Definition:Almost-Everywhere Equality Relation/Lebesgue Space/Definition 2

Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \closedint 1 \infty$.

Let $\map {\mathcal L^p} {X, \Sigma, \mu}$ be the Lebesgue $p$-space of $\struct {X, \Sigma, \mu}$.

We define the $\mu$-almost-everywhere equality relation $\sim$ on $\map {\mathcal L^p} {X, \Sigma, \mu}$ by:


 * $f \sim_\mu g$ $\map f x = \map g x$ for $\mu$-almost all $x \in X$.

That is:


 * $\map \mu {\set {x \in X : \map f x \ne \map g x} } = 0$