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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The space $L^{\infty} (μ)$ of essentially bounded measurable functions

Definition

Let $(X, A, μ)$ be a measure space. The class

$L^{\infty} (μ) : = {f : X \to R : f measurable and ∥ f ∥_{\infty} < \infty}$

of essentially bounded measurable functions is the page's $p = \infty$ space. Its size is measured by the essential-supremum functional of The essential supremum of a measurable function with respect to a measure.

The later theorem $L^{p}$ and $L^{\infty}$ are vector spaces for $p \geq 1$ proves that $L^{\infty} (μ)$ is a real vector space under pointwise addition and scalar multiplication.

Depends on

The essential supremum of a measurable function with respect 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, Section 7.1 (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, Section 7A (standard reference, not scraped)

The space L∞(μ)L∞(μ) of essentially bounded measurable functions