Ruziewicz problem

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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.

[edit] 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 MR0596890
• 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
• Survey of the area by Hee Oh