Summary
There exists a Borel set C of product Lebesgue measure one in the Hilbert cube having the property that, for every measure preserving transformationT of the unit interval, allT-orbits contained inC originate from a zero set. This settles an infinite dimensional version of a problem raised by Th. M. Rassias.
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References
Iwanik, A.,A property of doubly stochastic densities, Acta Univ. Carolin.—Math. Phys.30 (1989), 65–67.
Rassias, Th. M.,Problem 16, 1°, in:Report of the 27th International Symposium on Functional Equations (Problems and Remarks), Aequationes Math.39 (1990), 308.
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Iwanik, A. A large set containing few orbits of measure preserving transformations. Aeq. Math. 43, 156–158 (1992). https://doi.org/10.1007/BF01835697
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DOI: https://doi.org/10.1007/BF01835697