almost everywhere

Let (X,𝔅,μ) be a measure spaceMathworldPlanetmath. A condition holds almost everywhere on X if it holds “with probability 1,” i.e. if it holds everywhere except for a subset of X with measure 0. For example, let f and g be nonnegative functions on X. Suppose we want a sufficient condition on functions f(x) and g(x) such that the relationMathworldPlanetmath

Xf𝑑μ(x)Xg𝑑μ(x) (1)

holds. Certainly f(x)g(x) for all xX is a sufficient condition, but in fact it’s enough to have f(x)g(x) almost surely on X. In fact, we can loosen the above non-negativity condition to only require that f and g are almost surely nonnegative as well.

If X=[0,1], then g might be less than f on the Cantor setMathworldPlanetmath, an uncountable set with measure 0, and still satisfy the condition. We say that fg almost everywhere (often abbreviated a.e.).

Note that this is the of the “almost surely” from probabilistic measure .

Title almost everywhere
Canonical name AlmostEverywhere
Date of creation 2013-03-22 12:20:58
Last modified on 2013-03-22 12:20:58
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 60A10
Synonym almost surely
Synonym a.s.
Synonym a.e.
Synonym almost all