Summary
In 1959, H. Dye showed that any two ergodic, measure-preserving automorphisms of a Lebesgue measure algebra were weakly equivalent. In this paper, we study weak equivalence, for ergodic measure-preserving automorphisms on non-separable measure algebras. It is shown that, in general, Dye's Theorem does not hold, and in particular, it holds only on separable, i.e. Lebesgue, measure algebras.
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Choksi, J.R., Eigen, S.J.: An Automorphism of a Homogeneous Measure Algebra Which Does Not Factorize into a Direct Product, Conference in Modern Analysis and Probability, Contempory Mathematics vol. 26, A.M.S., Providence, 1982
Choksi, J.R., Prasad, V.S.: Ergodic Theory on Homogeneous Measure Algebras. Measure Theory Oberwolfach 1981. Springer Lecture Notes in Math. 945, pp. 366–405. Berlin-Heidelberg-New York: Springer 1982
Dye, H.A.: On Groups of Measure Preserving Transformatios I. Am. J. Math. 81, 119–159 (1959)
Hajian, A., Ito, Y., Kakutani, S.: Full Groups and a Theorem of Dye. Adv. Math. 17, 49–57 (1975)
Maharam, D.: On Homogeneous Measure Algebras. Proc.Natl. Acad. Sci. (Washington) 28, 108–111 (1942)
Maharam, D.: Automorphisms of Products of Measure Spaces. Proc. Am. Math. Soc. 9, 702–707 (1958)
Rohklin, V.A.: Lectures on Entropy Theory of Measure Prserving Transformations. Russ. Math. Surv. 22, 1–52 (1967)
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Eigen, S.J. A counter-example to Dye's Theorem for all non-separable measure algebras. Probab. Th. Rel. Fields 72, 471–475 (1986). https://doi.org/10.1007/BF00334197
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DOI: https://doi.org/10.1007/BF00334197