almost everywhere

Let $(X,𝔅,\mu )$ 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

$${\int }_{X}f𝑑\mu (x)\le {\int }_{X}g𝑑\mu (x)$$(1)

holds. Certainly $f(x)\le g(x)$ for all $x\in X$ is a sufficient condition, but in fact it’s enough to have $f(x)\le g(x)$ 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 a.e.).

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