Convergent set sequence

A convergent set sequence is a set sequence for which the limes superior of the set sequence and the limes inferior of the set sequence agree. Convergent set sequences occur, for example, in probability theory and measure theory .

definition

A sequence of sets is given from a basic set . The Limes superior to the sequence of quantities ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \Omega }$

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

is the set of all elements that are in infinitely many . The Limes inferior of the sequence of sets ${\displaystyle \Omega }$ ${\displaystyle A_{n}}$

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

is the set of all elements that are in almost all (ie in all but a finite number) . ${\displaystyle \Omega }$ ${\displaystyle A_{n}}$

The sequence of sets is then called convergent if its inferior limes and superior limes coincide, that is

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

is.

${\displaystyle \lim _{n\to \infty }A_{n}:=\limsup _{n\to \infty }A_{n}=\liminf _{n\to \infty }A_{n}}$

is then called the limit of the sequence or limit of the sequence . Then, they say that the amount of follow- up converges. ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \lim _{n\to \infty }A_{n}}$

Examples

As an example we consider the sequence of sets

${\displaystyle A_{n}:=[0;1{,}5+0{,}5(-1)^{n}]}$.

For anything is always ${\displaystyle n}$

${\displaystyle \bigcap _{m=n}^{\infty }A_{m}=[0;1]{\text{ und }}\bigcup _{m=n}^{\infty }A_{m}=[0;2]}$.

So is

${\displaystyle \liminf _{n\to \infty }A_{n}={\bigcup _{n=1}^{\infty }}[0;1]=[0;1]\neq [0;2]={\bigcap _{n=1}^{\infty }}[0;2]=\limsup _{n\to \infty }A_{n}}$.

Thus the limes superior and limes inferior do not coincide, so the sequence does not converge.

Convergence of monotonic set sequences

Monotonically decreasing set sequences , i.e. those with and monotonically growing set sequences , i.e. those with , always converge. A sequence of sets converges to ${\displaystyle A_{1}\supset A_{2}\supset A_{3}\cdots }$ ${\displaystyle A_{1}\subset A_{2}\subset A_{3}\cdots }$ ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$

${\displaystyle \lim _{n\to \infty }A_{n}=\bigcap _{n=1}^{\infty }A_{n}}$,

when it is monotonously falling, and against

${\displaystyle \lim _{n\to \infty }A_{n}=\bigcup _{n=1}^{\infty }A_{n}}$,

when it is monotonously growing. If the limit value is a monotonically falling sequence, one also writes . If the limit value is a monotonically increasing sequence, one also writes . ${\displaystyle A}$ ${\displaystyle A_{n}\downarrow A}$ ${\displaystyle A}$ ${\displaystyle A_{n}\uparrow A}$

See also

• Hausdorff convergence

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 .