Sigma-ideal

In mathematics, particularly measure theory, a sigma-ideal, or sigma ideal, of a sigma-algebra (sigma, 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 sigma-algebra of subsets of ). A subset of is a sigma-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 sigma-ideal is dual to that of a countably complete (sigma-) filter. If a measure is given on the set of -negligible sets ( such that ) is a sigma-ideal. The notion can be generalized to preorders with a bottom element as follows: is a sigma-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 sigma-ideal of a set is a sigma-ideal of the power set of That is, when no sigma-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 sigma-ideal generated by the collection of closed subsets with empty interior.