In mathematics, particularly measure theory, a π-ideal, or sigma ideal, of a Ο-algebra (π, 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.
[citation needed] Let
be a measurable space (meaning
is a π-algebra of subsets of
is a π-ideal if the following properties are satisfied: Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements.
The concept of π-ideal is dual to that of a countably complete (π-) filter.
If a measure
ΞΌ
The notion can be generalized to preorders
with a bottom element
and (iii') given a sequence
contains the bottom element, is downward closed, and satisfies a countable analogue of the property of being upwards directed.
A π-ideal of a set
is a π-ideal of the power set of
That is, when no π-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 π-ideal generated by the collection of closed subsets with empty interior.