Sigma-ideal


In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra 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 be a measurable space. A subset N of Σ is a σ-ideal if the following properties are satisfied:
Ø ∈ N;
When A ∈ N and B ∈ Σ, B ⊆ A ⇒ B ∈ 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 filter.
If a measure μ is given on, the set of μ-negligible sets is a σ-ideal.
The notion can be generalized to preorders with a bottom element 0 as follows: I is a σ-ideal of P just when
0 ∈ I,
x ≤ y & y ∈ I ⇒ x ∈ I, and
given a family xn ∈ I, there is y ∈ I such that xn ≤ y for each n
Thus I contains the bottom element, is downward closed, and satisfies a countable analogue of the property of being upwards directed.
A σ-ideal of a set X is a σ-ideal of the power set of X. That is, when no σ-algebra is specified, then one simply takes the full power set of the underlying set. For example, the meager subsets of a topological space are those in the σ-ideal generated by the collection of closed subsets with empty interior.