Space of Real-Valued Measurable Functions Identified by A.E. Equality is Vector Space

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 $\sim_\mu$ be the almost-everywhere equality relation on $\map {\mathcal M} {X, \Sigma, \R}$ with respect to $\mu$.

Let $+$ denote pointwise addition on $\map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$.

Let $\cdot$ be pointwise scalar multiplication on $\map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$.

Then:


 * $\struct {\map {\mathcal M} {X, \Sigma, \R}/\sim_\mu, +, \cdot}_\R$

forms a vector space.

Proof
We verify each of the vector space axioms.

Proof of $(\text V 0)$
This is shown in Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined.

Proof of $(\text V 1)$
Let $E_1, E_2 \in \map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$.

Write $E_1 = \eqclass f \sim$ and $E_2 = \eqclass g \sim$.

Then:

Proof of $(\text V 2)$
Let $E_1, E_2, E_3 \in \map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$.

Write $E_1 = \eqclass {f_1} \sim$, $E_2 = \eqclass {f_2} \sim$ and $E_3 = \eqclass {f_3} \sim$.

Then we have:

Proof of $(\text V 3)$
Set:


 * $\mathbf 0 = \eqclass 0 \sim$

Let $E \in \map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$.

Then, write $E = \eqclass f \sim$.

Then we have:

Combining this with $(\text V 2)$, we get:


 * $\mathbf 0 + E = E + \mathbf 0 = E$

Proof of $(\text V 4)$
Let $E \in \map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$.

Then, write $E = \eqclass f \sim$.

Let:


 * $-E = \paren {-1} \cdot E$

Then we have:

Proof of $(\text V 5)$
Let $E \in \map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$ and $\lambda, \mu \in \R$.

Then, write $E = \eqclass f \sim$.

Then, we have:

Proof of $(\text V 6)$
Let $E_1, E_2 \in \map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$ and $\lambda \in \R$.

Write $E_1 = \eqclass f \sim$ and $E_2 = \eqclass g \sim$.

Then:

Proof of $(\text V 7)$
Let $E \in \map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$ and $\lambda, \mu \in \R$.

Write $E = \sim f \sim$.

Then we have:

Proof of $(\text V 8)$
Let $E \in \map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$.

Write $E = \sim f \sim$.

Then we have: