In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Ruziewicz problem (sometimes Banach–Ruziewicz problem) in
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
asks whether the usual
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
on the
''n''-sphere is characterised, up to proportionality, by its properties of being
finitely additive
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity ...
, invariant under
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s, and defined on all
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
sets.
This was answered affirmatively and independently for ''n'' ≥ 4 by
Grigory Margulis
Grigory Aleksandrovich Margulis (russian: Григо́рий Алекса́ндрович Маргу́лис, first name often given as Gregory, Grigori or Gregori; born February 24, 1946) is a Russian-American mathematician known for his work on ...
and
Dennis Sullivan
Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Ce ...
around 1980, and for ''n'' = 2 and 3 by
Vladimir Drinfeld
Vladimir Gershonovich Drinfeld ( uk, Володи́мир Ге́ршонович Дрінфельд; russian: Влади́мир Ге́ршонович Дри́нфельд; born February 14, 1954), surname also romanized as Drinfel'd, is a renowne ...
(published 1984). It fails for the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
.
The problem is named after
Stanisław Ruziewicz
Stanisław Ruziewicz (29 August 1889 – 12 July 1941) was a Polish mathematician and one of the founders of the Lwów School of Mathematics.
He was a former student of Wacław Sierpiński, earning his doctorate in 1913 from the University of Lw ...
.
References
*.
*.
*.
*{{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 .
Survey of the area by Hee Oh
Measure theory