Sigma-ideal

Short description: Family closed under subsets and countable unions

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.

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

1. [math]\displaystyle{ \varnothing \in N }[/math];
2. When [math]\displaystyle{ A \in N }[/math] and [math]\displaystyle{ B \in \Sigma }[/math] then [math]\displaystyle{ B \subseteq A }[/math] implies [math]\displaystyle{ B \in N }[/math];
3. If [math]\displaystyle{ \left\{A_n\right\}_{n \in \N} \subseteq N }[/math] then [math]\displaystyle{ \bigcup_{n \in \N} A_n \in N. }[/math]

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 [math]\displaystyle{ \mu }[/math] is given on [math]\displaystyle{ (X, \Sigma), }[/math] the set of [math]\displaystyle{ \mu }[/math]-negligible sets ([math]\displaystyle{ S \in \Sigma }[/math] such that [math]\displaystyle{ \mu(S) = 0 }[/math]) is a 𝜎-ideal.

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

(i') [math]\displaystyle{ 0 \in I, }[/math]

(ii') [math]\displaystyle{ x \leq y \text{ and } y \in I }[/math] implies [math]\displaystyle{ x \in I, }[/math] and

(iii') given a sequence [math]\displaystyle{ x_1, x_2, \ldots \in I, }[/math] there exists some [math]\displaystyle{ y \in I }[/math] such that [math]\displaystyle{ x_n \leq y }[/math] for each [math]\displaystyle{ n. }[/math]

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

See also

• δ-ring – Ring closed under countable intersections
• Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
• 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
• Measurable function – Function for which the preimage of a measurable set is measurable
• π-system – Family of sets closed under intersection
• Ring of sets – Family closed under unions and relative complements
• Sample space – Set of all possible outcomes or results of a statistical trial or experiment
• 𝜎-algebra – Algebraic structure of set algebra
• 𝜎-ring – Ring closed under countable unions
• Sigma additivity

References

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