User:Caliburn/s/mt/Equality Almost Everywhere is Equivalence Relation

Measurable Functions

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\map {\mathcal M} {X, \Sigma}$ be the space of $\Sigma$-measurable functions on $\struct {X, \Sigma}$.

Let $\map {\mathcal M} {X, \Sigma, \R}$ be the space of real-valued $\Sigma$-measurable functions on $\struct {X, \Sigma}$.

Let $\mathcal S \in \set {\map {\mathcal M} {X, \Sigma}, \map {\mathcal M} {X, \Sigma, \R} }$

Let $\sim_\mu$ be the $\mu$-almost-everywhere equality relation on $\mathcal S$.

Then $\sim_\mu$ is an equivalence relation.

Lebesgue Space

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

Let $\map {\LL^p} {X, \Sigma, \mu}$ be the Lebesgue $p$-space on $\struct {X, \Sigma, \mu}$.

Let $\sim_\mu$ be the $\mu$-almost-everywhere equality relation on $\map {\LL^p} {X, \Sigma, \mu}$.

Then $\sim_\mu$ is an equivalence relation.