Definition:Lebesgue Space

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

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


 * $\map {\mathcal L^p} \mu := \set {f: X \to \R: f \in \map {\mathcal M} \Sigma, \displaystyle \int \size f^p \rd \mu < \infty}$

where $\map {\mathcal M} \Sigma$ denotes the space of $\Sigma$-measurable functions.

On $\map {\mathcal L^p} \mu$, we can introduce the $p$-seminorm $\norm {\, \cdot \,}_p$ by:


 * $\forall f \in \mathcal L^p: \norm f_p := \paren {\displaystyle \int \size f^p \rd \mu}^{1 / p}$

Next, define the equivalence $\sim$ by:


 * $f \sim g \iff \norm {f - g}_p = 0$

The resulting quotient space:
 * $\map {L^p} \mu := \map {\mathcal L^p} \mu / \sim$

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, for example, $\map {\mathcal L^p} X$ to emphasize $X$.

However, according to 's Topological Vector Spaces (1987) they were first introduced by 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