Sigma-ideal

In mathematics, particularly measure theory, a 𝜎-ideal, or sigma ideal, of a σ-algebra (𝜎, read "sigma") is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is in probability theory.[citation needed]

Let ( X , Σ ) {\displaystyle (X,\Sigma )} be a measurable space (meaning Σ {\displaystyle \Sigma } is a 𝜎-algebra of subsets of X {\displaystyle X}). A subset N {\displaystyle N} of Σ {\displaystyle \Sigma } is a 𝜎-ideal if the following properties are satisfied:

∅ ∈ N {\displaystyle \varnothing \in N};
When A ∈ N {\displaystyle A\in N} and B ∈ Σ {\displaystyle B\in \Sigma } then B ⊆ A {\displaystyle B\subseteq A} implies B ∈ N {\displaystyle B\in N};
If { A n } n ∈ N ⊆ N {\displaystyle \left\{A_{n}\right\}_{n\in \mathbb {N} }\subseteq N} then ⋃ n ∈ N A n ∈ N . {\textstyle \bigcup _{n\in \mathbb {N} }A_{n}\in N.}

Briefly, a sigma-ideal must contain the empty set and contain measurable 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 {\displaystyle \mu } is given on ( X , Σ ) , {\displaystyle (X,\Sigma ),} the set of μ {\displaystyle \mu }-negligible sets (S ∈ Σ {\displaystyle S\in \Sigma } such that μ ( S ) = 0 {\displaystyle \mu (S)=0}) is a 𝜎-ideal.

The notion can be generalized to preorders ( P , ≤ , 0 ) {\displaystyle (P,\leq ,0)} with a bottom element 0 {\displaystyle 0} as follows: I {\displaystyle I} is a 𝜎-ideal of P {\displaystyle P} just when

(i') 0 ∈ I , {\displaystyle 0\in I,}

(ii') x ≤ y and y ∈ I {\displaystyle x\leq y{\text{ and }}y\in I} implies x ∈ I , {\displaystyle x\in I,} and

(iii') given a sequence x 1 , x 2 , … ∈ I , {\displaystyle x_{1},x_{2},\ldots \in I,} there exists some y ∈ I {\displaystyle y\in I} such that x n ≤ y {\displaystyle x_{n}\leq y} for each n . {\displaystyle n.}

Thus I {\displaystyle 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 {\displaystyle X} is a 𝜎-ideal of the power set of X . {\displaystyle 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.

Sigma-ideal

See also