A note on some properties of $A$-functions

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 $X$.