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Proceedings of the American Mathematical Society

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On the Egoroff property of pointwise convergent sequences of functions


Authors: Andreas Blass and Thomas Jech
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
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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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DOI: https://doi.org/10.1090/S0002-9939-1986-0857955-3
Keywords: Egoroff property, continuum hypothesis, bounding number
Article copyright: © Copyright 1986 American Mathematical Society