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Sigma-ideal
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{{Short description|Family closed under subsets and countable unions}} In [[mathematics]], particularly [[measure theory]], a '''{{sigma}}-ideal''', or '''sigma ideal''', of a [[Ο-algebra]] ({{sigma}}, read "sigma") is a [[subset]] with certain desirable [[Closure (mathematics)|closure]] properties. It is a special type of [[Ideal (set theory)|ideal]]. Its most frequent application is in [[probability theory]].{{cn|date=December 2020}} Let <math>(X, \Sigma)</math> be a [[measurable space]] (meaning <math>\Sigma</math> is a {{sigma}}-algebra of subsets of <math>X</math>). A subset <math>N</math> of <math>\Sigma</math> is a {{sigma}}-ideal if the following properties are satisfied: # <math>\varnothing \in N</math>; # When <math>A \in N</math> and <math>B \in \Sigma</math> then <math>B \subseteq A</math> implies <math>B \in N</math>; # If <math>\left\{A_n\right\}_{n \in \N} \subseteq N</math> then <math display=inline>\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 {{sigma}}-ideal is [[Duality (order theory)|dual]] to that of a [[countably]] [[Completeness (order theory)|complete]] ({{sigma}}-) [[Filter (mathematics)|filter]]. If a [[Measure (mathematics)|measure]] <math>\mu</math> is given on <math>(X, \Sigma),</math> the set of <math>\mu</math>-[[negligible set]]s (<math>S \in \Sigma</math> such that [[Measure zero|<math>\mu(S) = 0</math>]]) is a {{sigma}}-ideal. The notion can be generalized to [[preorder]]s <math>(P, \leq, 0)</math> with a bottom element <math>0</math> as follows: <math>I</math> is a {{sigma}}-ideal of <math>P</math> just when (i') <math>0 \in I,</math> (ii') <math>x \leq y \text{ and } y \in I</math> implies <math>x \in I,</math> and (iii') given a sequence <math>x_1, x_2, \ldots \in I,</math> there exists some <math>y \in I</math> such that <math>x_n \leq y</math> for each <math>n.</math> Thus <math>I</math> contains the bottom element, is downward closed, and satisfies a countable analogue of the property of being [[Directed set|upwards directed]]. A '''{{sigma}}-ideal''' of a set <math>X</math> is a {{sigma}}-ideal of the power set of <math>X.</math> That is, when no {{sigma}}-algebra is specified, then one simply takes the full power set of the underlying set. For example, the [[Meagre set|meager subsets]] of a topological space are those in the {{sigma}}-ideal generated by the collection of closed subsets with empty interior. == See also == * {{annotated link|Delta-ring|{{delta}}-ring}} * {{annotated link|Field of sets}} * {{annotated link|Join (sigma algebra)}} * {{annotated link|Dynkin system|{{lambda}}-system (Dynkin system)}} * {{annotated link|Measurable function}} * {{annotated link|Pi-system|{{pi}}-system}} * {{annotated link|Ring of sets}} * {{annotated link|Sample space}} * {{annotated link|Ο-algebra|{{sigma}}-algebra}} * {{annotated link|Sigma-ring|{{sigma}}-ring}} * {{annotated link|Sigma additivity}} == References == * [[Heinz Bauer|Bauer, Heinz]] (2001): ''Measure and Integration Theory''. Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. [[Category:Measure theory]] [[Category:Families of sets]]
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