AlmostSurely 2002-02-16 10:40:46 2004-04-09 11:21:04 Definition \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 \newcommand{\mv}[1]{\mathbf{#1}} % matrix or vector \newcommand{\mvt}[1]{\mv{#1}^{\mathrm{T}}} \newcommand{\mvi}[1]{\mv{#1}^{-1}} \newcommand{\mpderiv}[1]{\frac{\partial}{\partial {#1}}} \newcommand{\borel}{\mathfrak{B}} \newcommand{\reals}{\mathbb{R}} \newcommand{\defined}{:=} \newcommand{\var}{\mathrm{var}} \newcommand{\cov}{\mathrm{cov}} \newcommand{\corr}{\mathrm{corr}} \newcommand{\set}[1]{\{#1\}} Let $(X, \borel, \mu)$ be a measure space. A condition holds \emph{almost everywhere} on $X$ if it holds ``with probability $1$,'' i.e. if it holds everywhere except for a subset of $X$ with measure $0$. For example, let $f$ and $g$ be nonnegative functions on $X$. Suppose we want a sufficient condition on functions $f(x)$ and $g(x)$ such that the relation \begin{equation} \int_{X} f d\mu(x) \le \int_{X} g d\mu(x) \end{equation} holds. Certainly $f(x)\leq g(x)$ for all $x\in X$ is a sufficient condition, but in fact it's enough to have $f(x)\leq g(x)$ almost surely on $X$. In fact, we can loosen the above non-negativity condition to only require that $f$ and $g$ are almost surely nonnegative as well. If $X = [0,1]$, then $g$ might be less than $f$ on the Cantor set, an uncountable set with measure $0$, and still satisfy the condition. We say that $f \le g$ almost everywhere (often abbreviated \emph{a.e.}). Note that this \PMlinkescapetext{term} is the \PMlinkescapetext{equivalent} of the \PMlinkescapetext{term} ``almost surely'' from probabilistic measure \PMlinkescapetext{theory}.