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 ${\displaystyle (X,\Sigma )}$ be a measurable space (meaning ${\displaystyle \Sigma }$ is a 𝜎-algebra of subsets of ${\displaystyle X}$). A subset ${\displaystyle N}$ of ${\displaystyle \Sigma }$ is a 𝜎-ideal if the following properties are satisfied:

1. ${\displaystyle \varnothing \in N}$;
2. When ${\displaystyle A\in N}$ and ${\displaystyle B\in \Sigma }$ then ${\displaystyle B\subseteq A}$ implies ${\displaystyle B\in N}$;
3. If ${\displaystyle \left\{A_{n}\right\}_{n\in \mathbb {N} }\subseteq N}$ then ${\textstyle \bigcup _{n\in \mathbb {N} }A_{n}\in N.}$

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 ${\displaystyle \mu }$ is given on ${\displaystyle (X,\Sigma ),}$ the set of ${\displaystyle \mu }$-negligible sets (${\displaystyle S\in \Sigma }$ such that ${\displaystyle \mu (S)=0}$) is a 𝜎-ideal.

The notion can be generalized to preorders ${\displaystyle (P,\leq ,0)}$ with a bottom element ${\displaystyle 0}$ as follows: ${\displaystyle I}$ is a 𝜎-ideal of ${\displaystyle P}$ just when

(i') ${\displaystyle 0\in I,}$

(ii') ${\displaystyle x\leq y{\text{ and }}y\in I}$ implies ${\displaystyle x\in I,}$ and

(iii') given a sequence ${\displaystyle x_{1},x_{2},\ldots \in I,}$ there exists some ${\displaystyle y\in I}$ such that ${\displaystyle x_{n}\leq y}$ for each ${\displaystyle n.}$

Thus ${\displaystyle I}$ 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 ${\displaystyle X}$ is a 𝜎-ideal of the power set of ${\displaystyle X.}$ 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.