In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a
non-empty
In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, whil ...
collection
Collection or Collections may refer to:
Computing
* Collection (abstract data type), the abstract concept of collections in computer science
* Collection (linking), the act of linkage editing in computing
* Garbage collection (computing), autom ...
of
sets is called a -ring (pronounced "") if it is
closed under
union,
relative complementation, and countable
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
. The name "delta-ring" originates from the German word for intersection, "Durchschnitt", which is meant to highlight the ring's closure under countable intersection, in contrast to a
-ring which is closed under countable unions.
Definition
A
family of sets
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F of subsets of a given set S is called a family of su ...
is called a -ring if it has all of the following properties:
#Closed under finite unions:
for all
#Closed under relative complementation:
for all
and
#Closed under countable intersections:
if
for all
If only the first two properties are satisfied, then
is a
ring of sets
(The) Ring(s) may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
but not a -ring. Every
-ring is a -ring, but not every -ring is a
-ring.
-rings can be used instead of
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with a ...
s in the development of
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 magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
if one does not wish to allow sets of infinite measure.
Examples
The family
is a -ring but not a
-ring because