Semialgebra

A semi algebra , also (quantity) Mid algebra is a set system , which in the measure theory , a branch of mathematics is used to certain amount of functions to define generalize the volume terms. Semi-algebras are closely related to semi-rings (in the sense of measure theory) , accordingly, analogously to these, one speaks of a semi-algebra in the narrower sense (i. E. S.) and a semi-algebra in the broader sense (iwS).

definition

Let a (non-empty) set be given . A non-empty system of sets is called a semialgebra (in a broader sense) if: ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {S}}\subset {\mathcal {P}}(\Omega )}$

1. Are , so also lies in (average stability). ${\displaystyle A,B\in {\mathcal {S}}}$ ${\displaystyle A\cap B}$ ${\displaystyle {\mathcal {S}}}$
2. The basic amount is contained in the system of amounts, so it applies . ${\displaystyle \Omega \in {\mathcal {S}}}$
3. For all there are pairwise disjoints in such that ${\displaystyle A,B\in {\mathcal {S}}}$ ${\displaystyle C_{1},\dots ,C_{n}}$ ${\displaystyle {\mathcal {S}}}$

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

is.

Calls, instead of the third, that for all with true that pairwise disjoint in there, so ${\displaystyle A,B\in {\mathcal {S}}}$ ${\displaystyle A\subset B}$ ${\displaystyle C_{1},\dots ,C_{n}}$ ${\displaystyle {\mathcal {S}}}$

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

applies and in addition

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

is for everyone , one speaks of a semialgebra (in the narrower sense) . ${\displaystyle k\in \{1,\dots ,n\}}$

comment

• A semialgebra (ieS / iwS) can be defined in a much more compact way than a semiring (ieS / iwS) that contains the basic set.
• The difference between the definition in the broader sense and that in the narrower sense is the following: In the broader sense, it is only required that the difference between two sets of the set system can be "filled" with disjoint sets of the set system. In a narrower sense, it is also required that one can "shimmy up" from the smaller amount using these amounts without leaving the system of amounts.

example

If one considers the set of all left open intervals that lie in, ie ${\displaystyle (0,1]}$

${\displaystyle {\mathcal {S}}=\{(a,b]\,|\,0<a<b\leq 1\}}$,

so this is a semialgebra in the broader sense. For one obtains the basic amount , likewise all cuts are again contained in the given interval and left open. Is now and , so is (although the sets can be empty because of ). Accordingly, the third requirement is also met. ${\displaystyle a=0,b=1}$ ${\displaystyle (0,1]}$ ${\displaystyle A=(a_{1},a_{2}],B=(b_{1},b_{2}]}$ ${\displaystyle A\subset B}$ ${\displaystyle B\setminus A=(b_{1},a_{1}]\cup (a_{2},b_{2}]}$ ${\displaystyle (c,c]:=\emptyset }$

use

Semi-algebras are used to define measures of probability . A positive σ-additive set function on a semi-algebra is called a probability measure if is. Usually, however , measures are defined in terms of σ-algebras . The procedure used here can be justified as follows: The set function is a premeasure on a semiring (since every semialgebra is a semiring) and can therefore be continued on the ring generated by the semiring. This ring is already an algebra here, since the basic set is contained in the system of sets. With Carathéodory's measure extension theorem , the premeasure of this algebra can be continued to a measure on a σ-algebra. Since, however , the premeasure is finite, i.e. in particular σ-finite and thus the continuation is unambiguous. Thus, every set function that fulfills the above conditions can be tacitly extended to a measure with this method. ${\displaystyle \mu }$ ${\displaystyle \mu (\Omega )=1}$ ${\displaystyle \mu }$ ${\displaystyle \mu (\Omega )=1}$

Relationship to other set systems

• Every σ-algebra and every algebra is a semi -algebra in the narrower sense and therefore also in the broader sense.
• 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 . ${\displaystyle \Omega =\{0,1,2,3,4\}}$

literature

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