L-infinity space

Builds on Measure space Measurable function Essential supremum Almost everywhere

Definition

Let $(X, \Sigma, \mu)$ be a measure space, and let $\mathbb{K}$ be $\R$ or $\C$ . A measurable function $f \colon X \to \mathbb{K}$ is essentially bounded if there is $M \ge 0$ with $\mu(\set{x \in X : \abs{f(x)} > M}) = 0$ . Its norm is the essential supremum $\norm{f}_\infty = \inf \set{M \ge 0 : \mu(\set{x : \abs{f(x)} > M}) = 0}.$ The space $L^\infty(\mu)$ is the set of essentially bounded measurable functions modulo equality almost everywhere, with the norm $\norm{\cdot}_\infty$ .

Remarks

Example: on $[0, 1]$ with Lebesgue measure, let $f(x) = x$ for rational $x$ and $f(x) = 0$ otherwise. The set where $f \ne 0$ is countable, so it has measure $0$ , and $\norm{f}_\infty = 0$ . So $f$ is the zero element of $L^\infty$ .

Non-example: $f(x) = 1/x$ on $(0, 1]$ is not essentially bounded. For each $M > 0$ , the set $\set{f > M}$ contains the interval $(0, \min(1, 1/M))$ , which has positive measure.

The infimum in the definition is attained: $\abs{f} \le \norm{f}_\infty$ almost everywhere. With the counting measure on a set $I$ , $L^\infty$ is the space $\ell^\infty(I)$ of bounded families with the supremum norm.

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Tag	`7PCHXZM5YE`
URI	`m2://7PCHXZM5YE`
Kind	definition
Writing	W0 stub
MSC	46E30
Defines	L-infinity space
Uses	Measure space, Measurable function, Essential supremum, Almost everywhere
Changed	2 November 2026

L-infinity space

Definition

Remarks

Reading path

Further reading

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