The Baire order of the functions continuous almost everywhere

Abstract:

Let $\Phi$ be the family of all real-valued functions defined on the unit interval $I$ which are continuous except for a set of Lebesgue measure zero. Let ${\Phi _0}$ be $\Phi$ and for each ordinal $\alpha$, let ${\Phi _\alpha }$ be the family of all pointwise limits of sequences taken from $\bigcup \nolimits _{\gamma < \alpha } {{\Phi _\gamma }}$ Then ${\Phi _{{\omega _1}}}$ is the Baire family generated by $\Phi$. It is proven here that if $0 < \alpha < {\omega _1}$, then ${\Phi _\alpha } \ne {\Phi _{{\omega _1}}}$. The proof is based upon the construction of a Borel measurable function $h$ from $I$ onto the Hilbert cube $Q$ such that if $x$ is in $Q$, then ${h^{ - 1}}(x)$ is not a subset of an ${F_\sigma }$ set of Lebesgue measure zero.