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

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 $\sim$ 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 $\sim$.

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


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

where $\eqclass f \sim, \eqclass g \sim \in \map {\mathcal M} {X, \Sigma, \R}/\sim$ and $f \cdot g$ denotes the usual pointwise product of $f$ and $g$.

Also see

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