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Meager sets on the hyperfinite time line

  • H. Jerome Keisler (a1) and Steven C. Leth (a2)

In this paper we study notions of a “meager subset” of a hyper-finite set. We work within an ω-saturated nonstandard universe and fix a hyperfinite natural number Є *N∖N. We shall consider subsets of the set = {1, 2, …,H}.

By analogy with the meager subsets of the real interval [0, 1], a notion of meager subset of should have the following properties.

1. Finite sets, countable unions of meager sets, subsets of meager sets, and translates of meager sets should be meager.

2. The Baire Category Theorem should hold; that is, should not be meager.

3. The internal analogue of the Cantor set should be meager.

4. The notion of a meager set should be with respect to a natural topology on .

5. There should exist meager subsets of of Loeb measure one.

6. Sierpiński and Lusin sets should have hyperfinite counterparts with properties similar to the classical case.

Property 5 is desirable so that, as with Lebesgue measure and Baire category on [0, 1], the topological and measure-theoretic notions of “large” and “small” sets are incomparable.

Property 6, while not as necessary as the other five, is desirable because of the strong interplay between measure and category in the classical results about Lusin and Sierpinski sets. Our objective is to find notions of meager set which have a relationship to Loeb measure similar to the classical relationship between meager sets and Lebesgue measure.

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[Ca] M. Canjar , Countable ultraproducts without CH, Annals of Pure and Applied Logic, vol. 37 (1988), pp. 179.

[G1] H. Gonshor , Remarks on the Dedekind completion of a nonstandard model of the reals, Pacific Journal of Mathematics, vol. 118 (1985), pp. 117132.

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[K1] H. J. Keisler , Monotone complete fields, Victoria symposium on nonstandard analysis (1974) ( A. Hurd and P. Loeb , editors). Lecture Notes in Mathematics, vol. 369, Springer-Verlag, Berlin, 1974, pp. 113115.

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[L1] S. Leth , Sequences in countable nonstandard models of the natural numbers, Studia Logica, vol. 47 (1988), pp. 6383.

[L2] S. Leth , Applications of nonstandard models and Lebesgue measure to sequences of natural numbers, Transactions of the American Mathematical Society, vol. 307 (1988), pp. 457468.

[Lo] P. Loeb , Conversion from nonstandard to standard measure spaces and applications in probability theory, Transactions of the American Mathematical Society, vol. 211 (1975), pp. 113122.

[M1] A. Miller , Special subsets of the real line. Handbook of set-theoretic topology ( K. Kunen and J. E. Vaughan , editors), North-Holland, Amsterdam, 1984, pp. 201234.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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