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

Definition

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:

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

\(=\)

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

Vector Space

Let $+$ denote pointwise addition on $\map \MM {X, \Sigma, \R}/\sim_\mu$.

Let $\cdot$ be pointwise scalar multiplication on $\map \MM {X, \Sigma, \R}/\sim_\mu$.

Then we define the vector space $\map \MM {X, \Sigma, \R}/\sim_\mu$ as:

$\struct {\map \MM {X, \Sigma, \R} / \sim_\mu, +, \cdot}_\R$

Also see

• Space of Real-Valued Measurable Functions Identified by A.E. Equality is Vector Space