Definition:Lebesgue Space

From ProofWiki
Jump to navigation Jump to search

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.


Lebesgue $\infty$-Space

The Lebesgue $\infty$-space for $\mu$, denoted $\mathcal{L}^\infty \left({\mu}\right)$, is defined as:

$\displaystyle \mathcal{L}^\infty \left({\mu}\right) := \left\{{f \in \mathcal M \left({\Sigma}\right): \text{$f$ is a.e. bounded}}\right\}$

and so consists of all $\Sigma$-measurable $f: X \to \R$ that are almost everywhere bounded, that is, subject to:

$\exists c \in \R: \mu \left({\left\{{\left\vert{f}\right\vert > c}\right\}}\right) = 0$


$\mathcal{L}^\infty \left({\mu}\right)$ can be endowed with the supremum seminorm $\left\Vert{\cdot}\right\Vert_\infty$ by:

$\displaystyle \forall f \in \mathcal{L}^\infty \left({\mu}\right): \left\Vert{f}\right\Vert_\infty := \inf \, \left\{{c \ge 0: \mu \left({\left\{{\left\vert{f}\right\vert > c}\right\}}\right) = 0}\right\}$


If, subsequently, we introduce the equivalence $\sim$ by:

$f \sim g \iff \left\Vert{f - g}\right\Vert_\infty = 0$

we obtain the quotient space $L^\infty \left({\mu}\right) := \mathcal{L}^\infty \left({\mu}\right) / \sim$, which is also called Lebesgue $\infty$-space for $\mu$.


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 $X$.


Source of Name

This entry was named for Henri Léon Lebesgue.

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

Also see

  • Results about Lebesgue spaces can be found here.


Sources