Definition:Lebesgue Space
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Definition
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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:
- $\exists c \in \R: \mu \left({\left\{{\left\vert{f}\right\vert > c}\right\}}\right) = 0$
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$\mathcal{L}^\infty \left({\mu}\right)$ 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 $\mu$.
Complex Lebesgue Space (first define complex integrals)
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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.
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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
- Results about Lebesgue spaces can be found here.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $12.5 \ \text{(ii)}$
- 2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control ... (previous) ... (next): $1.1$: Basic Definitions
