Measurable parametrizations of sets in product spaces

Abstract:

Various parametrization theorems are proved. In particular the following is shown: Let $B$ be a Borel subset of $I \times I$ (where $I = [0, 1]$) with uncountable vertical sections. Let $\sum \dot \cup N$ be the discrete (topological) union of $\sum$, the space of irrationals, and $N$, the set of natural numbers with discrete topology. Then there is a map $f:I \times (\sum \dot \cup N) \to I$ measurable with respect to the product of the analytic $\sigma$-field on $I$ (that is, the smallest $\sigma$-field on $I$ containing the analytic sets) and the Borel $\sigma$-field on $\sum \dot \cup N$ such that $f(t, \cdot ): \sum \dot \cup N \to I$ is a one-one continuous map of $\sum \dot \cup N$ onto $\{ x:(t, x) \in B\}$ for each $t \in T$. This answers a question of Cenzer and Mauldin.