Ring (quantity system)

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: ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$

1. ${\displaystyle {\mathcal {R}}\neq \emptyset \quad }$ ( is not empty). ${\displaystyle {\mathcal {R}}}$
2. ${\displaystyle A,B\in {\mathcal {R}}\implies A\cup B\in {\mathcal {R}}\quad }$ (Stability / isolation with regard to union ).
3. ${\displaystyle A,B\in {\mathcal {R}}\implies A\setminus B\in {\mathcal {R}}\quad }$ (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 . ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \emptyset }$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle A}$ ${\displaystyle \emptyset =A\setminus A\in {\mathcal {R}}}$

For equivalent definitions, see the relevant section below.

Examples

Power sets

Any power set is above any set ${\displaystyle \Omega }$

${\displaystyle {\mathcal {R}}={\mathcal {P}}(T)}$

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 . ${\displaystyle T\subseteq \Omega }$ ${\displaystyle {\mathcal {P}}(T)}$ ${\displaystyle {\mathcal {P}}(T)}$ ${\displaystyle T}$ ${\displaystyle \Omega }$

In particular, the power set is the largest set ring over , since it contains all subsets of . ${\displaystyle {\mathcal {P}}(\Omega )}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$

The power set of the empty set is in turn the smallest set ring over , because always is. ${\displaystyle {\mathcal {P}}(\emptyset )=\{\emptyset \}}$ ${\displaystyle \Omega }$ ${\displaystyle \emptyset \subseteq \Omega }$

System of all finite subsets

Is an arbitrary quantity and denotes the power of the quantity , then is the system ${\displaystyle \Omega }$ ${\displaystyle |A|}$ ${\displaystyle A\;(|\emptyset |=0)}$

${\displaystyle {\mathcal {R}}=\{A\subseteq \Omega \mid |A|\in \mathbb {N} _{0}\}}$

of all finite subsets of a set ring, because unions and differences of two finite sets are again finite. ${\displaystyle \Omega }$

Quantity ring of the d -dimensional figures

An important quantity ring in the application is the ring of -dimensional figures ${\displaystyle \mathbb {R} ^{d},\,d\in \mathbb {N} ,}$ ${\displaystyle d}$

${\displaystyle {\mathcal {R}}=\{[a_{1},b_{1})\cup \cdots \cup [a_{n},b_{n})\subset \mathbb {R} ^{d}\mid a_{i},b_{i}\in \mathbb {R} ^{d}{\text{ mit }}a_{i}\leq b_{i}{\text{ für }}i=1,\ldots ,n\}}$.

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 ${\displaystyle d}$

${\displaystyle {\mathcal {H}}=\{[a,b)\subset \mathbb {R} ^{d}\mid a,b\in \mathbb {R} ^{d}{\text{ mit }}a\leq b\}}$

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 : ${\displaystyle A,B}$ ${\displaystyle A\cap B=A\setminus (A\setminus B)}$ ${\displaystyle A\bigtriangleup B=(A\setminus B)\cup (B\setminus A)}$ ${\displaystyle {\mathcal {R}}}$

• ${\displaystyle A,B\in {\mathcal {R}}\implies A\cap B\in {\mathcal {R}}}$.
• ${\displaystyle A,B\in {\mathcal {R}}\implies A\bigtriangleup B\in {\mathcal {R}}}$.

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 : ${\displaystyle {\mathcal {R}}}$ ${\displaystyle n\in \mathbb {N} }$

• ${\displaystyle A_{1},\ldots ,A_{n}\in {\mathcal {R}}\implies A_{1}\cup \cdots \cup A_{n}\in {\mathcal {R}}\quad }$and . ${\displaystyle \quad \bigcup \emptyset =\emptyset \in {\mathcal {R}}}$
• ${\displaystyle A_{1},\ldots ,A_{n}\in {\mathcal {R}}\implies A_{1}\cap \cdots \cap A_{n}\in {\mathcal {R}}}$.
• ${\displaystyle A_{1},\ldots ,A_{n}\in {\mathcal {R}}\implies A_{1}\bigtriangleup \cdots \bigtriangleup A_{n}\in {\mathcal {R}}}$.

