Sigma-ideal

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.