On Borel measures and Baire’s class $3$

Abstract:

Let $S$ be a complete and separable metric space and $\mu$ a $\sigma$-finite, complete Borel measure on $S$. Let $\Phi$ be the family of all real-valued functions, continuous $\mu$-a.e. Let ${B_\alpha }(\Phi )$ be the functions of Baire’s class $\alpha$ generated by $\Phi$. It is shown that if $\mu$ is not a purely atomic measure whose set of atoms form a dispersed subset of $S$, then ${B_2}(\Phi ) \ne {B_{{\omega _1}}}(\Phi )$, where ${\omega _1}$ denotes the first uncountable ordinal.