math.LO — Logic

Borel Ruziewicz problem

Contributed by Konstantin Slutsky

We prove that normalized surface measure is the unique finitely additive rotation-invariant probability on the Borel subsets of $S^n$, $n\geq2$. We also prove that Lebesgue measure is the unique finitely additive measure on bounded Borel subsets of $\R^d$, $d\geq3$, 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.