N), there is y ∈ I such that x_n ≤ y for each n

Thus 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.

Reference

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