Definition:Space of Measurable Functions Identified by A.E. Equality

Definition

Real-Valued Function

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

Let $\map \MM {X, \Sigma, \R}$ be the set of real-valued $\Sigma$-measurable functions on $X$.

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

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

\(\ds \map \MM {X, \Sigma, \R}/\sim_\mu\)

\(=\)

\(\ds \set {\eqclass f {\sim_\mu} : f \in \map \MM {X, \Sigma, \R} }\)

Extended Real-Valued Function

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

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

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

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

\(\ds \map \MM {X, \Sigma, \R} / \sim_\mu\)

\(=\)

\(\ds \set {\eqclass f {\sim_\mu} : f \in \map \MM {X, \Sigma, \R} }\)