Definition:Lebesgue Space
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:
- $\map {\LL^p} \mu := \set {f: X \to \R: f \in \map \MM \Sigma, \ds \int \size f^p \rd \mu < \infty}$
where $\map \MM \Sigma$ denotes the space of $\Sigma$-measurable functions.
On $\map {\LL^p} \mu$, we can introduce the $p$-seminorm $\norm {\, \cdot \,}_p$ by:
- $\forall f \in \LL^p: \norm f_p := \paren {\ds \int \size f^p \rd \mu}^{1 / p}$
Next, define the equivalence $\sim$ by:
- $f \sim g \iff \norm {f - g}_p = 0$
The resulting quotient space:
- $\map {L^p} \mu := \map {\LL^p} \mu / \sim$
is also called (real) Lebesgue $p$-space.
Lebesgue $\infty$-Space
The Lebesgue $\infty$-space for $\mu$, denoted $\map {\LL^\infty} \mu$, is defined as:
- $\map {\LL^\infty} \mu := \set {f \in \map \MM \Sigma: f \text{ is essentially bounded} }$
and so consists of all $\Sigma$-measurable $f: X \to \R$ that are essentially bounded.
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$\map {\LL^\infty} \mu$ can be endowed with the supremum seminorm $\norm \cdot_\infty$ by:
- $\forall f \in \map {\LL^\infty} \mu: \norm f_\infty := \inf \set {c \ge 0: \map \mu {\set {\size f > c} } = 0}$
If, subsequently, we introduce the equivalence $\sim$ by:
- $f \sim g \iff \norm {f - g}_\infty = 0$
we obtain the quotient space $\map {L^\infty} \mu := \map {\LL^\infty} \mu / \sim$, which is also called Lebesgue $\infty$-space for $\mu$.
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Also known as
When the measure $\mu$ is clear, it is dropped from the notation, yielding $\LL^p$ and $L^p$.
If so desired, one can write, for example, $\map {\LL^p} X$ to emphasize $X$.
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.
Source of Name
This entry was named for Henri Léon Lebesgue.
Historical Note
While named for Henri Léon Lebesgue, the concept of the Lebesgue space was actually first introduced by Frigyes Riesz in $1910$.
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
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $1.1$: Normed and Banach spaces. Vector Spaces