Almost everywhere - Definition and Overview

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. If used for properties of the real numbers, the Lebesgue measure is assumed unless otherwise stated.

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 is a list of 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 every 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}{2e} \int_{x-e}^{x+e} f(t)dt<math>

converges to f(x) as e 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.

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

In probability theory, the phrases become almost surely, almost certain or almost always, corresponding to a probability of 1.