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

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The average of a locally integrable function over a Euclidean ball

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 $R^{n}$ ), let $x \in R^{n}$ , and let $r > 0$ . The ball average of $f$ over $B (x, r)$ is $A_{r} f (x) : = \frac{1}{λ (B (x, r))} \int_{B (x, r)} f (y) d λ (y) .$

The denominator is well defined because Euclidean balls have positive finite Lebesgue measure shows that every Euclidean ball has positive finite Lebesgue measure.

Depends on

Used by

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Sources

Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Section 3.4 (standard reference, not scraped)