# 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