Family closed under subsets and countable unions
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
). A subset
of
is a π-ideal if the following properties are satisfied:
;- When
and
then
implies
; - If
then ![{\textstyle \bigcup _{n\in \mathbb {N} }A_{n}\in N.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
is given on
the set of
-negligible sets (
such that
) is a π-ideal.
The notion can be generalized to preorders
with a bottom element
as follows:
is a π-ideal of
just when
(i') ![{\displaystyle 0\in I,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
(ii')
implies
and
(iii') given a sequence
there exists some
such that
for each ![{\displaystyle n.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Thus
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.
See also
- Ξ΄-ringΒ β Ring closed under countable intersections
- Field of setsΒ β Algebraic concept in measure theory, also referred to as an algebra of sets
- Join (sigma algebra)Β β Algebraic structure of set algebraPages displaying short descriptions of redirect targets
- π-system (Dynkin system)Β β Family closed under complements and countable disjoint unions
- Measurable functionΒ β Function for which the preimage of a measurable set is measurable
- Ο-systemΒ β Family of sets closed under intersection
- Ring of setsΒ β Family closed under unions and relative complements
- Sample spaceΒ β Set of all possible outcomes or results of a statistical trial or experiment
- π-algebraΒ β Algebraic structure of set algebra
- π-ringΒ β Family of sets closed under countable unions
- Sigma additivityΒ β Mapping functionPages displaying short descriptions of redirect targets
References
- Bauer, Heinz (2001): Measure and Integration Theory. Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.