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

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The space $L^{p} (μ)$ as the quotient by null functions

Definition

Let $(X, A, μ)$ be a measure space.

For $0 < p < \infty$ , define an equivalence relation on $L^{p} (μ)$ by $f \sim g ⟺ f - g \in N (μ),$ and write $L^{p} (μ)$ for the set of classes $[f]$ .
For $p = \infty$ , define the same relation on $L^{\infty} (μ)$ and again write $L^{\infty} (μ)$ for the set of classes.

Thus an element of $L^{p} (μ)$ is an almost-everywhere equivalence class of measurable representatives.

When $1 \leq p \leq \infty$ , the displayed set quotient agrees with the usual quotient-vector-space construction of The quotient vector space $V / W$ and its canonical projection.

For $0 < p < 1$ , the same class notation is used, but the later item The $L^{p}$ distance for $0 < p < 1$ is a complete translation-invariant metric supplies the metric structure rather than a normed-space structure.

Depends on

Used by

Dependency tree · two levels

11 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 7.17 (standard reference, not scraped)
John K. Hunter, Measure Theory, Section 7.4 (standard reference, not scraped)

The space Lp(μ)Lp(μ) as the quotient by null functions