For 𝑛>3 there is only one finitely additive rotationally invariant measure on the 𝑛-sphere defined on all Lebesgue measurable subsets

Sullivan, Dennis

doi:10.1090/S0273-0979-1981-14880-1

Bulletin of the American Mathematical Society

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ISSN 1088-9485 (online) ISSN 0273-0979 (print)

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For $n > 3$ there is only one finitely additive rotationally invariant measure on the $n$-sphere defined on all Lebesgue measurable subsets
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Bull. Amer. Math. Soc. 4 (1981), 121-123

References

D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen. 1 (1967), 71–74 (Russian). MR 0209390
Joseph Rosenblatt, Uniqueness of invariant means for measure-preserving transformations, Trans. Amer. Math. Soc. 265 (1981), no. 2, 623–636. MR 610970, DOI 10.1090/S0002-9947-1981-0610970-7