Almost-everywhere equality - Imported Knowlpedia Content

Reflexive: a ∼ a a\sim a a ∼ a .
Symmetric: if a ∼ b a\sim b a ∼ b , then b ∼ a b\sim a b ∼ a .
Transitive: if a ∼ b a\sim b a ∼ b and b ∼ c b\sim c b ∼ c , then a ∼ c a\sim c a ∼ c .

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.

Equivalent characterizations

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

Remarks

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.