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

Theorem
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 $\sim_\mu$ be the $\mu$-almost-everywhere equality relation on either $\map {\mathcal M} {X, \Sigma}$ or $\map {\mathcal M} {X, \Sigma, \R}$.

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