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 <math>(X, \Sigma)</math> be a measurable space (meaning <math>\Sigma</math> is a Template:Sigma-algebra of subsets of <math>X</math>). A subset <math>N</math> of <math>\Sigma</math> is a Template:Sigma-ideal if the following properties are satisfied:

<math>\varnothing \in N</math>;
When <math>A \in N</math> and <math>B \in \Sigma</math> then <math>B \subseteq A</math> implies <math>B \in N</math>;
If <math>\left\{A_n\right\}_{n \in \N} \subseteq N</math> then <math display=inline>\bigcup_{n \in \N} A_n \in N.</math>

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 <math>\mu</math> is given on <math>(X, \Sigma),</math> the set of <math>\mu</math>-negligible sets (<math>S \in \Sigma</math> such that <math>\mu(S) = 0</math>) is a Template:Sigma-ideal.

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

(i') <math>0 \in I,</math>

(ii') <math>x \leq y \text{ and } y \in I</math> implies <math>x \in I,</math> and

(iii') given a sequence <math>x_1, x_2, \ldots \in I,</math> there exists some <math>y \in I</math> such that <math>x_n \leq y</math> for each <math>n.</math>

Thus <math>I</math> 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 <math>X</math> is a Template:Sigma-ideal of the power set of <math>X.</math> 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.

ReferencesEdit

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

Sigma-ideal

See alsoEdit

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