In mathematics, particularly measure theory, a 𝜎-ideal, or sigma ideal, of a σ-algebra (𝜎, read "sigma") is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is in probability theory. Let (X, \Sigma) be a measurable space (meaning \Sigma is a 𝜎-algebra of subsets of X). A subset N of \Sigma is a 𝜎-ideal if the following properties are satisfied: 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 \mu is given on the set of \mu-negligible sets ( such that \mu(S) = 0) is a 𝜎-ideal. The notion can be generalized to preorders with a bottom element 0 as follows: I is a 𝜎-ideal of P just when (i') 0 \in I, (ii') implies x \in I, and (iii') given a sequence there exists some y \in I such that x_n \leq y for each n. Thus I 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 X is a 𝜎-ideal of the power set of X. 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.