In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
, a -ideal, or sigma ideal, of a
sigma-algebra (, read "sigma," means
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
in this context) is a
subset with certain desirable
closure properties. It is a special type of
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
. Its most frequent application is in
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
.
Let
be a
measurable space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
Definition
Consider a set X and a σ-algebra \mathcal A on X. Then the ...
(meaning
is a -algebra of subsets of
). A subset
of
is a -ideal if the following properties are satisfied:
#
;
# When
and
then
implies
;
# If
then
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
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
(-)
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
.
If a
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
is given on
the set of
-
negligible set
In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose.
As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integr ...
s (
such that
) is a -ideal.
The notion can be generalized to
preorders
with a bottom element
as follows:
is a -ideal of
just when
(i')
(ii')
implies
and
(iii') given a sequence
there exists some
such that
for each
Thus
contains the bottom element, is downward closed, and satisfies a countable analogue of the property of being
upwards directed.
A -ideal of a set
is a -ideal of the power set of
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.
See also
*
*
*
*
*
*
*
*
*
*
* {{annotated link, Sigma additivity
References
*
Bauer, Heinz (2001): ''Measure and Integration Theory''. Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.
Measure theory
Families of sets