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 by Drinfeld (published 1984) for "n" = 2 and 3, and for "n" ≥ 4 by Margulis and Dennis Sullivan around 1980. It fails for the circle.

The name is for Stanisław Ruziewicz.

References

* Alexander Lubotzky, "Discrete groups, expanding graphs and invariant measures". Progress in Mathematics, vol 125, Birkhäuser Verlag, Basel, 1994
* Grigory Margulis, "Some remarks on invariant means", Monatsh. Math. 90 (1980), no. 3, 233–235 MathSciNet|id=0596890
* Dennis Sullivan, "For n > 3 there is only one finitely additive rotationally invariant measure on the n-sphere on all Lebesgue measurable sets", Bull. AMS 1 (1981), 121–123
* [http://www.its.caltech.edu/~heeoh/compact.pdf Survey of the area by Hee Oh]