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.

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

1. $\varnothing \in N$;
2. When $A\in N$ and $B\in \Sigma$ then $B\subseteq A$ implies $B\in N$;
3. If $\left\{A_{n}\right\}_{n\in \mathbb {N} }\subseteq N$ then ${\textstyle \bigcup _{n\in \mathbb {N} }A_{n}\in N.}$

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 $\mu$ is given on $(X,\Sigma ),$ the set of $\mu$-negligible sets ($S\in \Sigma$ such that $\mu (S)=0$) is a 𝜎-ideal.

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

(i') $0\in I,$

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

(iii') given a sequence $x_{1},x_{2},\ldots \in I,$ there exists some $y\in I$ such that $x_{n}\leq y$ 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.

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
• Join (sigma algebra) – Algebraic structure of set algebraPages displaying short descriptions of redirect targets
• 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 – Family of sets closed under countable unions
• Sigma additivity – Mapping functionPages displaying short descriptions of redirect targets

References

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