Journal of Symbolic Logic 57 (2):528-533 (1992)

In an ω1-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U, a corresponding U-topology on the hyperintegers by letting O be U-open if for any x ∈ O there is a y greater than all the elements in U such that the interval $\lbrack x - y, x + y\rbrack \subseteq O$ . Let U be a cut in a hyperfinite time line H, which is a hyperfinite initial segment of the hyperintegers. A subset B of H is called a U-Lusin set in H if B is uncountable and for any Loeb-Borel U-meager subset X of H, B ∩ X is countable. Here a Loeb-Borel set is an element of the σ-algebra generated by all internal subsets of H. In this paper we answer some questions of Keisler and Leth about the existence of U-Lusin sets by proving the following facts. (1) If $U = x/\mathbb{N} = \{y \in \mathscr{H}: \forall n \in \mathbb{N}(y < x/n)\}$ for some x ∈ H, then there exists a U-Lusin set of power κ if and only if there exists a Lusin set of the reals of power κ. (2) If U ≠ x/N but the coinitiality of U is ω, then there are no U-Lusin sets if CH fails. (3) Under ZFC there exists a nonstandard universe in which U-Lusin sets exist for every cut U with uncountable cofinality and coinitiality. (4) In any ω2-saturated nonstandard universe there are no U-Lusin sets for all cuts U except U = x/N
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DOI 10.2307/2275287
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Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
Foundations of Infinitesimal Stochastic Analysis.K. D. Stroyan - 1988 - Journal of Symbolic Logic 53 (4):1261-1262.

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