Definition:Almost-Everywhere Equality Relation/Lebesgue Space/Definition 1
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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:
- $f \sim_\mu g$ if and only if $\norm {f - g}_p = 0$
where $\norm \cdot_p$ is the $p$-seminorm.