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Transactions of the American Mathematical Society

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Random gaps


Author: James Hirschorn
Journal: Trans. Amer. Math. Soc. 361 (2009), 19-39
MSC (2000): Primary 03E05; Secondary 03E40, 28E15, 60H30
DOI: https://doi.org/10.1090/S0002-9947-08-04614-X
Published electronically: August 12, 2008
MathSciNet review: 2439396
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Abstract: It is proved that there exists an $ (\omega_1,\omega_1)$ Souslin gap in the Boolean algebra $ (L^0(\nu)/\operatorname{Fin}, \subseteq_{\operatorname{ae}}^*)$ for every nonseparable measure $ \nu$. Thus a Souslin, also known as destructible, $ (\omega_1,\omega_1)$ gap in $ \mathcal{P}(\mathbb{N})/ \operatorname{Fin}$ can always be constructed from uncountably many random reals. We explain how to obtain the corresponding conclusion from the hypothesis that Lebesgue measure can be extended to all subsets of the real line (RVM).


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Additional Information

James Hirschorn
Affiliation: Graduate School of Science and Technology, Kobe University, Japan
Address at time of publication: Thornhill, Ontario, Canada
Email: j_hirschorn@yahoo.com

DOI: https://doi.org/10.1090/S0002-9947-08-04614-X
Keywords: Gap, destructible gap, random real, real-valued measurable cardinal, nonseparable measure.
Received by editor(s): May 24, 2006
Published electronically: August 12, 2008
Additional Notes: This research was primarily supported by the Lise Meitner Fellowship, Fonds zur Förderung der wissenschaftlichen Forschung, Project No. M749-N05; the first version was completed on October 15, 2003, with partial support of Consorcio Centro de Investigación Matemática, Spanish Government grant No. SB2002-0099. Revisions were made with the support of the Japanese Society for the Promotion of Science, Project No. P04301.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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