A note on some properties of $A$-functions

This note deals with (M, *) functions for various families M. It is shown that if M is the family of Borel sets of additive class a on a metric space X, then (M, *) functions are just the functions of the form supyg(x,y) where g: X x R -4 R is continuous in y and of class a in x. If M is the class of analytic sets in a Polish space X, then the (M, *) functions dominating a Borel function are just the functions supyg(x,y) where g is a real valued Borel function on X2. 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(C) e the a-algebra generated by the analytic sets on X.