# almost everywhere

Let $(X,\mathfrak{B},\mu)$ be a measure space. A condition holds 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

 $\int_{X}fd\mu(x)\leq\int_{X}gd\mu(x)$ (1)

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 $f\leq g$ almost everywhere (often abbreviated a.e.).

Note that this is the of the “almost surely” from probabilistic measure .

Title almost everywhere AlmostEverywhere 2013-03-22 12:20:58 2013-03-22 12:20:58 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 60A10 almost surely a.s. a.e. almost all