Definition:Space of Measurable Functions Identified by A.E. Equality/Extended Real-Valued Function

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

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

Let $\sim_\mu$ be the almost-everywhere equality relation on $\map {\mathcal M} {X, \Sigma}$ with respect to $\mu$.

We define the space of real-valued measurable functions identified by $\mu$-A.E. equality as the quotient set: