Algebra (set system)

In mathematics , (set) algebra is a basic term in measure theory . It describes a non-empty system of sets that is unification and complementary stable .

The branch of mathematics that deals with computing with sets is also known as set algebra. The term algebra , which is used for a subfield of mathematics and also for a special algebraic structure , is similarly ambiguous . The term set algebra used here is closely related to that of Boolean algebra , i.e. another special algebraic structure .

definition

Be any set. A system of subsets of is called a set algebra or algebra over if the following properties are fulfilled: ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$

1. ${\displaystyle {\mathcal {A}}\neq \emptyset }$   ( is not empty). ${\displaystyle {\mathcal {A}}}$
2. ${\displaystyle A,B\in {\mathcal {A}}\Rightarrow A\cup B\in {\mathcal {A}}}$   (Stability / isolation with regard to union ).
3. ${\displaystyle A\in {\mathcal {A}}\Rightarrow A^{\mathrm {c} }\in {\mathcal {A}}}$   (Stability / seclusion with regard to complement formation ). ${\displaystyle A^{\mathrm {c} }=\Omega \setminus A}$

Examples

• For any set , the smallest and the power set is the largest possible set algebra. ${\displaystyle \Omega }$ ${\displaystyle \{\emptyset ,\Omega \}}$ ${\displaystyle {\mathcal {P}}(\Omega )}$
• Every σ-algebra is a set algebra.
• For every set the set system is an algebra of sets. If is infinite, then there is no σ-algebra. ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {A}}=\{A\subseteq \Omega \mid A\ \mathrm {endlich\ oder} \ A^{\mathsf {c}}\ \mathrm {endlich} \}}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {A}}}$

properties

• Every set algebra about always contains and also the empty set , because it contains at least one element and thus are as well ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle \emptyset }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle A}$ ${\displaystyle \Omega =A\cup (\Omega \setminus A)=A\cup A^{\mathrm {c} }\in {\mathcal {A}}}$ ${\displaystyle \emptyset =\Omega \setminus \Omega =\Omega ^{\mathrm {c} }\in {\mathcal {A}}.}$
• The 6- tuple with the set algebra is a Boolean algebra in the sense of the lattice theory , whereby for all (stability / closure with respect to average ). The empty set corresponds to the zero element and the one element . ${\displaystyle ({\mathcal {A}},\cup ,\emptyset ,\cap ,\Omega ,{}^{\mathrm {c} })}$ ${\displaystyle {\mathcal {A}}\subseteq {\mathcal {P}}(\Omega )}$ ${\displaystyle A\cap B=(A^{\mathrm {c} }\cup B^{\mathrm {c} })^{\mathrm {c} }\in {\mathcal {A}}}$ ${\displaystyle A,B\in {\mathcal {A}}}$ ${\displaystyle \emptyset }$ ${\displaystyle \Omega }$

Conversely, if a system of sets is such that it is a Boolean algebra, then obviously there is also a set algebra. ${\displaystyle {\mathcal {A}}\subseteq {\mathcal {P}}(\Omega )}$ ${\displaystyle ({\mathcal {A}},\cup ,\emptyset ,\cap ,\Omega ,{}^{\mathrm {c} })}$ ${\displaystyle {\mathcal {A}}}$

• From the union and average stability it follows inductively that every finite union and every finite intersection of elements of set algebra is contained in it, i.e. H. applies to all : ${\displaystyle {\mathcal {A}}}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle A_{1},\dots ,A_{n}\in {\mathcal {A}}\Rightarrow A_{1}\cup \dots \cup A_{n}\in {\mathcal {A}}}$ and ${\displaystyle A_{1}\cap \dots \cap A_{n}\in {\mathcal {A}},}$

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

Equivalent Definitions

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

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

In addition, denotes the symmetrical difference of and so because of and and are equivalent: ${\displaystyle A\triangle B=(A\setminus B)\cup (B\setminus A)}$ ${\displaystyle A}$ ${\displaystyle B,}$ ${\displaystyle A\setminus B=A\cap B^{\mathrm {c} }}$ ${\displaystyle A\setminus B=A\triangle (A\cap B)}$ ${\displaystyle A\cup B=(A^{\mathrm {c} }\cap B^{\mathrm {c} })^{\mathrm {c} }}$

