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

Theorem

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

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 relation on $\map {\mathcal M} {X, \Sigma, \R}$.

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

Then pointwise addition on $\map {\mathcal M} {X, \Sigma, \R}/\sim$ is well-defined.

Proof

Let $E_1, E_2 \in \map {\mathcal M} {X, \Sigma, \R}/\sim$.

First, we show that if $E_1 = \eqclass f \sim$ and $E_2 = \eqclass g \sim$, that $\eqclass {f + g} \sim$ is well-understood.

This follows from Pointwise Sum of Measurable Functions is Measurable.

We now need to show that $E_1 + E_2$ is independent of the choice of representative for $E_1$ and $E_2$.

Suppose that:

$\eqclass f \sim = \eqclass F \sim = E_1$

and:

$\eqclass g \sim = \eqclass G \sim = E_2$

From Equivalence Class Equivalent Statements, we have:

$f \sim F$

and:

$g \sim G$

From Pointwise Addition preserves A.E. Equality, we have:

$f + g \sim F + G$

That is, from Equivalence Class Equivalent Statements:

$\eqclass {f + g} \sim = \eqclass {F + G} \sim$

which is what we aimed to show.

$\blacksquare$