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.
[citation needed] Let
be a measurable space (meaning
is a 𝜎-algebra of subsets of
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
μ
The notion can be generalized to preorders
with a bottom element
and (iii') given a sequence
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
is a 𝜎-ideal of the power set of
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.