• ${\displaystyle {\mathcal {A}}}$ is a set algebra.
• ${\displaystyle {\mathcal {A}}}$is a set lattice and: . ${\displaystyle A\in {\mathcal {A}}\Rightarrow A^{\mathrm {c} }\in {\mathcal {A}}}$
• ${\displaystyle ({\mathcal {A}},\cup ,\emptyset ,\cap ,\Omega ,{}^{\mathrm {c} })}$ is a Boolean algebra.
• ${\displaystyle {\mathcal {A}}}$is a set ring and . ${\displaystyle \Omega \in {\mathcal {A}}}$
• ${\displaystyle {\mathcal {A}}}$is a lot of half-ring and: . ${\displaystyle A,B\in {\mathcal {A}}\Rightarrow A\cup B\in {\mathcal {A}}}$
• ${\displaystyle ({\mathcal {A}},\triangle ,\cap ,\Omega )}$is a unitary ring in the sense of algebra with addition, multiplication and one . ${\displaystyle \triangle ,}$ ${\displaystyle \cap }$ ${\displaystyle \Omega }$
• ${\displaystyle ({\mathcal {A}},\triangle ,\cap ,\Omega )}$is a boolean ring .
• ${\displaystyle ({\mathcal {A}},\triangle ,\odot ,\cap ,\Omega )}$with the scalar multiplication is a unitary algebra in the sense of the algebra over the field . ${\displaystyle \odot \colon \mathbb {F} _{2}\times {\mathcal {A}}\to {\mathcal {A}},(0,A)\mapsto \emptyset ,(1,A)\mapsto A,}$ ${\displaystyle \mathbb {F} _{2}}$
• ${\displaystyle \Omega \in {\mathcal {A}}}$and: . ${\displaystyle A,B\in {\mathcal {A}}\Rightarrow A\setminus B\in {\mathcal {A}}}$
• ${\displaystyle {\mathcal {A}}\neq \emptyset }$and it applies: and . ${\displaystyle A,B\in {\mathcal {A}}\Rightarrow A\setminus B\in {\mathcal {A}}}$ ${\displaystyle A^{\mathrm {c} }\in {\mathcal {A}}}$
• ${\displaystyle {\mathcal {A}}\neq \emptyset }$and it applies: and . ${\displaystyle A,B\in {\mathcal {A}}\Rightarrow A\cap B\in {\mathcal {A}}}$ ${\displaystyle A^{\mathrm {c} }\in {\mathcal {A}}}$

Operations with algebras

Sections of algebras

Intersections of two algebras and , that is, the system of sets ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {A}}_{2}}$

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

are always an algebra. Because is exemplary , so is ${\displaystyle A\in {\mathcal {A}}_{1}\cap {\mathcal {A}}_{2}}$

• ${\displaystyle {\mathcal {\Omega }}\setminus A}$in , there is also in . ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {A}}_{1}}$
• ${\displaystyle {\mathcal {\Omega }}\setminus A}$in , there is also in . ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {A}}_{2}}$

Thus in , the intersection of the set systems is also complementary. The stability with regard to the other set operations follows analogously. ${\displaystyle {\mathcal {\Omega }}\setminus A}$ ${\displaystyle {\mathcal {A}}_{1}\cap {\mathcal {A}}_{2}}$

The statement also applies to the intersection of any number of algebras, since the above argument can then be extended to all of these algebras. Thus, if there is any index set and if algebras are all defined on the same base set , then the intersection of all these algebras is again an algebra : ${\displaystyle I}$ ${\displaystyle {\mathcal {A}}_{i}}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {A}}_{I}}$

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

Associations of algebras

The union of two algebras and , that is, the system of sets ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {A}}_{2}}$

