# Almost everywhere

In measure theory (a branch of mathematical analysis), a property holds **almost everywhere** if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of *almost surely* in probability theory.

More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero,^{[1]}^{[2]}^{[3]} or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated.

The term *almost everywhere* is abbreviated *a.e.*;^{[4]} in older literature *p.p.* is used, to stand for the equivalent French language phrase *presque partout*.^{[5]}

A set with **full measure** is one whose complement is of measure zero. In probability theory, the terms *almost surely*, *almost certain* and *almost always* refer to events with probability 1 not necessarily including all of the outcomes.^{[1]} These are exactly the sets of full measure in a probability space.

Occasionally, instead of saying that a property holds almost everywhere, it is said that the property holds for **almost all** elements (though the term almost all can also have other meanings).

As a consequence of the first two properties, it is often possible to reason about "almost every point" of a measure space as though it were an ordinary point rather than an abstraction.^{[citation needed]} This is often done implicitly in informal mathematical arguments. However, one must be careful with this mode of reasoning because of the third bullet above: universal quantification over uncountable families of statements is valid for ordinary points but not for "almost every point".

Outside of the context of real analysis, the notion of a property true almost everywhere is sometimes defined in terms of an ultrafilter. An ultrafilter on a set *X* is a maximal collection *F* of subsets of *X* such that:

A property *P* of points in *X* holds almost everywhere, relative to an ultrafilter *F*, if the set of points for which *P* holds is in *F*.

For example, one construction of the hyperreal number system defines a hyperreal number as an equivalence class of sequences that are equal almost everywhere as defined by an ultrafilter.

The definition of *almost everywhere* in terms of ultrafilters is closely related to the definition in terms of measures, because each ultrafilter defines a finitely-additive measure taking only the values 0 and 1, where a set has measure 1 if and only if it is included in the ultrafilter.