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.

• δ-ring – Ring closed under countable intersections
• Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
• Join (sigma algebra) – Algebraic structure of set algebraPages displaying short descriptions of redirect targets
• 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
• Measurable function – Kind of mathematical function
• π-system – Family of sets closed under intersection
• Ring of sets – Family closed under unions and relative complements
• Sample space – Set of all possible outcomes or results of a statistical trial or experiment
• 𝜎-algebra – Algebraic structure of set algebra
• 𝜎-ring – Family of sets closed under countable unions
• Sigma additivity – Mapping functionPages displaying short descriptions of redirect targets

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