Sigma-ideal

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In mathematics, particularly measure theory, a ๐œŽ-ideal, or sigma ideal, of a ฯƒ-algebra (๐œŽ, 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 ๐œŽ-algebra of subsets of ). A subset of is a ๐œŽ-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 ๐œŽ-ideal is dual to that of a countably complete (๐œŽ-) filter.

If a measure is given on the set of -negligible sets ( such that ) is a ๐œŽ-ideal.

The notion can be generalized to preorders with a bottom element as follows: is a ๐œŽ-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 ๐œŽ-ideal of a set is a ๐œŽ-ideal of the power set of That is, when no ๐œŽ-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 ๐œŽ-ideal generated by the collection of closed subsets with empty interior.

See also[edit]

  • δ-ring โ€“ Ring closed under countable intersections
  • Field of sets โ€“ Algebraic concept in measure theory, also referred to as an algebra of sets
  • Join (sigma algebra) โ€“ Algebraic structure of set algebra
  • ๐œ†-system (Dynkin system) โ€“ Family closed under complements and countable disjoint unions
  • Measurable function โ€“ Function for which the preimage of a measurable set is measurable
  • ฯ€-system โ€“ Family of sets closed under intersection
  • Ring of sets โ€“ Family closed under unions and relative complements
  • Sample space โ€“ Set of all possible outcomes or results of a statistical trial or experiment
  • ๐œŽ-algebra โ€“ Algebraic structure of set algebra
  • ๐œŽ-ring โ€“ Ring closed under countable unions
  • Sigma additivity โ€“ Mapping function

References[edit]

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