BorelCantelliLemma 2002-12-08 07:42:12 2004-02-23 14:49:35 Theorem % 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 random events in a probability space. \begin{enumerate} \item If $\sum_{n=1}^\infty P(A_n)<\infty$, then $P(A_n \operatorname{i.o.}) = 0$; \item If $A_1,A_2,\dots$ are independent, and $\sum_{n=1}^\infty P(A_n)=\infty$, then $P(A_n \operatorname{i.o.})=1$ \end{enumerate} where $A=[A_n \operatorname{i.o.}]$ represents the event ``$A_n$ happens for infinitely many values of $n$.'' Formally, $A = \limsup A_n$, which is a limit superior of sets.