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A Note on Some Properties of $A$-Functions

H. Sarbadhikari
Proceedings of the American Mathematical Society
Vol. 56, No. 1 (Apr., 1976), pp. 321-324
DOI: 10.2307/2041628
Stable URL: http://www.jstor.org/stable/2041628
Page Count: 4
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A Note on Some Properties of $A$-Functions
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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 $\sup_yg(x,y)$ where $g: X \times R \rightarrow R$ 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 $\sup_yg(x,y)$ 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 $f^{-1}(C) \not\in$ the $\sigma$-algebra generated by the analytic sets on $X$.

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