Half-ring (quantity system)

A (set) half-ring, also called (set) semiring , is a special set system in measure theory , a branch of mathematics, which forms the basis for modern integration theory and stochastics .

Due to their easy handling, half rings are used, for example, as areas of definition for content , which are then gradually expanded to dimensions . They are also popular producers of σ-algebras , especially Borel's σ-algebra , since according to the measure uniqueness theorem, a measure is already uniquely defined by its values ​​on a half-ring on the σ-algebra that is generated.

The definition was introduced by John von Neumann as a generalization of a set ring . The term half-ring used here differs fundamentally from that of a half-ring in the sense of algebra , i.e. a special algebraic structure . Both are not closely related!

definition

Be any set. A lot of system of subsets of is a lot of half-ring , or about half ring , if the following three criteria are fulfilled: ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$

1. ${\displaystyle {\mathcal {H}}}$ contains the empty set: ${\displaystyle \emptyset \in {\mathcal {H}}}$
2. ${\displaystyle {\mathcal {H}}}$is average stable , that is, if and , so is it ${\displaystyle A\in {\mathcal {H}}}$ ${\displaystyle B\in {\mathcal {H}}}$ ${\displaystyle A\cap B\in {\mathcal {H}}}$
3. The difference between two sets from can be represented as a finite union of disjoint sets from . So there are always pairwise disjoint sets from so ${\displaystyle A,B}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle C_{1},C_{2},\dotsc ,C_{n}}$ ${\displaystyle {\mathcal {H}}}$

${\displaystyle A\setminus B=\bigcup _{i=1}^{n}C_{i}}$.

Examples

Over any set , the smallest and the power set is the largest possible set half-ring. Both trivially contain the empty set. The half-ring is cut stable because the empty set cut with itself is again the empty set. The same applies to the difference between the empty set and itself. The statements for follow from the fact that the power set contains all subsets and is therefore stable with respect to all set operations. ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {H}}_{1}=\{\emptyset \}}$ ${\displaystyle {\mathcal {H}}_{2}={\mathcal {P}}(\Omega )}$ ${\displaystyle {\mathcal {H}}_{1}}$ ${\displaystyle {\mathcal {H}}_{2}}$

An important half-ring over the real numbers in the application is the set system of finite, right half-open intervals ${\displaystyle \mathbb {R} }$

${\displaystyle {\mathcal {I}}:=\{[a,b)\mid a,b\in \mathbb {R} ,a\leq b\}}$.

Half rings of this type are often referred to as producers for the Borel σ-algebra on selected, some with slight variations (left open, right closed intervals, only rational limits, etc.). ${\displaystyle \mathbb {R} }$

Half rings of this kind can also be formulated on the , where they also serve as generators for Borel's σ-algebra on . One sets for and as intervals ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle a=(a_{1},a_{2},\dotsc ,a_{n})\in \mathbb {R} ^{n}}$ ${\displaystyle b=(b_{1},b_{2},\dotsc ,b_{n})\in \mathbb {R} ^{n}}$

${\displaystyle [a,b)=\{x\in \mathbb {R} ^{n}\;\mid \;a_{1}\leq x_{1}<b_{1},\;a_{2}\leq x_{2}<b_{2},\;\dotsc ,a_{n}\leq x_{n}<b_{n}\}}$

and defined

${\displaystyle a\leq b}$iff for all , ${\displaystyle a_{i}\leq b_{i}}$ ${\displaystyle i=1,\dotsc ,n}$

so is

${\displaystyle {\mathcal {I}}^{n}:=\{[a,b)\mid a,b\in \mathbb {R} ^{n},a\leq b\}}$

a half-ring, which consists of -dimensional finite, right half-open intervals ( cuboids ). A special case of this are the dyadic unit cells . Here the corner points of the intervals are all on a grid. ${\displaystyle n}$

properties

From the stability of the average it follows inductively that every non-empty, finite intersection of elements of the set semicircle is also contained in it; This means that applies to all : ${\displaystyle {\mathcal {H}}}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle A_{1},\dotsc ,A_{n}\in {\mathcal {H}}\Rightarrow A_{1}\cap \dotsb \cap A_{n}\in {\mathcal {H}}.}$

Set half rings occur in particular as generating systems of σ-algebras . Due to the average stability of the half-rings, it follows from Dynkin's π-λ theorem that the σ-algebra generated by a half-ring is the same as the Dynkin system generated , so it applies

${\displaystyle \sigma ({\mathcal {H}})=\delta ({\mathcal {H}})}$.

Likewise, according to the principle of uniqueness of dimensions, dimensions are clearly determined by specifying their values ​​on the half-ring.

