Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T18:52:19.843Z Has data issue: false hasContentIssue false
  • English
  • Français

Vanishing Borel sets

Published online by Cambridge University Press:  12 March 2014

Kenneth Schilling
Kenneth Schilling*
Affiliation:
Mathematics Department, University of Michigan-Flint, Flint, MI 48503, USA, E-mail: ksch@umich.edu
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 1 , March 1998 , pp. 262 - 268
Copyright
Copyright © Association for Symbolic Logic 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Henson, C. W. and Ross, D., Analytic mappings on hyperfinite sets, Proceedings of the American Mathematical Society, vol. 118 (1993), pp. 587–596.CrossRefGoogle Scholar
[2]Keisler, H. J., Kunen, K., Miller, A., and Leth, S., Descriptive set theory over a hyperfinite set, this Journal, vol. 54 (1989), pp. 1167–1180.Google Scholar
[3]Keisler, H. J. and Leth, S., Meager sets on the hyperfinite time line, this Journal, vol. 56 (1991), pp. 71–102.Google Scholar
[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. 113–122.CrossRefGoogle Scholar
[5]Živaljević, B., Some results about Borel sets in descriptive set theory of hyperfinite sets, this Journal, vol. 55 (1990), pp. 604–614.Google Scholar