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, and let $p \in \closedint 1 \infty$.

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

Let $\sim$ be the almost-everywhere equality equivalence relation on $\map {\mathcal M} {X, \Sigma}$.

Let $\map {\mathcal M} {X, \Sigma}/\sim$ be the set of $\Sigma$-measurable functions identified by $\sim$.

Let $\alpha \in \R$.

We define pointwise scalar multiplication $\cdot$ on $\map {\mathcal M} {X, \Sigma}/\sim$ by:


 * $\alpha \cdot \eqclass f \sim = \eqclass {\alpha \cdot f} \sim$

where $\eqclass f \sim \in \map {\mathcal M} {X, \Sigma}/\sim$ and $\alpha \cdot f$ denotes the usual pointwise scalar multiple of $f$ by $\alpha$.

Also see

 * Pointwise Scalar Multiplication on Space of Measurable Functions Quotiented by A.E. Equality is Well-Defined