https://en.wikipedia.org/wiki/Sigma-ideal σ-ideal Sigma-ideal (Redirected from Σ-ideal) (抜粋) 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) O ∈ N; (ii) When A ∈ N and B ∈ Σ , B ⊆ A ⇒ B ∈ N; (iii) {A_n}_{n∈N }⊆ N→ ∪ _{n∈N }A_n∈ 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, (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 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.
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. (引用終り)
