Definition:Almost-Everywhere Equality Relation/Lebesgue Space/Definition 1

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 $\sim_\mu$ on $\map {\mathcal L^p} {X, \Sigma, \mu}$ by:


 * $f \sim_\mu g$ $\norm {f - g}_p = 0$

where $\norm \cdot_p$ is the $p$-seminorm.