Ruziewicz problem

In mathematics, the Ruziewicz problem (sometimes Banach–Ruziewicz problem) in measure theory asks whether the usual Lebesgue measure on the n-sphere is characterised, up to proportionality, by its properties of being finitely additive, invariant under rotations, and defined on all Lebesgue measurable sets.

This was answered affirmatively and independently for n ≥ 4 by Grigory Margulis and Dennis Sullivan around 1980, and for n = 2 and 3 by Vladimir Drinfeld (published 1984). It fails for the circle.

The problem is named after Stanisław Ruziewicz.

References

Lubotzky, Alexander (1994), Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, vol. 125, Basel: Birkhäuser Verlag, ISBN 0-8176-5075-X.
Drinfeld, Vladimir (1984), "Finitely-additive measures on S² and S³, invariant with respect to rotations", Funktsional. Anal. I Prilozhen., 18 (3): 77, MR 0757256.
Margulis, Grigory (1980), "Some remarks on invariant means", Monatshefte für Mathematik, 90 (3): 233–235, doi:10.1007/BF01295368, MR 0596890.
Sullivan, Dennis (1981), "For n > 3 there is only one finitely additive rotationally invariant measure on the n-sphere on all Lebesgue measurable sets", Bulletin of the American Mathematical Society, 4 (1): 121–123, doi:10.1090/S0273-0979-1981-14880-1, MR 0590825.
Survey of the area by Hee Oh