In mathematics, particularly measure theory, a -ideal, or sigma 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 in probability theory. Let be a measurable space (meaning is a -algebra of subsets of ). A subset of is a -ideal if the following properties are satisfied: ; When and then implies ; If then 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 the set of -negligible sets ( such that ) is a -ideal. The notion can be generalized to preorders with a bottom element as follows: is a -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 -ideal of a set is a -ideal of the power set of That is, when no -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 -ideal generated by the collection of closed subsets with empty interior. See also References Bauer, Heinz (2001): Measure and Integration Theory. Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. Measure theory Families of sets