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Sigma-ideal

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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:

  1. ;
  2. When and then implies ;
  3. 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[edit]

  • 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 algebra
  • 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 – Mapping function

References[edit]

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