Sequence of quantities

A set sequence is a term from set theory , a branch of mathematics . It is a generalization of a sequence of numbers for sets and is used, for example, in probability theory and measure theory .

definition

A sequence of sets on the basic set is formally defined as a mapping ${\displaystyle \Omega }$

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

which assigns a sequence member from the power set to each index from the set of natural numbers used as the index set ${\displaystyle i}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle A_{i}}$ ${\displaystyle {\mathcal {P}}(\Omega )}$

In other words, a set sequence is an ordered sequence of subsets of a common basic set.

example

The basic set is now the natural numbers : ${\displaystyle \mathbb {N} }$

${\displaystyle A_{i}:=\{i,\dots ,i^{2}\}\Rightarrow A_{1}=\{1\},\ A_{2}=\{2,3,4\},\ A_{3}=\{3,4,5,6,7,8,9\},\ \dots }$

Demarcation

In contrast to the set system , in a set sequence (as in any sequence) the order of the sequence members is important. In addition, the same element in the sequence can appear several times, but with a different index.

A set sequence is a special case of a set family if the natural numbers are chosen as the index set for the family. The difference between the sequence of sets and the set family is that a set family does not necessarily have an order relation on the indices. So there is no such thing as a smaller or a larger index. The indices of a sequence automatically carry this order via the natural order of the natural numbers.

properties

• A sequence of sets is called a monotonous sequence of sets whenever or is true. ${\displaystyle A_{1}\subset A_{2}\subset A_{3}\subset \cdots }$ ${\displaystyle A_{1}\supset A_{2}\supset A_{3}\supset \cdots }$
• As with number sequences, the limes superior and limes inferior can be defined by sequence of sets .
• With the help of the limes inferior and the limes superior, convergence for set sequences can also be defined. A sequence of sets converges if and only if the limes superior and the limes inferior coincide. For example, every monotonic sequence of sets converges.

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 .