Definition:Lebesgue Space

From ProofWiki
Jump to navigation Jump to search


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 $1910$.