Almost-everywhere equality

Two functions are a.e. equal if they differ only on a null set.

Almost-everywhere equality

An almost-everywhere equality (or a.e. equality) on a measure space $(X,\Sigma,\mu)$ is the relation on functions $f,g:X\to \overline{\mathbb R}$ defined by

f=g \text{ a.e.} \quad \Longleftrightarrow \quad \mu(\{x\in X: f(x)\neq g(x)\})=0.

Equivalently, the set where $f$ and $g$ disagree is a null set .

This formalizes equality almost everywhere and gives an equivalence relation on (for instance) the collection of measurable functions . Many constructions in integration theory and spaces such as Lp spaces treat functions as identical whenever they are a.e. equal.

Examples:

On $\mathbb R$ equipped with the Borel sigma-algebra and Lebesgue measure , the functions $f=0$ and $g=\mathbf{1}_{\{0\}}$ are a.e. equal.
If $E\subseteq X$ is a null set and $f$ is measurable, then $f$ and the function obtained by redefining $f$ arbitrarily on $E$ are a.e. equal.