Summary
In [1], an example was given of a measure-preserving dissipative transformation T in a σ-finite measure space (X, ℛ, Μ), such that T is conservative in the measure space (X, ℛ∞, Μ) where \(\mathcal{R}_\infty = \mathop \cap \limits_{n = 0}^\infty T^{ - n} \mathcal{R}\). Here we shall show that for this transformation we actually have R ∞={ØX}[μ].
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Reference
Helmberg, G., Simons, F. H.: On the conservative parts of the Markov processes induced by a measurable transformation. Z. Wahrscheinlichkeitstheorie verw. Geb. 11, 165–180 (1969).
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Simons, F.H. A dissipative transformation with a trivial tail-algebra. Z. Wahrscheinlichkeitstheorie verw Gebiete 15, 177–179 (1970). https://doi.org/10.1007/BF00534913
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DOI: https://doi.org/10.1007/BF00534913