Definition:Lebesgue Space

Definition
Let $\left({X, \Sigma, \mu}\right)$ be a measure space, and let $p \in \R$, $p \ge 1$.

The (real) Lebesgue $p$-space of $\mu$ is defined as:


 * $\mathcal{L}^p \left({\mu}\right) := \left\{{f: X \to \R: f \in \mathcal M \left({\Sigma}\right), \displaystyle \int \left\vert{f}\right\vert^p \, \mathrm d \mu < \infty}\right\}$

where $\mathcal M \left({\Sigma}\right)$ denotes the space of $\Sigma$-measurable functions.

On $\mathcal{L}^p \left({\mu}\right)$, we can introduce the $p$-seminorm $\left\Vert{\cdot}\right\Vert_p$ by:


 * $\forall f \in \mathcal{L}^p: \left\Vert{f}\right\Vert_p := \left({\displaystyle \int \left\vert{f}\right\vert^p \, \mathrm d \mu}\right)^{1 / p}$

Next, define the equivalence $\sim$ by:


 * $f \sim g \iff \left\Vert{f - g}\right\Vert_p = 0$

The resulting quotient space $L^p \left({\mu}\right) := \mathcal{L}^p \left({\mu}\right) / \sim$ is also called (real) Lebesgue $p$-space.

Also known as
When the measure $\mu$ is clear, it is dropped from the notation, yielding $\mathcal{L}^p$ and $L^p$.

If so desired, one can write eg. $\mathcal{L}^p \left({X}\right)$ to emphasize $X$.

However, according to Bourbaki's Topological Vector Spaces (1987) they were first introduced by Frigyes Riesz in 1910.

Also see

 * Definition:$p$-Sequence Space, a very important special kind of Lebesgue space
 * Definition:Space of Integrable Functions, the special case $p = 1$
 * Definition:$p$-Seminorm
 * Definition:$p$-Norm, induced on $L^p$ by the $p$-seminorm