Sigma-ideal

In mathematics, particularly measure theory, a "σ"-ideal of a sigma-algebra ("σ", 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 perhaps in probability theory.

Let ("X",Σ) be a measurable space (meaning Σ is a "σ"-algebra of subsets of "X"). A subset "N" of Σ is a "σ"-ideal if the following properties are satisfied:

(i) Ø ∈ "N";

(ii) When "A" ∈ "N" and "B" ∈ Σ , "B" ⊆ "A" ⇒ "B" ∈ "N";

(iii)

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 "μ" is given on ("X",Σ), the set of "μ"-negligible sets ("S" ∈ Σ such that "μ"("S") = 0) is a "σ"-ideal.

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

(i') 0 ∈ "I",

(ii') "x" ≤ "y" & "y" ∈ "I" ⇒ "x" ∈ "I", and

(iii') given a family "x"_"n" ∈ "I" ("n" ∈ N), there is "y" ∈ "I" such that "x"_"n" ≤ "y" for each "n"

Thus "I" contains the bottom element, is downward closed, and is closed under countable suprema (which must exist). It is natural in this context to ask that "P" itself have countable suprema.

References

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