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In mathematics, particularly measure theory, a 𝜎-ideal, or sigma ideal, of a sigma-algebra (𝜎, read "sigma," means countable in this context) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is in probability theory. 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 If a measure is given on the set of -negligible sets ( such that ) is a 𝜎-ideal. (i') (ii') implies and

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rdfs:label	Sigma-ideal (en)
rdfs:comment	In mathematics, particularly measure theory, a 𝜎-ideal, or sigma ideal, of a sigma-algebra (𝜎, read "sigma," means countable in this context) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is in probability theory. 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 If a measure is given on the set of -negligible sets ( such that ) is a 𝜎-ideal. (i') (ii') implies and (en)
dcterms:subject	Families of sets Measure theory
Wikipage page ID	1567410 (xsd:integer)
Wikipage revision ID	1106927343 (xsd:integer)
Link from a Wikipage to another Wikipage	Preorder Mathematics Measure (mathematics) Measure theory Negligible set Measure (mathematics) Closure (mathematics) Completeness (order theory) Meagre set Measurable space Duality (order theory) Countable Filter (mathematics) Directed set Probability theory Heinz Bauer Families of sets Measure theory Measure zero Ideal (set theory) Subset Countably
sameAs	Sigma-ideal Sigma-ideal Sigma-ideal Sigma-ideal
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has abstract	In mathematics, particularly measure theory, a 𝜎-ideal, or sigma ideal, of a sigma-algebra (𝜎, read "sigma," means countable in this context) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is in probability theory. 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. (en)
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foaf:isPrimaryTopicOf	wikipedia-en:Sigma-ideal
is Link from a Wikipage to another Wikipage of	Σ-ideal Nowhere dense set Negligible set Meagre set Null set S-ideal Sigma ideal
is Wikipage redirect of	Σ-ideal S-ideal Sigma ideal
is foaf:primaryTopic of	wikipedia-en:Sigma-ideal

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