Definition:Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality

Definition

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

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

Let $\map \MM {X, \Sigma, \R} / \sim_\mu$ be the space of real-valued $\Sigma$-measurable functions identified by $\mu$-A.E. equality.

Let $\alpha \in \R$.

We define pointwise scalar multiplication $\cdot$ on $\map \MM {X, \Sigma, \R}/\sim_\mu$ by:

$\alpha \cdot \eqclass f {\sim_\mu} = \eqclass {\alpha \cdot f} {\sim_\mu}$

where:

$\eqclass f \sim \in \map \MM {X, \Sigma, \R}/\sim$

$\alpha \cdot f$ denotes the usual pointwise scalar multiple of $f$ by $\alpha$.

Also see

• Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined