# Sigma-ideal

In mathematics, particularly measure theory, a ** σ-ideal** of a sigma-algebra (

*σ*, 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 perhaps in probability theory.

Let (*X*,Σ) be a measurable space (meaning Σ is a *σ*-algebra of subsets of *X*). A subset *N* of Σ is a *σ*-ideal if the following properties are satisfied:

(i) Ø ∈ *N*;

(ii) When *A* ∈ *N* and *B* ∈ Σ , *B* ⊆ *A* ⇒ *B* ∈ *N*;

(iii)

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* is given on (*X*,Σ), the set of *μ*-negligible sets (*S* ∈ Σ such that *μ*(*S*) = 0) is a *σ*-ideal.

The notion can be generalized to preorders (*P*,≤,0) with a bottom element 0 as follows: *I* is a *σ*-ideal of *P* just when

(i') 0 ∈ *I*,

(ii') *x* ≤ *y* & *y* ∈ *I* ⇒ *x* ∈ *I*, and

(iii') given a family *x*_{n} ∈ *I* (*n* ∈ **N**), there is *y* ∈ *I* such that *x*_{n} ≤ *y* for each *n*

Thus *I* contains the bottom element, is downward closed, and is closed under countable suprema (which must exist). It is natural in this context to ask that *P* itself have countable suprema.

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.

## References

- Bauer, Heinz (2001):
*Measure and Integration Theory*. Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.