Measure theory | Families of sets

Sigma-ideal

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. (Wikipedia).

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Subset | Filter (mathematics) | Preorder | Mathematics | Probability theory | Closure (mathematics) | Measure (mathematics) | Duality (order theory) | Directed set | Meagre set | Negligible set | Measurable space | Ideal (set theory) | Completeness (order theory)