DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-09-24 (gpt-6-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Spherical averages and local ball means in Rn

Definition

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), under which the cited Lebesgue and polar surface measures and their integrals are supplied.

Let $n \geq 1$ , let $Ω \subseteq R^{n}$ be open, and let $u : Ω \to R$ be locally Lebesgue integrable. Assume also that $θ \mapsto u (x + r θ)$ is $σ$ -integrable on $S^{n - 1}$ whenever $\overline{B_{r} (x)} \subseteq Ω$ . Put $ω_{n - 1} : = σ (S^{n - 1})$ . For $x \in Ω$ and $r > 0$ with $\overline{B_{r} (x)} \subseteq Ω$ , define $M_{u} (x, r) : = \frac{1}{ω_{n - 1}} \int_{S^{n - 1}} u (x + r θ) d σ (θ), A_{u} (x, r) : = \frac{1}{∣ B_{r} (x) ∣} \int_{B_{r} (x)} u (y) d y .$ The spherical (respectively ball) mean-value property says $u (x) = M_{u} (x, r)$ (respectively $u (x) = A_{u} (x, r)$ ) for every such ball. The polar formula Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma makes the normalizations meaningful.

Depends on

Dependency tree · 2 levels

Sources

PDE source treatment