CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-sol)

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Ball mean-value property for harmonic functions under Countable Choice

Statement

Assume the Axiom of Countable Choice. Under Spherical mean-value property for harmonic functions, $u (x) = A_{u} (x, r)$ for every $B_{r} (x) ⋐ Ω$ .

Proof

Given: Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), $u$ is harmonic and $B_{r} (x) ⋐ Ω$ .

1.1

Under Countable Choice, polar coordinates express $\int_{B_{r} (x)} u = ω_{n - 1} \int_{0}^{r} t^{n - 1} M_{u} (x, t) d t$ [given].

1.2

The spherical identity makes this $ω_{n - 1} u (x) r^{n} / n = u (x) ∣ B_{r} ∣$ [given, algebra].

2.1

Divide by the positive ball volume from Sphere and ball measures scale in Rn [step 1.2]. ∎

Depends on

Used by

Entire harmonic functions with bounded gradient are affine Corollary
Removable singularity for bounded harmonic functions under Countable Choice Corollary
Harnack constant from the Poisson kernel ratio under Countable Choice Example
Derivative estimate proof of one sided harmonic liouville Lemma
Harmonic functions are real analytic Theorem
Harnack inequality on a ball Theorem
Interior derivative estimates for harmonic functions Theorem
Locally uniform limits of harmonic functions are harmonic Theorem
Weyl's lemma for the Laplacian Theorem

Dependency tree · two levels

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Sources

PDE source treatment (standard reference, not scraped)