Definition:Lp Space

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

Let $\map {\mathcal M} {X, \Sigma, \R}$ be the set of real-valued $\Sigma$-measurable functions on $X$.

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

Let $\sim_\mu$ be the restriction of the $\mu$-almost-everywhere equality relation on $\map {\mathcal M} {X, \Sigma, \R}$ to $\map {\mathcal L^p} {X, \Sigma, \mu}$.

We define the $L^p$ space $\map {L^p} {X, \Sigma, \mu}$ as the quotient set:

Health Warning
In most discussions of $L^p$ spaces, a function 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, 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.

Also see

 * Definition:Integral on $L^1$ Space
 * Definition:$L^p$ Norm
 * Definition:$L^2$ Inner Product
 * Definition:Weak $L^p$ Space