Definition:Pointwise Addition 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 $\map {\mathcal M} {X, \Sigma, \R}$ be the set of real-valued $\Sigma$-measurable functions on $X$.

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

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

We define pointwise addition $+$ on $\map {\mathcal M} {X, \Sigma, \R}/\sim$ by:


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

where $\eqclass f \sim, \eqclass g \sim \in \map {\mathcal M} {X, \Sigma, \R}/\sim$.

Also see

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