Definition:Lebesgue Space/L-Infinity
Definition
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.
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$\map {\LL^\infty} \mu$ 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
It is common to name $\map {\LL^\infty} \mu$ after its symbol, that is: 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 $\LL^\infty$.
Also see
Source of Name
This entry was named for Henri Léon Lebesgue.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $12.15$