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 be a measurable space (meaning is a 𝜎-algebra of subsets of ). A subset of is a 𝜎-ideal if the following properties are satisfied:
- ;
- When and then implies ;
- If then
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 is given on the set of -negligible sets ( such that ) is a 𝜎-ideal.
The notion can be generalized to preorders with a bottom element as follows: is a 𝜎-ideal of just when
(i')
(ii') implies and
(iii') given a sequence there exists some such that for each
Thus 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 is a 𝜎-ideal of the power set of 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.