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Borel Ruziewicz problem

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Contributed by Konstantin Slutsky ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We prove that normalized surface measure is the unique finitely additive rotation-invariant probability on the Borel subsets of SnS^n, n≥2n\geq2. We also prove that Lebesgue measure is the unique finitely additive measure on bounded Borel subsets of Rd\R^d, d≥3d\geq3, invariant under orientation-preserving rigid motions and normalized on the unit cube. The main step is the annihilation of Borel null sets. A spectral gap first gives bounded divergence certificates for compact null sets, chosen in a jointly Borel way. A shared random threshold then combines these certificates to cover arbitrary null sets. For Euclidean space, the construction is applied to families of spheres about two centers; spherical averaging completes the identification of the measure.

Tools used

OpenAI
Codex
Anthropic
Claude Fable

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Version history

  1. v1 · 12 pagesSubmitted by Konstantin SlutskyInitial depositCurrentNov 11, 2026