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Proceedings of the American Mathematical Society

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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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A note on some properties of $A$-functions
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by H. Sarbadhikari PDF
Proc. Amer. Math. Soc. 56 (1976), 321-324 Request permission

Abstract:

This note deals with $({\mathbf {M}},\ast )$ functions for various families ${\mathbf {M}}$. It is shown that if ${\mathbf {M}}$ is the family of Borel sets of additive class $\alpha$ on a metric space $X$, then $({\mathbf {M}},\ast )$ functions are just the functions of the form ${\sup _y}g(x,y)$ where $g:X \times R \to R$ is continuous in $y$ and of class $\alpha$ in $x$. If ${\mathbf {M}}$ is the class of analytic sets in a Polish space $X$, then the $({\mathbf {M}},\ast )$ functions dominating a Borel function are just the functions ${\sup _y}g(x,y)$ where $g$ is a real valued Borel function on ${X^2}$. It is also shown that there is an $A$-function $f$ defined on an uncountable Polish space $X$ and an analytic subset $C$ of the real line such that ${f^{ - 1}}(C) \notin$ the $\sigma$-algebra generated by the analytic sets on $X$.
References
    Felix Hausdorff, Mengenlehre, de Gruyter, Berlin, 1937; English transl., Set theory, Chelsea, New York, 1957. MR 19, 111. K. Kunugui, Sur un théorème d’existence dans la théorie des ensembles projectifs, Fund. Math. 29 (1937), 167-181.
  • K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
  • E. Sélivanowski, Sur une classe d’ensembles définis par une infinité dénombrable de conditions, C.R. Acad. Sci. Paris 184 (1927), 1311.
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 56 (1976), 321-324
  • MSC: Primary 26A42
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0407213-X
  • MathSciNet review: 0407213