Infinite products of Borel measurable functions (Q952601)

For a nonzero ordinal \(\alpha\), by \({\mathcal B}_\alpha\) the corresponding Baire class is denoted. For a family of functions \({\mathcal F}\subset \mathbb R^{\mathbb R}\) by \({}^\pi{\mathcal B}_1({\mathcal F})\) denote the class of all functions \(f:\mathbb R\to \mathbb R\) for which there is \(\{f_n\}_{n\in \mathbb N}\subset\mathbb F\) such that \(f(x) =\prod_{n=1}^\infty f_n(x)\), for each \(x\in \mathbb R\). Let \(M\) be the family of all functions \(f:\mathbb R\to\mathbb R\) with the following property: \[ (\forall a<b) \quad \left( f(a)b(f)<0 \Rightarrow (a,b) \cap \{f=0\}\neq \emptyset\right). \] In the paper it is proved that \[ {}^\pi {\mathcal B}_1({\mathcal B}_0)=M\cap {\mathcal B}_1; \qquad {}^\pi {\mathcal B}_1\biggl(\bigcup_{\beta<\alpha} {\mathcal B}_\beta\biggr)={\mathcal B}_\alpha \quad \text{for any ordinal \(\alpha>1\)}. \]