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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the Egoroff property of pointwise convergent sequences of functions
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by Andreas Blass and Thomas Jech PDF
Proc. Amer. Math. Soc. 98 (1986), 524-526 Request permission

Abstract:

The space $\mathcal {L}(x)$ of real-valued functions on $X$ has the Egoroff property if for any $\{ {f_{nk}}\}$ such that $0 \leqslant {f_{nk}}{ \uparrow _k}f$ (for every $n$), there exists ${g_m} \uparrow f$ such that, for each $m$ and $n$, ${g_m}{ \leqslant _{nk}}$ for some $k$. We show that $\mathcal {L}(X)$ has the Egoroff property if and only if the cardinality of $X$ is smaller than the minimum cardinality of an unbounded family of functions from the set of natural numbers to itself. Therefore, the statement that there is an uncountable set $X$ such that $\mathcal {L}(X)$ has the Egoroff property is independent of the axioms of set theory.
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 98 (1986), 524-526
  • MSC: Primary 54A35; Secondary 03E35, 54C35
  • DOI: https://doi.org/10.1090/S0002-9939-1986-0857955-3
  • MathSciNet review: 857955