On piecewise injective measurable maps (Q1022255)

Let \(X\) be a Polish space and let \(\mu\) be the completion of a positive finite Borel measure on \(X\). Denote by \({\mathcal B}_\mu(X)\) the domain of \(\mu\). The author calls a \(\mu\)-measurable map \(\alpha: X\to Y\), where \(Y\) is another Polish space, almost piecewise injective if there exists a countable family \(\{X_i: i\in I\}\) in \({\mathcal B}_\mu(X)\) such that \(\mu(X\setminus\bigcup_{i\in I}X_i)= 0\) and \(\alpha|X_i\) is injective for all \(i\in I\). It is shown that this condition is equivalent to each of the following ones: (1) there exists \(\widetilde X\) in \({\mathcal B}_\mu(X)\) such that \(\mu(X\setminus\widetilde X)= 0\) and \(\alpha^{-1}(y)\cap\widetilde X\) is countable for all \(y\in Y\); (2) there exists \(\widetilde X\) in \({\mathcal B}_\mu(X)\) such that \(\mu(X\setminus\widetilde X)= 0\) and for every \(A\subset\widetilde X\) with \(\mu(A)= 0\) we have \(\mu(\alpha^{-1}(\alpha(A)))= 0\). The proofs are based on some classical results from the descriptive theory of Polish spaces mainly due to Luzin and Mazurkiewicz. Unfortunately, the English of the paper is rather bad and some proofs are lengthy.