Definition:Lp Space
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Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \closedint 1 \infty$.
Let $\map \MM {X, \Sigma, \R}$ be the set of real-valued $\Sigma$-measurable functions on $X$.
Let $\map {\LL^p} {X, \Sigma, \mu}$ be the Lebesgue $p$-space of $\struct {X, \Sigma, \mu}$.
Let $\sim_\mu$ be the restriction of the $\mu$-almost-everywhere equality relation on $\map \MM {X, \Sigma, \R}$ to $\map {\LL^p} {X, \Sigma, \mu}$.
We define the $L^p$ space $\map {L^p} {X, \Sigma, \mu}$ as the quotient set:
| \(\ds \map {L^p} {X, \Sigma, \mu}\) | \(=\) | \(\ds \map {\LL^p} {X, \Sigma, \mu} / \sim_\mu\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \set {\eqclass f {\sim_\mu}: f \in \map {\LL^p} {X, \Sigma, \mu} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \set {\set {g \in \map {\LL^p} {X, \Sigma, \mu} : f = g \, \mu\text{-almost everywhere} }: f \in \map {\LL^p} {X, \Sigma, \mu} }\) |
Vector Space
Let $\map \MM {X, \Sigma, \R} / \sim_\mu$ be the space of real-valued measurable functions identified by $\mu$-A.E. equality.
Let $+$ denote pointwise addition on $\map \MM {X, \Sigma, \R} / \sim_\mu$.
Let $\cdot$ be pointwise scalar multiplication on $\map \MM {X, \Sigma, \R} / \sim_\mu$.
Then we define the vector space $\map {L^p} {X, \Sigma, \mu}$ as:
- $\struct {\map {L^p} {X, \Sigma, \mu}, +, \cdot}_\R$
Normed Vector Space
Let $\norm \cdot_p$ be the $L^p$ norm on $\map {L^p} {X, \Sigma, \mu}$.
From $L^p$ norm is norm, $\norm \cdot_p$ is a norm on $\map {L^p} {X, \Sigma, \mu}$.
So, we can define the normed vector space $\struct {\map {L^p} {X, \Sigma, \mu}, \norm \cdot_p}$ by:
- $\struct {\map {L^p} {X, \Sigma, \mu}, \norm \cdot_p} = \struct {\struct {\map {L^p} {X, \Sigma, \mu}, +, \cdot}_\R, \norm \cdot_p}$
Warning
In most discussions of $L^p$ spaces, a mapping is not distinguished from the equivalence class it occupies.
That is, for $f \in \map {\LL^p} {X, \Sigma, \mu}$, no distinction is made between the objects $f$ and $\eqclass f {\sim_\mu}$.
On $\mathsf{Pr} \infty \mathsf{fWiki}$, we maintain the distinction for foundational results, but in advanced results it may be too cumbersome to do so, so this is not required for high-level results.
It should however always be noted where this lack of distinction may cause confusion, or is very important to the result.