Almost Everywhere

Definition §

Let $\mathcal{P}(x)$ be a predicate/property, i.e. $\mathcal{P}(x)$ holds true for some $x$ and is false for other values of $x$. We say $\mathcal{P}(x)$ holds true almost everywhere (shorthand a.e.) on a set $S$ if $E=\{x\in S\mid \mathcal{P}(x)\text{ is false}\}$ is a null set.