Almost Everywhere

measure-theory analysis

Definition

Almost Everywhere

In measure theory, a property is said to hold almost everywhere (often abbreviated as a.e.) if the set of points for which the property does not hold is a null set (a set of measure zero).

Formally, given a measure space $(X,\Sigma ,\mu )$, a property $P$ holds almost everywhere in $X$ if there exists a set $N\in \Sigma$ such that $\mu (N)=0$ and the property $P(x)$ holds for all $x\in X\setminus N$.

Examples

• Functions: Two functions $f$ and $g$ are said to be equal almost everywhere ($f=g$ a.e.) if $\mu (\{x\mid f(x)\ne g(x)\})=0$.
• Continuity: The Dirichlet function (indicator function of the rationals) is discontinuous everywhere, but it is equal to the constant zero function almost everywhere with respect to the Lebesgue measure (since the rationals are a countable null set).
• Integration: In the Lebesgue integral, if $f=g$ a.e., then $\int fd\mu =\int gd\mu$.

Related Concepts

• Almost All: In probability theory, the equivalent term is “almost surely” (a.s.).
• Null Set: A set that can be covered by a sequence of intervals with arbitrarily small total length.