Finitely additive measures on \(S^ 2\) and \(S^ 3\), invariant with respect to rotations
A finitely additive rotationally invariant measure on the ring of all Lebesgue measurable subsets in the n-sphere \(S^ n\) is called Ruziewicz measure. [Oeuvres avec des commentaires, vol. 1, 318-322 (1967; Zbl 0163.056)] proved, that there exist a Ruziewicz measure on \(S^ 1\), which is not proportional to Lebesgue measure. \textit{G. A. Margulis} [Monatsh. Math. 90, 233-236 (1980; Zbl 0436.43002)] and \textit{D. Sullivan} [Bull. Am. Math. Soc. New Ser. 4, 121-123 (1981; Zbl 0459.28009)] proved that every Ruziewicz measure on \(S^ n\) is proportional to Lebesgue measure if \(n\geq 4\). The author proves that every Ruziewicz measure on \(S^ 2\) and \(S^ 3\) is also proportional to Lebesgue measure.
finitely additive rotationally invariant measure
scientific article; zbMATH DE number 3920860
scientific article