An analog of the Baire classification of measurable and standard functions (Q2276614)

Let \(L\) be a topological space and \({\mathcal B}_0(L)\) its \(\sigma\)-algebra of Baire sets. It is well-known that every \({\mathcal B}_0(L)\)-measurable function \(f:L\to\mathbb{R}\) on \(L\) is a Baire function of type \(\alpha\) for some ordinal number \(\alpha\). The author generalizes this fact, in particular, replacing \((L,{\mathcal B}_0(L))\) by a measurable space \((L,\Lambda)\) and \(\mathbb{R}\) by a separable metric space \(Y\). The obtained generalization is then used to get information about ``standard'' functions, which are defined as follows: Let \((T, {\mathcal T})\) be a measurable space, \(X\) a metric space, \({\mathcal B}(X)\) its Borel \(\sigma\)-algebra and \(\Lambda:={\mathcal T}\times{\mathcal B}(X)\) be the \(\sigma\)-algebra on \(L:=T\times X\) generated by the sets \(A\times B\) where \(A \in{\mathcal T}\) and \(B\in{\mathcal B}(X)\). Let \({\mathcal R}=\{T_0\subseteq T:A\in {\mathcal I}\) for any \(A \subseteq T\setminus T_0\}\). Then a function \(f:L\to Y\) with values in a metric space is called standard if for a certain \(T_0\in {\mathcal R}\) the restriction \(f|_{T_0\times X}\) is \((\Lambda,{\mathcal B}(Y))\)-measurable.