Set family

This article deals with sets indexed with any index set. For sets of subsets of a superset, see set system .

A set family is a term from set theory , a branch of mathematics . A set is assigned to each element of this index set for any index set. Thus a set family is a special case of a family and in turn contains the set sequences as a special case. Set families are one of the basic concepts of mathematics and have a wide range of applications, for example in measure theory and probability theory .

definition

A set family on the basic set is a mapping of any index set into the power set of the basic set . ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {P}}(\Omega )}$ ${\displaystyle \Omega }$

${\displaystyle {\begin{matrix}A\colon &I&\to &{\mathcal {P}}(\Omega )\\&i&\mapsto &A_{i},\end{matrix}}}$

Any subset of the basic set is assigned to each element of the index set . ${\displaystyle I}$ ${\displaystyle \Omega }$

Examples

An example of a set family with a countably infinite index set is the set sequence

${\displaystyle A_{i}:=\{1,\dots ,i^{2}\}}$.

Here are both the base set and the index set . ${\displaystyle \Omega =I=\mathbb {N} }$

An example with an uncountable index set would be the interval as the index set and the family defined as ${\displaystyle [0,1]=I}$

${\displaystyle A_{i}=[0,i]}$.

The superset could then be, for example, the interval or all of the real numbers . ${\displaystyle \Omega }$ ${\displaystyle [0,1]}$ ${\displaystyle \mathbb {R} }$

The choice of the index set is completely free. One can also choose the unit square as the index set and define the family, for example, by , where . Each element of this family of sets is then of the form for . As a superset, you can again choose the interval or all of the real numbers. ${\displaystyle I:=[0,1]\times [0,1]\subset \mathbb {R} ^{2}}$ ${\displaystyle A_{i}=[0,x_{1}+x_{2}]}$ ${\displaystyle i=(x_{1},x_{2})\in I}$ ${\displaystyle [0,p]}$ ${\displaystyle p\in [0,2]}$ ${\displaystyle [0,2]}$

Features and remarks

• A set family with the natural numbers as an index set is a set sequence .
• The index set can be completely without structure. This is also the main difference to the set sequence: the set sequence has, by definition, a natural order in the index set. This is not required for the index set of the set family. For example, the third of the above examples does not have any natural ordering structure on the index set.
• In contrast to the set system , a subset of the superset can appear any number of times in a set family, but then with a different index.

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 .