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 . ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \mathrm {I} :=\bigcup {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}}$

Any power set

${\displaystyle {\mathcal {R}}={\mathcal {P}}(T)}$

a set is a set ring over with one element . ${\displaystyle T\subseteq \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle \mathrm {I} =T}$

The set system is against it

${\displaystyle {\mathcal {R}}=\{\emptyset \}\cup \{\{n_{1},\ldots ,n_{m}\}\mid n_{1},\ldots ,n_{m}\in \mathbb {N} ,\;m\in \mathbb {N} \}}$

of all finite subsets of an example of a set ring without one, because . ${\displaystyle \Omega =\mathbb {N} }$ ${\displaystyle \bigcup {\mathcal {R}}=\mathbb {N} \notin {\mathcal {R}}}$

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 . ${\displaystyle ({\mathcal {R}},\bigtriangleup ,\cap )}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \emptyset }$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \mathrm {I} =\bigcup {\mathcal {R}}}$ ${\displaystyle ({\mathcal {R}},\bigtriangleup ,\cap )}$

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 . ${\displaystyle {\mathcal {R}}}$ ${\displaystyle ({\mathcal {R}},\bigtriangleup ,\cap )}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle A\cup B=(A\bigtriangleup B)\bigtriangleup (A\cap B)\in {\mathcal {R}}}$ ${\displaystyle A\setminus B=A\bigtriangleup (A\cap B)\in {\mathcal {R}}}$ ${\displaystyle A,B\in {\mathcal {R}}}$

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. ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \emptyset }$

Equivalent Definitions

If is a system of subsets of and if are sets, then because of and the following two statements are equivalent : ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \Omega }$ ${\displaystyle A,B}$ ${\displaystyle A\cap B=A\setminus (A\setminus B)}$ ${\displaystyle A\setminus B=A\setminus (A\cap B)}$

• ${\displaystyle A,B\in {\mathcal {R}}\implies A\setminus B\in {\mathcal {R}}}$.
• ${\displaystyle A,B\in {\mathcal {R}}\implies A\cap B\in {\mathcal {R}}}$and if also . ${\displaystyle B\subseteq A}$ ${\displaystyle A\setminus B\in {\mathcal {R}}}$

If besides , then because of and as well as for every set with are also equivalent: ${\displaystyle {\mathcal {R}}\neq \emptyset }$ ${\displaystyle A\setminus B=(A\cup B)\bigtriangleup B}$ ${\displaystyle A\cup B=(A\setminus B)\cup B}$ ${\displaystyle A\cup B=C\setminus ((C\setminus A)\cap (C\setminus B))}$ ${\displaystyle C}$ ${\displaystyle A\cup B\subseteq C}$

• ${\displaystyle {\mathcal {R}}}$ is a quantity ring.
• ${\displaystyle {\mathcal {R}}}$is a differential stable set .
• ${\displaystyle {\mathcal {R}}}$is a union- stable set half-ring .
• ${\displaystyle {\mathcal {R}}}$is stable with respect to symmetrical difference and average . ${\displaystyle \bigtriangleup }$ ${\displaystyle \cap }$
• ${\displaystyle ({\mathcal {R}},\bigtriangleup )}$is an Abelian group and is a semigroup . ${\displaystyle ({\mathcal {R}},\cap )}$
• ${\displaystyle ({\mathcal {R}},\bigtriangleup ,\cap )}$is a ring in the sense of algebra with addition and multiplication . ${\displaystyle \bigtriangleup }$ ${\displaystyle \cap }$
• ${\displaystyle ({\mathcal {R}},\bigtriangleup ,\cap )}$is an idempotent ( commutative ) ring in the sense of algebra.
• ${\displaystyle {\mathcal {R}}}$is stable with respect to symmetric difference and union . ${\displaystyle \bigtriangleup }$ ${\displaystyle \cup }$
• ${\displaystyle A,B\in {\mathcal {R}}\implies A\setminus B\in {\mathcal {R}}}$and if there is a with . ${\displaystyle A\cap B=\emptyset }$ ${\displaystyle C\in {\mathcal {R}}}$ ${\displaystyle A\cup B\subseteq C}$
• ${\displaystyle A,B\in {\mathcal {R}}\implies A\setminus B\in {\mathcal {R}}}$and there is a with . ${\displaystyle C\in {\mathcal {R}}}$ ${\displaystyle A\cup B\subseteq C}$

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. ${\displaystyle {\mathcal {R}}_{1}\cap {\mathcal {R}}_{2}}$ ${\displaystyle {\mathcal {R}}_{1}}$ ${\displaystyle {\mathcal {R}}_{2}}$ ${\displaystyle A,B\in {\mathcal {R}}_{1}\cap {\mathcal {R}}_{2}}$ ${\displaystyle A,B\in {\mathcal {R}}_{1}}$ ${\displaystyle A,B\in {\mathcal {R}}_{2}}$ ${\displaystyle A\cup B\in {\mathcal {R}}_{1}}$ ${\displaystyle A\cup B\in {\mathcal {R}}_{2}}$ ${\displaystyle A\cup B}$ ${\displaystyle {\mathcal {R}}_{1}\cap {\mathcal {R}}_{2}}$

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 : ${\displaystyle I}$ ${\displaystyle {\mathcal {R}}_{i}}$ ${\displaystyle i\in I}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$

${\displaystyle {\mathcal {R}}_{I}:=\bigcap _{i\in I}{\mathcal {R}}_{i}}$.

Association of Rings

The union of two set rings and is generally no longer a set ring. For example, consider the two rings ${\displaystyle {\mathcal {R}}_{1}\cup {\mathcal {R}}_{2}}$ ${\displaystyle {\mathcal {R}}_{1}}$ ${\displaystyle {\mathcal {R}}_{2}}$

${\displaystyle {\mathcal {R}}_{1}=\{\emptyset ,\{1\},\{2,3\},\{1,2,3\}\}}$

such as

${\displaystyle {\mathcal {R}}_{2}=\{\emptyset ,\{2\},\{1,3\},\{1,2,3\}\}}$,

so is

${\displaystyle {\mathcal {R}}_{1}\cup {\mathcal {R}}_{2}=\{\emptyset ,\{1\},\{2\},\{1,3\},\{2,3\},\{1,2,3\}\}}$.

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. ${\displaystyle \{1\}\cup \{2\}=\{1,2\}}$ ${\displaystyle \{1,3\}\setminus \{1\}=\{3\}}$

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 ${\displaystyle {\mathcal {S}}_{1}}$ ${\displaystyle \Omega _{1}}$ ${\displaystyle {\mathcal {S}}_{2}}$ ${\displaystyle \Omega _{2}}$ ${\displaystyle {\mathcal {S}}_{1}}$ ${\displaystyle {\mathcal {S}}_{2}}$

${\displaystyle {\mathcal {S}}_{1}\boxdot {\mathcal {S}}_{2}:=\{A\!\times \!B\mid A\in {\mathcal {S}}_{1},\;B\in {\mathcal {S}}_{2}\}}$

over . ${\displaystyle \Omega _{1}\times \Omega _{2}}$

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

${\displaystyle {\mathcal {R}}={\mathcal {P}}(\{1,2\})=\{\emptyset ,\{1\},\{2\},\{1,2\}\}}$,

so the system of sets contains the sets ${\displaystyle {\mathcal {R}}\boxdot {\mathcal {R}}}$

${\displaystyle A=\{1\}\times \{1\}=\{(1,1)\}}$

and

${\displaystyle B=\{1,2\}\times \{1,2\}=\{(1,1),(1,2),(2,1),(2,2)\}}$.

The amount

${\displaystyle B\setminus A=\{(1,2),(2,1),(2,2)\}}$

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. ${\displaystyle {\mathcal {R}}\boxdot {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}\boxdot {\mathcal {R}}}$

The ring product of two sets of rings over and over is therefore defined as their tensor product ${\displaystyle {\mathcal {R}}_{1}}$ ${\displaystyle \Omega _{1}}$ ${\displaystyle {\mathcal {R}}_{2}}$ ${\displaystyle \Omega _{2}}$

${\displaystyle {\mathcal {R}}_{1}\boxtimes {\mathcal {R}}_{2}:=\{A_{1}\!\times \!B_{1}\cup \cdots \cup A_{n}\!\times \!B_{n}\mid A_{1},\ldots ,A_{n}\in {\mathcal {R}}_{1},\;B_{1},\ldots ,B_{n}\in {\mathcal {R}}_{2},\;n\in \mathbb {N} \}}$,

so that this is again a quantity ring above , namely the ring generated by (see below). ${\displaystyle \Omega _{1}\times \Omega _{2}}$ ${\displaystyle {\mathcal {R}}_{1}\boxdot {\mathcal {R}}_{2}}$

