Definition:Lebesgue Space/L-Infinity

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Definition

Let $\left({X, \Sigma, \mu}\right)$ be a measure 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:

$\exists c \in \R: \mu \left({\left\{{\left\vert{f}\right\vert > c}\right\}}\right) = 0$


$\mathcal{L}^\infty \left({\mu}\right)$ 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 $\mu$.


Also known as

It is common to name $\mathcal{L}^\infty \left({\mu}\right)$ after its symbol, i.e. L-infinity or L-infinity for $\mu$.

A more descriptive term is space of essentially bounded functions for $\mu$, cf. essentially bounded function.


When $\mu$ is clear from the context, it may be dropped from the notation, yielding $\mathcal{L}^\infty$.


Also see


Source of Name

This entry was named for Henri Léon Lebesgue.


Sources