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.
$\Box$
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:
| \(\ds E_1 + E_2\) | \(=\) | \(\ds \eqclass f \sim + \eqclass g \sim\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {f + g} \sim\) | Definition of Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {g + f} \sim\) | Real Addition is Commutative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass g \sim + \eqclass f \sim\) | Definition of Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality |
$\Box$
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:
| \(\ds \paren {E_1 + E_2} + E_3\) | \(=\) | \(\ds \paren {\eqclass {f_1} \sim + \eqclass {f_2} \sim} + \eqclass {f_3} \sim\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {f_1 + f_2} \sim + \eqclass {f_3} \sim\) | Definition of Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {\paren {f_1 + f_2} + f_3} \sim\) | Definition of Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {f_1 + \paren {f_2 + f_3} } \sim\) | Real Addition is Associative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {f_1} \sim + \paren {\eqclass {f_2 + f_3} \sim}\) | Definition of Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {f_1} \sim + \paren {\eqclass {f_2} \sim + \eqclass {f_3} \sim}\) | Definition of Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds E_1 + \paren {E_2 + E_3}\) |
$\Box$
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:
| \(\ds \mathbf 0 + E\) | \(=\) | \(\ds \eqclass 0 \sim + \eqclass f \sim\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {0 + f} \sim\) | Definition of Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass f \sim\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds E\) |
Combining this with $(\text V 2)$, we get:
- $\mathbf 0 + E = E + \mathbf 0 = E$
$\Box$
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:
| \(\ds E + \paren {-E}\) | \(=\) | \(\ds \eqclass f \sim + \paren {-1} \cdot \eqclass f \sim\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass f \sim + \eqclass {-f} \sim\) | Definition of Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {f - f} \sim\) | Definition of Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass 0 \sim\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \mathbf 0\) |
$\Box$
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:
| \(\ds \paren {\lambda + \mu} \cdot E\) | \(=\) | \(\ds \paren {\lambda + \mu} \cdot \eqclass f \sim\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {\paren {\lambda + \mu} f} \sim\) | Definition of Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {\lambda f + \mu f} \sim\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {\lambda f} \sim + \eqclass {\mu f} \sim\) | Definition of Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lambda \cdot \eqclass f \sim + \mu \cdot \eqclass f \sim\) | Definition of Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lambda \cdot E + \mu \cdot E\) |
$\Box$
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:
| \(\ds \lambda \cdot \paren {E_1 + E_2}\) | \(=\) | \(\ds \lambda \cdot \paren {\eqclass f \sim + \eqclass g \sim}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lambda \cdot \eqclass {f + g} \sim\) | Definition of Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {\lambda \paren {f + g} } \sim\) | Definition of Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {\lambda f + \lambda g} \sim\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {\lambda f} \sim + \eqclass {\lambda g} \sim\) | Definition of Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lambda \cdot \eqclass f \sim + \lambda \cdot \eqclass g \sim\) | Definition of Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lambda \cdot E_1 + \lambda \cdot E_2\) |
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:
| \(\ds \lambda \cdot \paren {\mu \cdot E}\) | \(=\) | \(\ds \lambda \cdot \paren {\mu \cdot \eqclass f \sim}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lambda \cdot \eqclass {\mu f} \sim\) | Definition of Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {\paren {\lambda \mu} f} \sim\) | Definition of Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\lambda \mu} \cdot \eqclass f \sim\) | Definition of Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality |
$\Box$
Proof of $(\text V 8)$
Let $E \in \map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$.
Write $E = \sim f \sim$.
Then we have:
| \(\ds 1 \cdot E\) | \(=\) | \(\ds 1 \cdot \eqclass f \sim\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {1 \cdot f} \sim\) | Definition of Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass f \sim\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds E\) |
$\blacksquare$