Trace of a ring

The trace of a ring over in a set , so the set system ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \Omega }$ ${\displaystyle T\subseteq \Omega }$

${\displaystyle {\mathcal {R}}|_{T}:=\{A\cap T\mid A\in {\mathcal {R}}\}}$,

is always a quantity ring over and over . ${\displaystyle \Omega }$ ${\displaystyle T}$

Creation of rings

Since any sections of quantity rings are back rings (s. O.), Can be for any amount system over by ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \Omega }$

${\displaystyle \varrho ({\mathcal {S}}):=\bigcap \;\{{\mathcal {R}}\mid {\mathcal {R}}{\text{ ist ein Ring über }}\Omega {\text{ mit }}{\mathcal {S}}\subseteq {\mathcal {R}}\}}$

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 . ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {S}}}$

Sometimes the generated ring can be specified directly. Thus the ring created by a set half -ring is of the shape ${\displaystyle {\mathcal {H}}}$

${\displaystyle \varrho ({\mathcal {H}})=\{A_{1}\cup \cdots \cup A_{n}\mid A_{1},\ldots ,A_{n}\in {\mathcal {H}}{\text{ sind paarweise disjunkt}},\;n\in \mathbb {N} \}}$.

An explicit example of this form is the above example of the set ring of -dimensional figures ${\displaystyle d}$ .

The same applies to the product of two quantity rings discussed above and : ${\displaystyle {\mathcal {R}}_{1}}$ ${\displaystyle {\mathcal {R}}_{2}}$

${\displaystyle {\mathcal {R}}_{1}\boxtimes {\mathcal {R}}_{2}=\varrho ({\mathcal {R}}_{1}\boxdot {\mathcal {R}}_{2})}$.

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

${\displaystyle {\mathcal {H}}=\{\emptyset ,\{1\},\{2\},\{3\},\{1,2,3\}\}}$

is not a set ring because it is neither union nor difference stable. ${\displaystyle {\mathcal {H}}}$

Mass association

A set ring is always a (differential-stable) set , but not every set is a set ring:

The mass association

${\displaystyle {\mathcal {V}}=\{\{1\}\}}$

is not a quantity ring because it is not differential stable. ${\displaystyle {\mathcal {V}}}$

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. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega \in {\mathcal {A}}}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \Omega =\bigcup {\mathcal {A}}}$

So is the quantity ring

${\displaystyle {\mathcal {R}}={\mathcal {P}}(\{1\})=\{\emptyset ,\{1\}\}}$

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 . ${\displaystyle \Omega =\{1,2\}}$ ${\displaystyle \Omega \notin {\mathcal {R}}}$ ${\displaystyle \mathrm {I} }$ ${\displaystyle \mathrm {I} =\{1\}\in {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \mathrm {I} }$

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 ${\displaystyle {\mathcal {R}}}$ ${\displaystyle A_{1},\ldots ,A_{n}\in {\mathcal {R}}}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle B_{n}:=\bigcup _{i=1}^{n}A_{i}\in {\mathcal {R}}}$.

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 ${\displaystyle B_{n}}$

${\displaystyle \lim _{n\to \infty }B_{n}=\bigcup _{i=1}^{\infty }A_{i}\in {\mathcal {R}}}$.

${\displaystyle {\mathcal {R}}}$is thus closed with respect to countable unions. Thus the monotonic class generated by a ring is always a σ-ring. ${\displaystyle {\mathcal {R}}}$

See also

• Topological space

literature

• Heinz Bauer : Measure and integration theory . 2., revised. Edition. De Gruyter , Berlin / New York 1992, ISBN 3-11-013626-0 .
• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer , Berlin / Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Ernst Henze : Introduction to Dimension Theory . 2. revised Edition. Bibliographisches Institut , Mannheim / Zurich 1985, ISBN 3-411-03102-6 .

Individual evidence

1. ↑ Felix Hausdorff: Fundamentals of set theory . Veit & Comp., Leipzig 1914, p. 14 . Hausdorff referred to the association as "sum".
2. ↑ Hausdorff called such a "body" ( Fundamentals of set theory . P. 15) .
3. ↑ Peter Eichelsbacher: Probability Theory . Ruhr University Bochum, S. 5 f . ( [1] [PDF; accessed October 30, 2018] Lecture notes for the winter semester 2016/17).