\begin{definition}[Definition:Lebesgue Space/L-Infinity]
Let $\struct {X, \Sigma, \mu}$ be a measure 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.
{{refactor|The following can be split off into separate pages, as they need proving in their own right.}}
$\map {\LL^\infty} \mu$ 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 $\mu$'''.
\end{definition}