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Proceedings of the American Mathematical Society

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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?)

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Keywords: <!– MATH $({\mathbf {M}},\ast )$ –> <IMG WIDTH="62" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img5.gif" ALT="$({\mathbf {M}},\ast )$"> functions, <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$A$">-functions, <IMG WIDTH="19" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$\alpha$">-functions, complete ordinary function system, functions of class <IMG WIDTH="19" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\alpha$">, operation <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img4.gif" ALT="$A$">
Article copyright: © Copyright 1976 American Mathematical Society