Sigma-ring
Encyclopedia
In mathematics
, a nonempty collection of sets
is called a σ-ring if it is closed
under countable union
and relative complementation
:
From these two properties we immediately see that
if 
for all 
This is simply because
.
If the first property is weakened to closure under finite union (i.e.,
whenever 
) but not countable union, then 
is a ring but not a σ-ring.
σ-rings can be used instead of σ-fields
in the development of measure
and integration
theory, if one does not wish to require that the universal set
be measurable. Every σ-field is also a σ-ring, but a σ-ring need not be a σ-field.
A σ-ring induces a σ-algebra. If
is a σ-ring over the set 
, then define 
to be the collection of all subsets of X that are elements of 
or whose complements are elements of 
. We see that 
is a σ-algebra over the set
X.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a nonempty collection of sets
is called a σ-ring if it is closedClosure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
under countable union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
and relative complementation
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
:
if 
for all 
if 
From these two properties we immediately see that
if for all This is simply because
.If the first property is weakened to closure under finite union (i.e.,
whenever ) but not countable union, then is a ring but not a σ-ring.σ-rings can be used instead of σ-fields
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...
in the development of measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
and integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
theory, if one does not wish to require that the universal set
Universe (mathematics)
In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains all the entities one wishes to consider in a given situation...
be measurable. Every σ-field is also a σ-ring, but a σ-ring need not be a σ-field.
A σ-ring induces a σ-algebra. If
is a σ-ring over the set , then define to be the collection of all subsets of X that are elements of or whose complements are elements of . We see that is a σ-algebra over the setX.