Almost Everywhere

Definition (Almost Everywhere)

We say that a property $P(x)$ depending on $x\in X$ holds almost everywhere (abbreviated by a.e.) or for almost all $x\in X$ if the set of points where it does not hold has measure zero.

In other words, $P(x)$ a.e. iff $\mu (\{x\in X:\neg P(x)\})=0$.

Let $f:X\to \overline{\mathbb{R}}$ be a nonnegative measurable function. Then

$${\int }_{X}fd\mu =0$$

holds if and only if $f$ vanishes almost everywhere.