${\displaystyle {\mathcal {A}}_{1}\cup {\mathcal {A}}_{2}=\{A\subset \Omega \;|\;A\in {\mathcal {A}}_{1}{\text{ oder }}A\in {\mathcal {A}}_{2}\}}$

is generally no longer algebra. For example, consider the two algebras

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

such as

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

on so is ${\displaystyle \Omega =\{1,2,3\}}$

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

However, this system of sets is not union-stable, since it does not contain, and thus also no algebra. ${\displaystyle \{1\}\cup \{3\}=\{1,3\}}$

Products of algebras

Are and systems of sets on and and is the product of and defined as ${\displaystyle {\mathcal {M}}_{1}}$ ${\displaystyle {\mathcal {M}}_{2}}$ ${\displaystyle \Omega _{1}}$ ${\displaystyle \Omega _{2}}$ ${\displaystyle {\mathcal {M}}_{1}}$ ${\displaystyle {\mathcal {M}}_{2}}$

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

so the product of two algebras is generally no longer an algebra , but just a half ring . Because if you look at algebra ${\displaystyle \Omega _{1}\times \Omega _{2}}$

${\displaystyle {\mathcal {A}}=\{\emptyset ,\{1\}\{2\}\{1,2\}\}}$,

about , the system of sets contains both the sets ${\displaystyle \Omega =\{1,2\}}$ ${\displaystyle {\mathcal {A}}\times {\mathcal {A}}}$

${\displaystyle M_{1}=\{1,2\}\times \{1,2\}=\{(1,1),(1,2),(2,1),(2,2)\}}$as well . ${\displaystyle M_{2}=\{2\}\times \{2\}=\{(2,2)\}}$

The amount

${\displaystyle M_{1}\setminus M_{2}=M_{2}^{\mathrm {c} }=\{(1,1),(1,2),(2,1)\}}$

is not included in, however, because it cannot be represented as the Cartesian product of two sets of . Thus the product of the system of sets is not complementary stable and consequently it cannot be an algebra. ${\displaystyle {\mathcal {A}}}$

However, if one defines the product of two systems of sets as

${\displaystyle {\mathcal {M}}_{1}\boxtimes {\mathcal {M}}_{2}:={\Biggl \{}\bigcup _{i=1}^{n}A_{i}\times B_{i}\,|\,A_{i}\in {\mathcal {M}}_{1},B_{i}\in {\mathcal {M}}_{2}{\Biggl \}}}$,

so the product of two algebras is again an algebra. It is also used, among other things, to define the product σ-algebra .

Trace of an algebra

The trace of an algebra with respect to a set , i.e. the set system ${\displaystyle {\mathcal {A}}}$ ${\displaystyle U}$

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

is always an algebra regardless of the choice of . ${\displaystyle U}$

The generated algebra

Since arbitrary cuts of algebras are again algebras, the envelope operator

${\displaystyle {\mathcal {A}}({\mathcal {E}}):=\bigcap _{{\mathcal {E}}\subseteq {\mathcal {A}}_{i} \atop {\mathcal {A}}_{i}{\text{ Algebra}}}{\mathcal {A}}_{i}}$

define. It is by definition the smallest algebra (in terms of set inclusion) that the set system contains and is called the algebra generated by . ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}}$

Relationship to related structures

Hierarchy of the quantity systems used in measure theory

• The set algebras are precisely the set rings that contain the basic set . If one understands set rings as a ring in the sense of algebra with the symmetrical difference as addition and the average as multiplication, then the set algebras are precisely the unitary rings (i.e. with one element) of this form. ${\displaystyle \Omega }$
• Since set algebras are rings, they are automatically also set lattices and half rings
• If a set algebra is even closed with respect to the union of a countably infinite number of its elements, then one obtains a σ- (set) algebra .
• The monotonic class generated by an algebra corresponds to the -algebra generated by the algebra ${\displaystyle \sigma }$
• Every algebra is a semialgebra in both a narrow and a broader sense.

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 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer, Berlin / Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .

Individual evidence

1. ↑ Kusolitsch: Measure and probability theory. 2014, p. 19.