In mathematics, particularly measure theory, a -ideal, or sigma ideal, of a σ-algebra (, 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 <math>(X, \Sigma)</math> be a measurable space (meaning <math>\Sigma</math> is a -algebra of subsets of <math>X</math>). A subset <math>N</math> of <math>\Sigma</math> is a -ideal if the following properties are satisfied:

  1. <math>\varnothing \in N</math>;
  2. When <math>A \in N</math> and <math>B \in \Sigma</math> then <math>B \subseteq A</math> implies <math>B \in N</math>;
  3. 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 measurable subsets and countable unions of its elements. The concept of -ideal is dual to that of a countably complete (-) 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 -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 -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 -ideal of a set <math>X</math> is a -ideal of the power set of <math>X.</math> That is, when no -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 -ideal generated by the collection of closed subsets with empty interior.

See also

References

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