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:

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 ${\displaystyle n.}$