Definition:Lebesgue Space/L-Infinity

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

The Lebesgue $\infty$-space for $\mu$, denoted $\map {\LL^\infty} \mu$, is defined as:


 * $\displaystyle \map {\LL^\infty} \mu := \set $

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:


 * $\displaystyle \forall f \in \map {\LL^\infty} \mu: \norm f_\infty := \inf \, \set $

If, subsequently, we introduce the equivalence $\sim$ by:


 * $f \sim g \iff \norm {f - g}_\infty = 0$

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

 * Definition:Lebesgue Space
 * Definition:Supremum Seminorm
 * Definition:Supremum Norm