Sigma-ideal

In mathematics, particularly measure theory, a -ideal, or sigma ideal, of a sigma-algebra (, read "sigma," means countable in this context) 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 -algebra of subsets of $X$). A subset $N$ of $\Sigma$ is a -ideal if the following properties are satisfied:

# $\varnothing \in N$;
# When $A \in N$ and $B \in \Sigma$ then $B \subseteq A$ implies $B \in N$;
# If $\left\_ \subseteq N$ then $\bigcup_ A_n \in N.$

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

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

(i') $0 \in I,$

(ii') $x \leq y \text y \in I$ 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 $y.$

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

A -ideal of a set $X$ is a -ideal of the power set of $X.$ 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

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* {{annotated link, Sigma additivity

References

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

Measure theory
Families of sets