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

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 $\struct {X, \Sigma, \mu}$.

Then:

$\map {L^p} {X, \Sigma, \mu} \subseteq \map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$

In particular:

if $f \in \map {\LL^p} {X, \Sigma, \mu}$ and $f \sim_\mu F$ we have $F \in \map {\LL^p} {X, \Sigma, \mu}$

Proof

Let $f \in \map {\LL^p} {X, \Sigma, \mu}$.

Denote by $\eqclass f {\sim_\mu}$ the $\sim_\mu$-equivalence class of $f$ in $\map {\LL^p} {X, \Sigma, \mu}$.

Let $\eqclass f {\sim_\mu}^\ast$ be the $\sim_\mu$-equivalence class of $f$ in $\map {\mathcal M} {X, \Sigma, \R}$.

We aim to show that:

$\eqclass f {\sim_\mu} = \eqclass f {\sim_\mu}^\ast$

Since $\map {\LL^p} {X, \Sigma, \mu} \subseteq \map {\mathcal M} {X, \Sigma, \R}$, we have:

$\eqclass f {\sim_\mu} \subseteq \eqclass f {\sim_\mu}^\ast$

Now let $F \in \eqclass f {\sim_\mu}^\ast$.

Then:

$F \sim_\mu f$

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

$\size F^p \sim_\mu \size f^p$

Then, from A.E. Equal Positive Measurable Functions have Equal Integrals: Corollary 1, we have:

$\ds \int \size F^p \rd \mu = \int \size f^p \rd \mu$

so that:

$\ds \paren {\int \size F^p \rd \mu}^{1/p} = \paren {\int \size f^p \rd \mu}^{1/p}$

so:

$\norm F_p = \norm f_p < \infty$

We therefore finally have:

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

giving:

$\eqclass f {\sim_\mu} = \eqclass f {\sim_\mu}^\ast$

$\blacksquare$