Sigma-ideal explained

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

(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 ;

1. When

A \in N and B \in \Sigma then B \subseteq A implies B \in N ;

1. If

\left\{A _{n\right\}} _{n} \subseteq N then $\backslash bigcup\_\; A\_n\; \backslash in\; N.$

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

\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 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 -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.

References

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