## Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. 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

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
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 $X$.