Lp Space is Vector Space

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \closedint 1 \infty$.

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

Let $\map {L^p} {X, \Sigma, \mu}$ be the $L^p$ space of $\struct {X, \Sigma, \mu}$.

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:


 * $\struct {\map {L^p} {X, \Sigma, \mu}, +, \cdot}_\R$

forms a vector space.

Proof
It is enough to show that $\struct {\map {L^p} {X, \Sigma, \mu}, +, \cdot}_\R$ is a vector subspace of $\struct {\map {\mathcal M} {X, \Sigma, \R}/\sim_\mu, +, \cdot}_\R$.

Let $\eqclass f {\sim_\mu}, \eqclass g {\sim_\mu} \in \map {L^p} {X, \Sigma, \mu}$ and $\lambda \in \R$.

Then, we have $f, g \in \map {\LL^p} {X, \Sigma, \mu}$ where $\map {\LL^p} {X, \Sigma, \mu}$ is the Lebesgue $p$-space $\struct {X, \Sigma, \mu}$.

From the definition of pointwise addition on $\map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$, we have:


 * $\eqclass f {\sim_\mu} + \lambda \eqclass g {\sim_\mu} = \eqclass {f + \lambda g} {\sim_\mu}$

From Lebesgue Space is Vector Space, we have:


 * $f + \lambda g \in \map {\LL^p} {X, \Sigma, \mu}$

Further, from A.E. Equal Positive Measurable Functions have Equal Integrals: Corollary 1, if:


 * $F \sim_\mu f + \lambda g$

then:


 * $F \in \map {\LL^p} {X, \Sigma, \mu}$

so that:


 * $\eqclass {f + \lambda g} {\sim_\mu} \subseteq \map {\LL^p} {X, \Sigma, \mu}$

We conclude that:


 * $\eqclass {f + \lambda g} {\sim_\mu} \in \map {L^p} {X, \Sigma, \mu}$.

So, from the One-Step Vector Subspace Test, we have that $\struct {\map {L^p} {X, \Sigma, \mu}, +, \cdot}_\R$ is a vector subspace of $\struct {\map {\mathcal M} {X, \Sigma, \R}/\sim_\mu, +, \cdot}_\R$.

So $\struct {\map {L^p} {X, \Sigma, \mu}, +, \cdot}_\R$ forms a vector space as required.