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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Random gaps
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by James Hirschorn PDF
Trans. Amer. Math. Soc. 361 (2009), 19-39 Request permission


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
  • MR Author ID: 633758
  • Email:
  • 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.
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 19-39
  • MSC (2000): Primary 03E05; Secondary 03E40, 28E15, 60H30
  • DOI:
  • MathSciNet review: 2439396