Limes superior and limes inferior of sequence of sets

In mathematics , the Limes superior and Limes inferior of a sequence of sets are terms from measure theory and probability theory , which generalize the terms of Limes superior and Limes inferior of sequences of numbers and sequences of functions for sequences of sets . In stochastics, for example, they are used to model events that occur infinitely often or to define convergent set sequences . The term goes back to Émile Borel .

definition

A set sequence is given in the superset . Then is called ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \Omega }$

${\displaystyle \liminf _{n\rightarrow \infty }A_{n}={\bigcup _{n=1}^{\infty }}\left({\bigcap _{m=n}^{\infty }}A_{m}\right)}$

the Limes inferior of the sequence of sets and

${\displaystyle \limsup _{n\rightarrow \infty }A_{n}={\bigcap _{n=1}^{\infty }}\left({\bigcup _{m=n}^{\infty }}A_{m}\right)}$

the Limes superior to the sequence of sets . Alternative spellings are for the Limes inferior or for the Limes superior. ${\displaystyle \varliminf _{n\to \infty }}$ ${\displaystyle \varlimsup _{n\to \infty }}$

example

Consider as an example the sequence of sets with ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$

${\displaystyle A_{n}:=[-1,n]}$

on the basic set . It is now ${\displaystyle \Omega =\mathbb {R} }$

${\displaystyle B_{n}:=\bigcap _{m=n}^{\infty }A_{m}=[-1,n]}$.

It follows directly from this

${\displaystyle \liminf _{n\rightarrow \infty }A_{n}=\bigcup _{n=1}^{\infty }B_{n}=\bigcup _{n=1}^{\infty }[-1,n]=[-1,\infty ).}$

Analogously follows for the Limes superior

${\displaystyle C_{n}:=\bigcup _{m=n}^{\infty }A_{m}=[-1,\infty )}$

and thus

${\displaystyle \limsup _{n\rightarrow \infty }A_{n}=\bigcap _{n=1}^{\infty }C_{n}=\bigcap _{n=1}^{\infty }[-1,\infty )=[-1,\infty ).}$

interpretation

The limes superior and inferior can be interpreted as follows:

${\displaystyle \limsup _{n\to \infty }A_{n}=\{\omega \in \Omega \,|\,\omega {\text{ ist in unendlich vielen der Mengen }}A_{n}{\text{ enthalten }}\}}$

${\displaystyle \liminf _{n\to \infty }A_{n}=\{\omega \in \Omega \,|\,\omega {\text{ ist in allen bis auf endlich viele der Mengen }}A_{n}{\text{ enthalten }}\}}$

You can make this clear from the formulas when you write out the outer set operation. It is then

${\displaystyle \liminf _{n\to \infty }A_{n}={\bigcap _{m=1}^{\infty }}A_{m}\cup {\bigcap _{m=2}^{\infty }}A_{m}\cup {\bigcap _{m=3}^{\infty }}A_{m}\dots }$

Each of the quantities is written out

${\displaystyle C_{N}:={\bigcap _{m=N}^{\infty }}A_{m}=\{\omega \in \Omega \,|\,\omega \in A_{n}{\text{ für alle }}n\geq N\}}$.

If one now combines all of the to form the Limes inferior, then the union contains all elements of the superset that are contained in at least one . This is equivalent to having an index for each element so that each contains if is. But this can only be the case if all but a finite number are contained in all . ${\displaystyle C_{N}}$ ${\displaystyle C_{N}}$ ${\displaystyle \omega }$ ${\displaystyle N}$ ${\displaystyle \omega }$ ${\displaystyle A_{n}}$ ${\displaystyle n\geq N}$ ${\displaystyle \omega }$ ${\displaystyle A_{n}}$

The same results for the Limes superior

${\displaystyle \limsup _{n\to \infty }A_{n}={\bigcup _{m=1}^{\infty }}A_{m}\cap {\bigcup _{m=2}^{\infty }}A_{m}\cap {\bigcup _{m=3}^{\infty }}A_{m}\dots }$

Then are the individual union sets

${\displaystyle D_{N}={\bigcup _{m=N}^{\infty }}A_{m}=\{\omega \in \Omega \,|\,{\text{ Es gibt ein }}n\geq N,{\text{ so dass }}\omega \in A_{n}{\text{ ist }}\}}$

If you now cut all of them to form the limes superior, the intersection contains all that lie in each one . But then these are exactly the elements that lie in an infinite number . ${\displaystyle D_{N}}$ ${\displaystyle \omega }$ ${\displaystyle D_{N}}$ ${\displaystyle A_{n}}$

Connection with characteristic functions

The characteristic functions of the limes inferior or limes superior of sets are the pointwise limes inferior or limes superior of the characteristic functions of the individual sets: Off

${\displaystyle \chi _{A}(x)=\sup _{n}\chi _{A_{n}}(x)}$ For ${\displaystyle A=\bigcup _{n}A_{n}}$

and

${\displaystyle \chi _{A}(x)=\inf _{n}\chi _{A_{n}}(x)}$ For ${\displaystyle A=\bigcap _{n}A_{n}}$

follows

${\displaystyle \chi _{\bigcup _{n}\bigcap _{m\geq n}A_{m}}(x)=\sup _{n}\chi _{\bigcap _{m\geq n}A_{m}}(x)=\sup _{n}\inf _{m\geq n}\chi _{A_{m}}(x),}$

analogously for lim sup.

Overall, then

${\displaystyle \chi _{\limsup \limits _{n\to \infty }A_{n}}=\limsup _{n\to \infty }\chi _{A_{n}}}$

and

${\displaystyle \chi _{\liminf \limits _{n\to \infty }A_{n}}=\liminf _{n\to \infty }\chi _{A_{n}}}$.

use

The superior limit of set sequences is used in probability theory, for example, in the Borel-Cantelli lemma or in Kolmogorow's zero-one law , where they are typical examples of terminal events . More generally, limes superior and inferior are used to define convergence of set sequences . A sequence of sets converges when Limes inferior and superior coincide. This is the case, for example, if there is an index for each , so that either applies to all or to all . Convergent set sequences occur, for example, in measure theory . ${\displaystyle \omega }$ ${\displaystyle N=N(\omega )}$ ${\displaystyle \omega \in A_{n}}$ ${\displaystyle n\geq N}$ ${\displaystyle \omega \notin A_{n}}$ ${\displaystyle n\geq N}$

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 .
• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 .