# 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,**125**, Basel: Birkhäuser Verlag, ISBN 0-8176-5075-X. - Drinfeld, Vladimir (1984), "Finitely-additive measures on S
^{2}and S^{3}, 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 590825. - Survey of the area by Hee Oh

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