Sigma-ideal

Template:Short description In mathematics, particularly measure theory, a Template:Sigma-ideal, or sigma ideal, of a σ-algebra (Template:Sigma, 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.Template:Cn

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

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

Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. The concept of Template:Sigma-ideal is dual to that of a countably complete (Template:Sigma-) filter.

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

The notion can be generalized to preorders $(P,\le ,0)$ with a bottom element $0$ as follows: $I$ is a Template:Sigma-ideal of $P$ just when

(i') $0\in I,$

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

(iii') given a sequence $x_1,x_2,\dots \in I,$ there exists some $y\in I$ such that $x_n\le y$ for each $n.$

Thus $I$ contains the bottom element, is downward closed, and satisfies a countable analogue of the property of being upwards directed.

A Template:Sigma-ideal of a set $X$ is a Template:Sigma-ideal of the power set of $X.$ That is, when no Template:Sigma-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 Template:Sigma-ideal generated by the collection of closed subsets with empty interior.

See also

• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link

References

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