almost everywhere

“Almost everywhere” is the measure-theorist’s way of saying “everywhere that matters.” A statement holds almost everywhere if the set of points where it fails is so small — measure zero — that it can be ignored for every purpose that integration cares about. The exceptional points are there, but they carry no weight.

Formally, given a measure mu, a property P(x) holds mu-almost everywhere (abbreviated a.e.) if the set of x for which P(x) is false is contained in a set of measure zero. Two functions f and g are equal almost everywhere if {x : f(x) is not g(x)} has measure zero. This is an equivalence relation, and the Lebesgue theory systematically identifies functions that agree almost everywhere.

This notion is what makes Lebesgue integration robust: changing a function on a null set changes none of its integrals, and theorems about convergence (monotone convergence, dominated convergence) ask only for convergence almost everywhere, not at every single point. The honest warning is that “almost everywhere” depends on the measure — a.e. with respect to Lebesgue measure differs from a.e. with respect to a measure concentrated on a single point.