Sigma-ideal


In mathematics, particularly measure theory, a -ideal, or sigma ideal, of a σ-algebra 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. A subset of is a -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 -ideal is dual to that of a countably complete filter.
If a measure is given on the set of -negligible sets is a -ideal.
The notion can be generalized to preorders with a bottom element as follows: is a -ideal of just when

implies and
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.