Almost everywhere

In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i.e. is a set with measure zero, or in cases where the measure is not complete, contained within a set of measure zero. If used for properties of the real numbers, the Lebesgue measure is assumed unless otherwise stated. It is abbreviated a. e.; in older literature one can find p. p. instead, which stands for the equivalent French language phrase presque partout. A set with full measure is one whose complement is of measure zero. Occasionally, instead of saying that a property holds almost everywhere, one also says that the property holds for almost all elements, though the term almost all also has other meanings. Here are some theorems that involve the term "almost everywhere":

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

<math>\int f(x) \, dx \geq 0.</math>

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

<math>\int_a^b |f(x)| \, dx < \infty</math>

for all real numbers a < b, then there exists a null set E (depending on f) such that, if x is not in E, the Lebesgue mean

<math>\frac{1}{2\epsilon} \int_{x-\epsilon}^{x+\epsilon} f(t)\,dt</math>

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

• 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.

Outside of the context of real analysis, the notion of a property true almost everywhere can be defined in terms of an ultrafilter. 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. In probability theory, the phrases become almost surely, almost certain or almost always, corresponding to a probability of 1.

References

• Billingsley, Patrick (1995). Probability and measure, 3rd edition, 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.