Sigma-ideal

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


 * 1) $$\varnothing \in N$$;
 * 2) When $$A \in N$$ and $$B \in \Sigma$$ then $$B \subseteq A$$ implies $$B \in N$$;
 * 3) If $$\left\{A_n\right\}_{n \in \N} \subseteq N$$ then $\bigcup_{n \in \N} A_n \in N.$

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 $$\mu$$ is given on $$(X, \Sigma),$$ the set of $$\mu$$-negligible sets ($$S \in \Sigma$$ such that $\mu(S) = 0$) is a 𝜎-ideal.

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

(i') $$0 \in I,$$

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

(iii') given a sequence $$x_1, x_2, \ldots \in I,$$ there exists some $$y \in I$$ such that $$x_n \leq y$$ 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 𝜎-ideal of a set $$X$$ is a 𝜎-ideal of the power set of $$X.$$ 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.