Almost everywhere

Remarks

Let $\lambda$ be Lebesgue measure on $\R$ and let $f(x) = 1$ for rational $x$ and $f(x) = 0$ otherwise. Then $f = 0$ holds $\lambda$-almost everywhere: take $N = \Q$, a countable set, so $\lambda(\Q) = 0$. The statement $x \ne 0 \Rightarrow x < 0$ does not hold $\lambda$-almost everywhere, since it fails on $(0, \infty)$, and every measurable set containing $(0, \infty)$ has measure $\infty$. The set where $P$ fails is only required to lie inside a null set; it need not be measurable itself. The usual abbreviation is a.e., or $\mu$-a.e. On a probability space, almost everywhere is called almost surely.