# limit superior of sets

Let $A_{1},A_{2},\dots$ be a sequence of sets. The limit superior of sets is defined by

 $\limsup A_{n}=\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_{k}.$

It is easy to see that $x\in\limsup A_{n}$ if and only if $x\in A_{n}$ for infinitely many values of $n$. Because of this, in probability theory the notation $[A_{n}\operatorname{i.o.}]$ is often used to refer to $\limsup A_{n}$, where i.o. stands for infinitely often.

The limit inferior of sets is defined by

 $\liminf A_{n}=\bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}A_{k},$

and it can be shown that $x\in\liminf A_{n}$ if and only if $x$ belongs to $A_{n}$ for all but finitely many values of $n$.

Title limit superior of sets LimitSuperiorOfSets 2013-03-22 13:13:22 2013-03-22 13:13:22 Koro (127) Koro (127) 8 Koro (127) Definition msc 28A05 msc 60A99 limit inferior of sets infinitely often i.o.