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 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
- {{Citation | last = Lubotzky | first = Alexander | author-link = Alexander Lubotzky |title=Discrete groups, expanding graphs and invariant measures |series=Progress in Mathematics |volume=125 |publisher=Birkhäuser Verlag |location=Basel |year=1994 |isbn=0-8176-5075-X }}.
- {{Citation |first=Vladimir |last=Drinfeld |title=Finitely-additive measures on S2 and S3, invariant with respect to rotations |journal=Funktsional. Anal. I Prilozhen. |volume=18 |year=1984 |issue=3 |pages=77 |mr=0757256 }}.
- {{Citation |first=Grigory |last=Margulis |title=Some remarks on invariant means |journal=Monatshefte für Mathematik |volume=90 |year=1980 |issue=3 |pages=233–235 |mr=0596890 |doi=10.1007/BF01295368 }}.
- {{Citation |first=Dennis |last=Sullivan |title=For n > 3 there is only one finitely additive rotationally invariant measure on the n-sphere on all Lebesgue measurable sets |journal=Bulletin of the American Mathematical Society |volume=4 |year=1981 |issue=1 |pages=121–123 |mr=590825 |doi=10.1090/S0273-0979-1981-14880-1 |doi-access=free }}.
- [https://gauss.math.yale.edu/~ho2/doc/compact.pdf Survey of the area by Hee Oh]