Almost everywhere

In measure theory (a branch of mathematical analysis), a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero (Halmos 1974). In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is assumed unless otherwise stated.

The term almost everywhere is abbreviated a.e.; in older literature p.p. is used, to stand for the equivalent French language phrase presque partout.

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 sets with probability 1, which 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 also has other meanings).

Properties

If f : R → R is a Lebesgue integrable function and f(x) ≥ 0 almost everywhere, then

$\int_a^b f(x) \, dx \geq 0$

for all real numbers a < b with equality iff $f(x)=0$ almost everywhere.

If f : [a, b] → R is a monotonic function, then f is differentiable almost everywhere.
If f : R → R is Lebesgue measurable and

$\int_a^b |f(x)| \, dx < \infty$

for all real numbers a < b, then there exists a set E (depending on f) such that, if x is in E, the Lebesgue mean

$\frac{1}{2\epsilon} \int_{x-\epsilon}^{x%2B\epsilon} f(t)\,dt$

converges to f(x) as $\epsilon$ decreases to zero. The set E is called the Lebesgue set of f. Its complement can be proved to have measure zero. In other words, the Lebesgue mean of f converges to f almost everywhere.

If f(x,y) is Borel measurable on R² then for almost every x, the function y→f(x,y) is Borel measurable.

A bounded function f : [a, b] -> R is Riemann integrable if and only if it is continuous almost everywhere.

Definition using ultrafilters

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:

If U ∈ F and U ⊆ V then V ∈ F
The intersection of any two sets in F is in F
The empty set is not in F

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.

References

Billingsley, Patrick (1995). Probability and measure (3rd edition ed.). New York: John Wiley & sons. ISBN 0-471-00710-2.
Halmos, Paul R. (1974). Measure Theory. New York: Springer-Verlag. ISBN 0-387-90088-8.