The Baire order of the functions continuous almost everywhere

Abstract:

Let $S$ be a complete and separable metric space and $\mu$ a $\sigma$-finite, complete Borel measure on $S$ with $\mu (S) > 0$. Let $\Phi$ be the family of all real-valued functions defined on $S$ whose set of points of discontinuity is of $\mu$-measure 0. Let ${B_\alpha }(\Phi )$ be the functions of Baire’s class $\alpha$ generated by $\Phi$. It is shown that ${B_1}(\Phi ) = {B_2}(\Phi )$ if and only if $\mu$ is a purely atomic measure whose set of atoms forms a scattered subset of $S$ and that if ${B_1}(\Phi ) \ne {B_2}(\Phi )$, then the Baire order of $\Phi$ is ${\omega _1}$; in other words, if $0 \leq \alpha < {\omega _1}$, then ${B_\alpha }(\Phi ) \ne {B_{\alpha + 1}}(\Phi )$. This answers a generalized version of a problem raised by Sierpinski and Felsztyn. An example is given of a normal space with Borel order 2 and Baire order ${\omega _1}$.