Operations

Cut of half-rings

In contrast to most of the set systems of measure theory, the intersection of half rings, i.e. the set system

${\displaystyle {\mathcal {H}}_{1}\cap {\mathcal {H}}_{2}=\{A\subset \Omega \;|\;A\in {\mathcal {H}}_{1}{\text{ und }}A\in {\mathcal {H}}_{2}\}}$

generally not a half-ring. The opposite example are the half rings

${\displaystyle {\mathcal {H}}_{1}=\{\emptyset ,\{1\},\{4\},\{2,3\},\{1,2,3,4\}\}}$

and

${\displaystyle {\mathcal {H}}_{2}=\{\emptyset ,\{1\},\{2\},\{3,4\},\{1,2,3,4\}\}}$.

Then

${\displaystyle {\mathcal {H}}_{1}\cap {\mathcal {H}}_{2}=\{\emptyset ,\{1\},\{1,2,3,4\}\}}$

no half-ring.

Products from half rings

If one defines for two set systems and on and the product of these set systems as ${\displaystyle {\mathcal {M}}_{1}}$ ${\displaystyle {\mathcal {M}}_{2}}$ ${\displaystyle \Omega _{1}}$ ${\displaystyle \Omega _{2}}$

${\displaystyle {\mathcal {M}}_{1}\times {\mathcal {M}}_{2}:=\{A\times B\subset \Omega _{1}\times \Omega _{2}\;|\;A\in {\mathcal {M}}_{1},\;B\in {\mathcal {M}}_{2}\}}$,

so the product of two half-rings is again a half-ring. For are half rings and as well as , so are and are included in. Here but ${\displaystyle {\mathcal {H}}_{1},{\mathcal {H}}_{2}}$ ${\displaystyle A_{1},B_{1}\in {\mathcal {H}}_{1}}$ ${\displaystyle A_{2},B_{2}\in {\mathcal {H}}_{2}}$ ${\displaystyle A_{1}\times A_{2}}$ ${\displaystyle B_{1}\times B_{2}}$ ${\displaystyle {\mathcal {H}}_{1}\times {\mathcal {H}}_{2}}$

${\displaystyle (A_{1}\times A_{2})\cap (B_{1}\times B_{2})=(A_{1}\cap B_{1})\times (A_{2}\cap B_{2})}$

applies, in is and in , is , so the product is stable. An analogous consideration using ${\displaystyle A_{1}\cap B_{1}}$ ${\displaystyle {\mathcal {H}}_{1}}$ ${\displaystyle A_{2}\cap B_{2}}$ ${\displaystyle {\mathcal {H}}_{2}}$ ${\displaystyle (A_{1}\times A_{2})\cap (B_{1}\times B_{2})\in {\mathcal {H}}_{1}\times {\mathcal {H}}_{2}}$

${\displaystyle (A_{1}\times A_{2})\setminus (B_{1}\times B_{2})=\lbrack (A_{1}\setminus B_{1})\times (A_{2}\setminus B_{2})\rbrack \cup \lbrack (A_{1}\setminus B_{1})\times (A_{2}\cap B_{2})\rbrack \cup \lbrack (A_{2}\setminus B_{2})\times (A_{1}\cap B_{1})\rbrack }$

provides the differential property of a half ring for the products. Example of the stability of half-rings with product formation are the set systems of the half-open intervals in the above example, for which applies. ${\displaystyle {\mathcal {I}}^{2}={\mathcal {I}}\times {\mathcal {I}}}$

For many other set systems in measure theory such as rings, algebras and σ-algebras, it is generally not true that a product of these set systems is again a set system of the same kind. However, if set systems each contain a half ring, the product is always at least one half ring. Typical examples are rings or algebras. The half-ring created as a product is then partly used as a generating system in order to obtain a system of sets with a corresponding structure that contains the Cartesian products of all sets contained in the individual sets of sets. An example of this would be the product σ-algebra or the product of rings defined here . ${\displaystyle {\mathcal {R}}_{1}\boxtimes {\mathcal {R}}_{2}}$

Trace of a half ring

The trace of a half ring with respect to a set , i.e. the set system ${\displaystyle {\mathcal {H}}}$ ${\displaystyle U}$

${\displaystyle {\mathcal {H}}|_{U}:=\{A\cap U\;|\;A\in {\mathcal {H}}\}}$

is always a half ring, regardless of the choice of . ${\displaystyle U}$

Equivalent Definitions

${\displaystyle {\mathcal {H}}}$be a system of subsets of . If sets are and if the symmetric difference of denotes, then because of and and the following statements are equivalent : ${\displaystyle \Omega }$ ${\displaystyle A,B}$ ${\displaystyle A\triangle B=(A\setminus B)\cup (B\setminus A)}$ ${\displaystyle A,B}$ ${\displaystyle \bigcup \emptyset =\emptyset }$ ${\displaystyle A\setminus B=A\triangle (A\cap B)}$ ${\displaystyle A\setminus B=A\setminus (A\cap B)}$

