A set ring , also simply called a ring for short , is a special set system in measure theory and thus a set of sets. Rings and their extensions to more complex set systems such as σ-algebras play an important role in the axiomatic structure of probability theory and integration theory .
Felix Hausdorff called a set “ring” because of “an approximate analogy” to the algebraic structure of a ring in algebraic number theory . In the theory of measure today, a ring is usually understood to be a system of quantities as defined here.
The term ring used here also differs from that of a ring in the sense of algebra, but both are related.
definition
Be any set. A set system over , i.e. a set of subsets of , is called a set ring or ring over if the following properties are met:
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( is not empty).
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(Stability / isolation with regard to union ).
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(Stability / seclusion with regard to difference ).
Each set ring contains a zero element or a zero with the empty set , because it contains at least one element and therefore is .
For equivalent definitions, see the relevant section below.
Examples
Power sets
Any power set is above any set
a set ring from a set . Because it is not empty and stable with respect to all set operations, since by definition it contains all subsets of which are also subsets of .
In particular, the power set is the largest set ring over , since it contains all subsets of .
The power set of the empty set is in turn the smallest set ring over , because always is.
System of all finite subsets
Is an arbitrary quantity and denotes the power of the quantity , then is the system
of all finite subsets of a set ring, because unions and differences of two finite sets are again finite.
Quantity ring of the d -dimensional figures
An important quantity ring in the application is the ring of -dimensional figures
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.
It consists of all sets, which can be represented as finite unions of right-open dimensional intervals , and is that of the set half-ring
generated ring (see below).
properties
Stability with respect to set operations
For any two sets we always have and . Therefore, every volume ring is also stable / closed with regard to average and symmetrical difference :
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.
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.
From the stability with respect to union, intersection and symmetrical difference it follows inductively that all finite unions as well as all non-empty, finite averages and symmetrical differences of elements of the set ring are contained in it, i.e. H. applies to all :
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and .
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.
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.
Quantity ring with one
Since every set ring is union and average stable , it is also a set . If, as such, it also contains a single element or one , then a quantity ring with one or, for short, a ring with one .
Any power set
a set is a set ring over with one element .
The set system is against it
of all finite subsets of an example of a set ring without one, because .
Relationship to the ring in the sense of algebra
The triple with the set ring is a ring in the sense of algebra and the empty set is its zero element . If a lot of ring with unity, is also the identity of .
Conversely, if a system of sets is such that a ring is in the sense of algebra, then there is always a set ring because of and for all .
So that each set ring can be represented as a ring in the sense of algebra, it must not be empty, because the empty set cannot contain a zero element and therefore cannot be a carrier set of a ring in the sense of algebra.
Equivalent Definitions
If is a system of subsets of and if are sets, then because of and the following two statements are equivalent :
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.
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and if also .
If besides , then because of and as well as for every set with are also equivalent:
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is a quantity ring.
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is a differential stable set .
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is a union- stable set half-ring .
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is stable with respect to symmetrical difference and average .
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is an Abelian group and is a semigroup .
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is a ring in the sense of algebra with addition and multiplication .
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is an idempotent ( commutative ) ring in the sense of algebra.
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is stable with respect to symmetric difference and union .
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and if there is a with .
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and there is a with .
Operations with rings
Cut of rings
The intersection of two quantity rings and is always a ring again. Because are , so are and , so as well . Thus, also in , the cut is consequently stable with respect to union. The stability with regard to the difference follows analogously.
The statement also applies to the intersection of any number of set rings, since the above argument can then be extended to all of these rings. Thus:
If there is an arbitrary index set and if they are all over the same basic set for quantity rings , then the intersection of all these rings is again a quantity ring over :
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.
Association of Rings
The union of two set rings and is generally no longer a set ring. For example, consider the two rings
such as
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,
so is
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.
However, this system of sets is neither union-stable, since it does not contain, nor is it differential- stable because it does not contain, and thus also no set ring.
Product of rings
Let there be a set system over and a set system over . The direct product of and is defined as the system of sets
over .
However, the direct product of two set rings is generally no longer a set ring, but only a set half- ring .
Consider the power set ring as a counterexample
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,
so the system of sets contains the sets
and
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.
The amount
is not included in, however , because it cannot be represented as the Cartesian product of two sets of . Thus, the direct product is not differential stable and therefore also not a quantity ring.
The ring product of two sets of rings over and over is therefore defined as their tensor product
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,
so that this is again a quantity ring above , namely the ring generated by (see below).
Trace of a ring
The trace of a ring over in a set , so the set system
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,
is always a quantity ring over and over .
Creation of rings
Since any sections of quantity rings are back rings (s. O.), Can be for any amount system over by
define an envelope . By definition, this is the smallest volume ring above that contains the system of quantities and is called the ring generated by .
Sometimes the generated ring can be specified directly. Thus the ring created by a set half -ring is of the shape
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.
An explicit example of this form is the above example of the set ring of -dimensional figures .
The same applies to the product of two quantity rings discussed above and :
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.
Related set systems
Hierarchy of the quantity systems used in measure theory
Generalizations
- Quantity half-ring
Every set ring is a (union- stable ) set half- ring , but not every set half-ring is also a set ring:
Because the quantity half-ring
is not a set ring because it is neither union nor difference stable.
- Mass association
A set ring is always a (differential-stable) set , but not every set is a set ring:
The mass association
is not a quantity ring because it is not differential stable.
Special quantity rings
- Set algebra
A set ring over a set with is called a set algebra over . So every set algebra is a set ring with the one , but not every set ring is a set algebra.
So is the quantity ring
no set algebra over the basic set , there . If, on the other hand, one takes its one as the basic set, then a set algebra is and thus is over .
The assumed basic set is therefore essential for the concept of set algebra.
- δ ring
A set ring that is closed with respect to a countable number of sections is called a δ-ring .
- σ ring
A ring of sets that is closed with respect to a countable number of unions is called a σ-ring .
- Monotonous classes
Every ring that is a monotonic class is a σ-ring (and therefore also a δ-ring). Because if everyone is for , so is the properties of the ring
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.
However, the sets form a monotonically increasing sequence of sets , which is why its limit value is due to the properties of the monotonous class
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.
is thus closed with respect to countable unions. Thus the monotonic class generated by a ring is always a σ-ring.
See also
literature
Individual evidence
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↑ Felix Hausdorff: Fundamentals of set theory . Veit & Comp., Leipzig 1914, p. 14 . Hausdorff referred to the association as "sum".
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↑ Hausdorff called such a "body" ( Fundamentals of set theory . P. 15) .
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↑
Peter Eichelsbacher: Probability Theory . Ruhr University Bochum, S. 5 f . ( [1] [PDF; accessed October 30, 2018] Lecture notes for the winter semester 2016/17).