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Vanishing Borel sets

  • Kenneth Schilling (a1)

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Henson and Ross [1] answered the question of when two hyperfinite sets A, B in an ℵ1-saturated nonstandard universe are bijective by a Borel function: precisely when ∣A∣/∣B∣ ≈ 1. Živaljević [5] generalized this result to nonvanishing Borel sets. He defined a set to be nonvanishing if it is Loeb-measurable and has finite, non-zero measure with respect to some Loeb counting measure. He then showed that two nonvanishing Borel sets are Borel bijective just in case they have the same finite, non-zero measure with respect to some Loeb counting measure.

Here we shall complete the cycle, for Borel sets at least. For N ∈ *N, let λN be the internal counting measure given by λN(A) = ∣A∣/N for A internal. Then for vanishing Loeb-measurable sets B, it is natural to consider the Dedekind cut (BL, BR) on *N consisting of those N for which B has 0λN-measure infinity and zero, respectively. We show that, for all vanishing Borel sets B, B and BL are Borel bijective. It follows that vanishing Borel sets B and C are Borel bijective if, and only if, BL = CL. Combined with Živaljević's result, we can characterize when arbitrary Borel sets are Borel bijective: precisely when they have the same measure with respect to all Loeb counting measures.

In the final section, we generalize in a similar way results of [2] and [5] to characterize when two Borel sets are bijective by a countably determined function: precisely when, for all N, one has 0λN-measure 0 if and only if the other also has 0λN-measure 0.

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[1]Henson, C. W. and Ross, D., Analytic mappings on hyperfinite sets, Proceedings of the American Mathematical Society, vol. 118 (1993), pp. 587596.
[2]Keisler, H. J., Kunen, K., Miller, A., and Leth, S., Descriptive set theory over a hyperfinite set, this Journal, vol. 54 (1989), pp. 11671180.
[3]Keisler, H. J. and Leth, S., Meager sets on the hyperfinite time line, this Journal, vol. 56 (1991), pp. 71102.
[4]Loeb, P. A., Conversion from nonstandard to standard measure spaces and applications in probability theory, Transactions of the American Mathematical Society, vol. 211 (1975), pp. 113122.
[5]Živaljević, B., Some results about Borel sets in descriptive set theory of hyperfinite sets, this Journal, vol. 55 (1990), pp. 604614.

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Vanishing Borel sets

  • Kenneth Schilling (a1)

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