Definition:Lebesgue Space

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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.


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, $\mathcal L^p \left({X}\right)$ 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


Sources