DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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Lebesgue points and the Lebesgue set of an $L_{l o c}^{1}$ class

Definition

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Let $[f] \in L_{l o c}^{1} (R^{n})$ (A locally integrable function on $R^{n}$ ).

For a representative $f$ of that class, a point $x \in R^{n}$ is a Lebesgue point of $f$ if $\lim_{r \to 0^{+}} \frac{1}{λ (B (x, r))} \int_{B (x, r)} ∣ f (y) - f (x) ∣ d λ (y) = 0.$

The Lebesgue set of the class $[f]$ is the set $Leb ([f]) : = {x \in R^{n} : some representative of [f] has x as a Lebesgue point} .$

The later theorem on Lebesgue points shows that different representatives of the same $L_{l o c}^{1}$ class change this set only by a null set.

Depends on

Used by

Dependency tree · two levels

12 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

Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Theorem 3.20 (standard reference, not scraped)
Walter Rudin, Real and Complex Analysis, 3rd ed., Theorem 7.7 (standard reference, not scraped)

Lebesgue points and the Lebesgue set of an Lloc1Lloc1​ class