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

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

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

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

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

where:

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

$f \cdot g$ denotes the usual pointwise product of $f$ and $g$.

Also see

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