• Source: 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.
Let



(
X
,
Σ
)


{\displaystyle (X,\Sigma )}

be a measurable space (meaning



Σ


{\displaystyle \Sigma }

is a 𝜎-algebra of subsets of



X


{\displaystyle X}

). A subset



N


{\displaystyle N}

of



Σ


{\displaystyle \Sigma }

is a 𝜎-ideal if the following properties are satisfied:




∅
∈
N


{\displaystyle \varnothing \in N}

;
When



A
∈
N


{\displaystyle A\in N}

and



B
∈
Σ


{\displaystyle B\in \Sigma }

then



B
⊆
A


{\displaystyle B\subseteq A}

implies



B
∈
N


{\displaystyle B\in N}

;
If





{

A

n


}


n
∈

N



⊆
N


{\displaystyle \left\{A_{n}\right\}_{n\in \mathbb {N} }\subseteq N}

then




⋃

n
∈

N




A

n


∈
N
.


{\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



μ


{\displaystyle \mu }

is given on



(
X
,
Σ
)
,


{\displaystyle (X,\Sigma ),}

the set of



μ


{\displaystyle \mu }

-negligible sets (



S
∈
Σ


{\displaystyle S\in \Sigma }

such that



μ
(
S
)
=
0


{\displaystyle \mu (S)=0}

) is a 𝜎-ideal.
The notion can be generalized to preorders



(
P
,
≤
,
0
)


{\displaystyle (P,\leq ,0)}

with a bottom element



0


{\displaystyle 0}

as follows:



I


{\displaystyle I}

is a 𝜎-ideal of



P


{\displaystyle P}

just when
(i')



0
∈
I
,


{\displaystyle 0\in I,}


(ii')



x
≤
y

and

y
∈
I


{\displaystyle x\leq y{\text{ and }}y\in I}

implies



x
∈
I
,


{\displaystyle x\in I,}

and
(iii') given a sequence




x

1


,

x

2


,
…
∈
I
,


{\displaystyle x_{1},x_{2},\ldots \in I,}

there exists some



y
∈
I


{\displaystyle y\in I}

such that




x

n


≤
y


{\displaystyle x_{n}\leq y}

for each



n
.


{\displaystyle n.}


Thus



I


{\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



X


{\displaystyle X}

is a 𝜎-ideal of the power set of



X
.


{\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.




See also


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




References


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

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