DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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The essential supremum of a measurable function with respect to a measure

Definition

Let $(X, A, μ)$ be a measure space, and let $f : X \to R$ be measurable. The essential supremum of $∣ f ∣$ with respect to $μ$ is

$∥ f ∥_{\infty, μ} : = \inf {M \in [0, \infty] : ∣ f ∣ \leq M μ -almost everywhere} .$

When the measure is fixed from context, this is written $∥ f ∥_{\infty}$ . The phrase essentially bounded means $∥ f ∥_{\infty} < \infty$ .

The later proposition The essential supremum is attained as the least essential bound proves that when $∥ f ∥_{\infty} < \infty$ , the inequality $∣ f ∣ \leq ∥ f ∥_{\infty}$ itself holds almost everywhere and that this bound is the least essential bound.

Depends on

Measure-null sets and almost-everywhere statements relative to a measure

Used by

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John K. Hunter, Measure Theory, Definition 7.3 (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, Section 7A (standard reference, not scraped)