# Definition:Lebesgue Space

## Definition

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.

$\map {\LL^\infty} \mu$ 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 $\mu$.

## 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

• 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 $1910$.