Borel parametrizations

Abstract:

Let X and Y be uncountable Polish spaces and B a Borel subset oi $X \times Y$ such that for each x, ${B_x}$ is uncountable. A Borel parametrization of B is a Borel isomorphism, g, of $X \times E$ onto B where E is a Borel subset of Y such that for each x, $g\left ( {x, \cdot } \right )$ maps E onto ${B_x} = \left \{ {y: \left ( {x, y} \right ) \in B} \right \}$. It is shown that B has a Borel parametrization if and only if B contains a Borel set M such that for each x, ${M_x}$ is a nonempty compact perfect set, or, equivalently, there is an atomless conditional probability distribution, $\mu$, so that for each x, $\mu \left ( {x, {B_x}} \right ) > 0$. It is also shown that if Y is dense-in-itself and ${B_x}$ is not meager, for each x, then B has a Borel parametrization.