DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-31

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The null subspace of measurable functions that vanish almost everywhere

Definition

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

$N (μ) : = {f : X \to R : f measurable and f = 0 μ -almost everywhere} .$

This is the family of null representatives that will be quotiented out of each $L^{p} (μ)$ and of $L^{\infty} (μ)$ . The linear-subspace claim implicit in that phrasing is proved in Null functions form a linear subspace and are exactly the zero-seminorm class.

Depends on

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

Used by

The space Lᵖ(μ) as the quotient by null functions Definition
Null functions form a linear subspace and are exactly the zero-seminorm class Proposition

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

Sheldon Axler, Measure, Integration & Real Analysis, Definition 7.15 (standard reference, not scraped)
John K. Hunter, Measure Theory, Section 7.1 (standard reference, not scraped)