• ${\displaystyle {\mathcal {H}}}$ is a set half-ring.
• ${\displaystyle ({\mathcal {H}},\cap )}$is a semi-lattice and the following applies: There are pairwise disjoint with ${\displaystyle A,B\in {\mathcal {H}}\Rightarrow }$ ${\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}$ ${\displaystyle A\setminus B=C_{1}\cup \dotsb \cup C_{n}.}$
• ${\displaystyle \emptyset \in {\mathcal {H}}}$and the following applies: and there exists a finite subsystem whose elements are pairwise disjoint, with . can also be empty. ${\displaystyle A,B\in {\mathcal {H}}\Rightarrow A\cap B\in {\mathcal {H}}}$ ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {H}},}$ ${\displaystyle A\setminus B=\bigcup {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$
• ${\displaystyle {\mathcal {H}}\neq \emptyset }$and the following applies: and there are pairwise disjoint with ${\displaystyle A,B\in {\mathcal {H}}\Rightarrow A\cap B\in {\mathcal {H}}}$ ${\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}$ ${\displaystyle A\triangle B=C_{1}\cup \dotsb \cup C_{n}.}$
• ${\displaystyle {\mathcal {H}}\neq \emptyset }$and it holds: and if holds, there are pairwise disjoint with ${\displaystyle A,B\in {\mathcal {H}}\Rightarrow A\cap B\in {\mathcal {H}}}$ ${\displaystyle B\subseteq A}$ ${\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}$ ${\displaystyle A\setminus B=C_{1}\cup \dotsb \cup C_{n}.}$

In addition, it results inductively:

• ${\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}$ are pairwise disjoint ${\displaystyle \Rightarrow C_{1}\cup \dots \cup C_{n}=C_{1}\triangle \dotsb \triangle C_{n}.}$

Half rings in the narrower sense

Some authors call the set system defined above a semiring / half-ring in the broader sense (i. W. S.) and also define a semiring / half-ring in the narrower sense (i. E. S.) as a set system , ${\displaystyle {\mathcal {W}}}$

1. that contains the empty set,
2. that is stable,
3. where applies that for all with a exists, so pairwise disjoint from exist for the ${\displaystyle A,B\in {\mathcal {W}}}$ ${\displaystyle A\subset B}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle C_{1},\dotsc ,C_{n}}$ ${\displaystyle {\mathcal {W}}}$

${\displaystyle B\setminus A=\bigcup _{i=1}^{n}C_{i}}$

applies and in addition

${\displaystyle A\cup \bigcup _{i=1}^{k}C_{i}\in {\mathcal {W}}}$

for everyone . ${\displaystyle k\in \{1,\dots ,n\}}$

Related set systems

Quantity rings

Every set ring is a set ring , but not every set ring is a set ring: Above the base set , the set system is a half ring, but not a set ring, because it is not differential stable . If a half-ring is used to create a ring, the ring produced has the shape ${\displaystyle \Omega =\{0,1,2,3,4\}}$ ${\displaystyle {\mathcal {H}}=\{\emptyset ,\{1\},\{2\},\{3\},\{1,2,3\}\}}$ ${\displaystyle {\mathcal {H}}}$

${\displaystyle {\mathcal {R}}=\left\{\left.\bigcup _{j=1}^{n}A_{j}\,\right|\,A_{1},\dotsc ,A_{n}\in {\mathcal {H}},\quad A_{j}{\text{ paarweise disjunkt}}\right\}}$.

Semi-algebras

By definition, every half ring (in the narrower sense / in the broader sense) is a semialgebra (in the narrower sense / in the broader sense) if and only if it contains the superset . An example of a half ring that is not semialgebra would be the half ring ${\displaystyle \Omega }$

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

on the basic set . ${\displaystyle \Omega =\{0,1,2,3,4\}}$

Further quantity systems

Since every set ring is a half ring, set algebras , σ-rings , δ-rings and σ-algebras are always also half rings, since they are all rings too. The reverse is generally not true, as the above example shows.

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin / Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Ernst Henze : Introduction to Dimension Theory. 2nd, revised edition. Bibliographisches Institut, Mannheim et al. 1985, ISBN 3-411-03102-6 .
• Caratheodory's Extension. Script at: probability.net. (PDF file; 115 kB).

Individual evidence

1. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 20.
2. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 20.
3. ↑ Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin / Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 13 , doi : 10.1007 / 978-3-642-45387-8 .