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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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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


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 $X$.
    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:
  • MathSciNet review: 0407213