almost everywhere

Let $(X, \borel, \mu)$ 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 \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 a.e.).

Note that this term is the equivalent of the term ``almost surely'' from probabilistic measure theory.