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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A note on some properties of $ A$-functions


Author: H. Sarbadhikari
Journal: Proc. Amer. Math. Soc. 56 (1976), 321-324
MSC: Primary 26A42
MathSciNet review: 0407213
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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 [Enhancements On Off] (What's this?)

  • [1] Felix Hausdorff, Mengenlehre, de Gruyter, Berlin, 1937; English transl., Set theory, Chelsea, New York, 1957. MR 19, 111.
  • [2] K. Kunugui, Sur un théorème d'existence dans la théorie des ensembles projectifs, Fund. Math. 29 (1937), 167-181.
  • [3] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York, 1966. MR 0217751 (36 #840)
  • [4] 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

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0407213-X
PII: S 0002-9939(1976)0407213-X
Keywords: $ ({\mathbf{M}},\ast)$ functions, $ A$-functions, $ \alpha $-functions, complete ordinary function system, functions of class $ \alpha $, operation $ A$
Article copyright: © Copyright 1976 American Mathematical Society