Definition:Lebesgue Space

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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