High Quality Content by WIKIPEDIA articles! In mathematics, particularly measure theory, a ?-ideal of a sigma-algebra (?, 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 perhaps in probability theory. Let (X,?) be a measurable space (meaning ? is a ?-algebra of subsets of X). A subset N of ? is a ?-ideal if the following properties are satisfied: (i) ? N; (ii) When A ? N and B ? ? , B ? A ? B ? N; (iii) left{A_n right}_{ninmathbb{N}} subseteq N Rightarrow bigcup_{ninmathbb{N}} A_nin N. Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. The concept of ?-ideal is dual to that of a countably complete (?-) filter. If a measure ? is given on (X,?), the set of ?-negligible sets (S ? ? such that ?(S) = 0) is a ?-ideal.

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