LimitSuperiorOfSets 2002-12-08 07:59:11 2005-02-06 13:33:15 Definition limit inferior of sets infinitely often i.o. % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 \textit{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 $n$.