sigma-ideal

{{Short description|Family closed under subsets and countable unions}}

In mathematics, particularly measure theory, a {{sigma}}-ideal, or sigma ideal, of a σ-algebra ({{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.{{cn|date=December 2020}}

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

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

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

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

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

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

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

(iii') given a sequence $x\_1,\; x\_2,\; \backslash ldots\; \backslash in\; I,$ there exists some $y\; \backslash in\; I$ such that $x\_n\; \backslash leq\; y$ 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 {{sigma}}-ideal of a set $X$ is a {{sigma}}-ideal of the power set of $X.$ That is, when no {{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 {{sigma}}-ideal generated by the collection of closed subsets with empty interior.